Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Nonlinear interferometric surface-plasmon-resonance sensor

Hailong Wang, Zhongxing Fu, Zhihao Ni, Xiong Zhang, Chunliu Zhao, Shangzhong Jin, and Jietai Jing

Open Access Open Access

Abstract

A nonlinear interferometer can be constructed by replacing the beam splitter in the Mach-Zehnder interferometer with four-wave mixing (FWM) process. Meanwhile, the conventional surface plasmon resonance (SPR) sensors can be extensively used to infer the information of refractive index of the sample to be measured via either angle demodulation technique or intensity demodulation technique. Combined with a single FWM process, a quantum SPR sensor has been realized, whose noise floor is reduced below standard quantum limit with sensitivity unobtainable with classical SPR sensor. Therefore, in this work we have theoretically proposed a nonlinear interferometric SPR sensor, in which a conventional SPR sensor is placed inside nonlinear interferometer, which is called as I-type nonlinear interferometric SPR sensor. We demonstrate that near resonance angle I-type nonlinear interferometric SPR sensor has the following advantages: its degree of intensity-difference squeezing, estimation precision ratio, and signal-noise-ratio are improved by the factors of 4.6 dB, 2.3 dB, and 4.6 dB respectively than that obtained with a quantum SPR sensor based on a single FWM process. In addition, the theoretical principle of this work can also be expanded to other types of sensing, such as bending, pressure, and temperature sensors based on a nonlinear interferometer.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

When light is illuminated to the prism$|$thin metallic film$|$sample system, the intensity of the reflected light will experience an attenuation process when a propagating surface plasmon wave is excited at the interface between the metallic film and the sample. The excitation conditions depend sensitively on the refractive index of the sample to be measured [1]. In other words, the information of refractive index of the sample can be identified by observing the variation in either angle or intensity of the reflected light. Following this basic principle, various surface plasmon resonance (SPR) sensors for detecting the tiny change of refractive index of the sample have also found extensive applications in catalytic analysis [2], monitoring of various pollutants [3–6], dengue virus E-protein [7], and heavy metal ion [8–10], detection of biological and chemical molecule [11], and environmental and water analysis [12]. However, the ultimate sensitivity of classical SPR sensors is always bounded by standard quantum limit (SQL) due to Heisenberg uncertainty principle [13–15].

On one hand, by increasing the incident power to boost the sensitivity is not desirable because an excessive increase may cause optical damage to the measured biological sample [16–18] or other unwanted thermal effects [19]. On the other hand, the use of quantum states of light will assist their performance to surpass the constraint imposed by SQL. In the discrete-variable regime, it has been experimentally shown that the estimation errors of the SPR sensor with heralded single photons as the input are reduced below SQL even in the presence of various experimental imperfections [20]. Meanwhile, in the continuous-variable regime, it has been experimentally demonstrated that the sensitivities of quantum enhanced SPR sensors operating with two-mode squeezed states are also beyond the most sensitive classical SPR sensors [21–27]. Especially, the two-mode squeezed states used in the aforementioned SPR sensors are produced from four-wave mixing (FWM) process based on a double$-\Lambda$ configuration in hot $^{85}$Rb atoms [28–42], where the seed beam is amplified and a new beam called idler beam is generated on the other side of the pump beam simultaneously. The quantum properties shared by the two beams generated from single FWM process can be efficiently enhanced by cascading one more FWM process, i. e., a nonlinear interferometer consisting of two identical FWM processes can further boost the degree of intensity-difference squeezing (DS) of the two beams generated from the first FWM process [43–53]. Therefore, in this work we propose two types of nonlinear interferometric SPR sensors in which a conventional SPR sensor is placed inside or outside nonlinear interferometer, which is called as I-type or II-type nonlinear interferometric SPR sensors respectively. Then the sensing properties for nonlinear interferometric SPR sensors can be characterized by DS [54], estimation precision ratio [55], and signal-noise-ratio (SNR) of the two output beams. The analytical results show that the sensing performance from I-type nonlinear interferometric SPR sensor are all superior to the ones of a quantum SPR senor based on a single FWM process and II-type nonlinear interferometric SPR sensor.

This work is organized as follows: In Sec. 2, we describe the operational principle for the conventional SPR sensors using the Kretschmann configuration; In Sec. 3, we introduce I-type and II-type nonlinear interferometric SPR sensors and quantify the dependence of the corresponding DS values on refractive index of the sample to be measured; In Sec. 4 and Sec. 5, we also characterize estimation precision ratio and SNR from both I-type and II-type nonlinear interferometric SPR sensors respectively; In Sec. 6, we give a brief summary of this work.

2. Operational principle of the conventional SPR sensors

Before nonlinear interferometric SPR sensors based on a nonlinear interferometer are discussed, the operational principle for the conventional SPR sensors should be exhibited firstly. As shown in Fig. 1(a), an attenuated total reflection prism setup using the Kretschmann configuration consists of three layers: prism, gold film, and sample. When the surface plasmon wave is excited, the reflection coefficient $r_{spr}$ of the reflected light from the Kretschmann configuration can be expressed as

$$r_{spr}=\frac{r_{pg}+r_{gs}e^{2ik_{gz}d}}{1+r_{pg}r_{gs}e^{2ik_{gz}d}},$$
where $d$ is the thickness of gold film and the reflection coefficient between the $l$th and $m$th layers $r_{lm}$ can be given by
$$r_{lm}=\frac{k_{lz}\varepsilon_{m}-k_{mz}\varepsilon_{l}}{k_{lz} \varepsilon_{m}+k_{mz}\varepsilon_{l}},$$
where $l, m$=(p (prism), g (gold), and s (sample)), $k_{lz}$ is the normal component of the wave vector in the gold film layer and can be written as
$$k_{lz}=\frac{2\pi}{\lambda}\sqrt{\varepsilon_{l}-\varepsilon_{p}\sin \theta^{2}},$$
with the incident angle $\theta$, the permittivities of prism $\varepsilon _{p}$ and gold film $\varepsilon _{g}$ for a 795 nm incident light are $1.511^{2}$ and $(0.181+i5.126)^{2}$ respectively. To choose an optimal thickness of the gold film, the intensity of the reflected light as a function of the incident angle $\theta$ for the three different thicknesses of gold film: $d$=40 nm (red curve), $d$=50 nm (cyan curve), and $d$=60 nm (blue curve) is depicted in Fig. 1(b), the reflection coefficient $|r_{spr}|^{2}$ for $d$= 40 nm shows the highest attenuation contrast compared with the other two thicknesses, thus $d$ is set to equal to 40 nm in the following analysis. With a fixed thickness of gold film, the intensity of the reflected light as a function of the incident angle $\theta$ for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve) is shown in Fig. 1(c), from this figure it can be seen that a change of the refractive index can be identified by observing a shift of the resonance angle $\theta$, or alternatively the change of the intensity of the reflected light is observed for a given incident angle $\theta =62^{\circ }$ in the refractive index range of $1.288\leq n_{s}\leq 1.319$ [see Fig. 1(d)], which also indicates that there is a point-to-point relationship between the reflected intensity and refractive index. Therefore, Fig. 1(c) and Fig. 1(d) are angle demodulation technique and intensity demodulation technique respectively for the conventional SPR sensors, because the information of refractive index of the sample can be identified by observing the change of resonant angle and reflected intensity respectively, which constitute the basic operational principle of the conventional SPR sensors.

3. Nonlinear interferometric SPR sensors

The previous section describes the basic operational principle of the conventional SPR sensors, nonlinear interferometric SPR sensors based on a nonlinear interferometer will be discussed in this section. As shown in Fig. 2(a), the amplified seed beam from FWM$_{1}$ process with the power gain $G_{1}$ interacts with the sample via the conventional SPR sensors, the reflected beam and idler beam are both seeded into the FWM$_{2}$ process with the power gain $G_{2}$, the two output beams are analyzed by using different evaluation parameters. Different from the case in Fig. 2(a), the output beam $\hat {a}_{1}$ from nonlinear interferometer is directly illuminated to the Kretschmann configuration, and then is also analyzed with the help of the other beam $\hat {a}_{2}$. The former and latter schemes are called as I-type and II-type nonlinear interferometric SPR sensors respectively. For the simplicity of the following discussions, the reflection coefficient $|r_{spr}|^{2}$ can be replaced with $\eta _{1}$, meanwhile $\eta _{2}$ represents the attenuation for the other beam $\hat {a}_{2}$, which can be easily realized by inserting variable neutral density filter in the corresponding optical path. To obtain the quantitative description of nonlinear interferometric SPR sensors, the input-output relation of Fig. 2(a) can be expressed as

$$\begin{aligned}\hat{a}_{1} &=(\sqrt{G_{2}g_{1}\eta_{2}}e^{i\varphi }+\sqrt{G_{1}g_{2}\eta_{1} })\hat{a}_{in}^{\dagger}+(\sqrt{G_{1}G_{2}\eta_{2}}e^{i\varphi }+\sqrt{ g_{1}g_{2}\eta_{1}})\hat{v}_{0}+\\ &\quad\quad\sqrt{G_{2}(1-\eta_{2})}e^{i\varphi }\hat{v}_{1}+\sqrt{ g_{2}(1-\eta_{1})}\hat{v}_{2}^{\dagger},\\ \hat{a}_{2} &=(\sqrt{G_{1}G_{2}\eta_{1}}e^{i\varphi }+\sqrt{g_{1}g_{2}\eta_{2} })\hat{a}_{in}+(\sqrt{G_{2}g_{1}\eta_{1}}e^{i\varphi }+\sqrt{ G_{1}g_{2}\eta_{2}})\hat{v}_{0}^{\dagger}+\\ &\quad\quad\sqrt{g_{2}(1-\eta_{2})}\hat{v}_{1}^{\dagger}+\sqrt{ G_{2}(1-\eta_{1})}e^{i\varphi }\hat{v}_{2}, \end{aligned}$$
where $G_{i}$ is the power gain for FWM$_{i}$ process, and $g_{i}=G_{i}-1~(i=1, 2)$, $\varphi$ is interference phase inside nonlinear interferometer. $\hat {v}_{1}$ and $\hat {v}_{2}$ are the vacuum fields due to the introduction of the attenuations $\eta _{1}$ and $\eta _{2}$.

 figure: Fig. 1.

Fig. 1. (a) An attenuated total reflection prism setup using the Kretschmann configuration for the conventional SPR sensors. (b) The intensity of the reflected light as a function of the incident angle $\theta$ for the three different thicknesses of gold film: $d$=40 nm (red curve), $d$=50 nm (cyan curve), and $d$=60 nm (blue curve). Here the refractive index is set to equal to 1.301. (c) The intensity of the reflected light as a function of the incident angle $\theta$ for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). Here the thickness of gold film is set to equal to 40 nm. (d) The intensity of the reflected light as a function of refractive index of the sample in the range of $1.288\leq n_{s}\leq 1.319$. Here the incident angle and thickness of gold film are set to equal to 62$^{\circ }$ and 40 nm respectively. Other details can be seen in the main text.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. Nonlinear interferometric SPR sensors. (a) I-type nonlinear interferometric SPR sensor; (b) II-type nonlinear interferometric SPR sensor. $\hat {a}_{in}$ is coherent input beam, $\hat {v}_{0}$ is vacuum input, $\hat {a}_{1}$ and $\hat {a}_{2}$ are the two output beams. $G_{i}$ is the power gain for FWM$_{i}$ process, $\eta _{i}$ is the attenuation for the beams $\hat {a}_{i}$ ($i=1, 2$). For the sake of comparison, a quantum SPR sensor based on a single FWM process with the power gain $G_{s}$ is also shown in the red dashed box in Fig. 2(a).

Download Full Size | PDF

Let us evaluate the first parameter, i. e., the DS between the two output beams, which is given by the ratio of the variance on the intensity difference Var[$\hat {N}_{1}$–$\hat {N}_{2}$] to the same variance at SQL Var[$\hat {N}_{1}$–$\hat {N}_{2}$]$_{SQL}$=$\langle \hat {N}_{1}+\hat {N}_{2}\rangle$, a ratio value smaller than 1 means the presence of quantum correlation between the two output beams [56]. In I-type nonlinear interferometric SPR sensor given by Fig. 2(a), the DS between the two output beams can be calculated as

$$DS_{a}=\frac{ (G\eta_{1}-g\eta_{2})^{2}+Gg(\eta_{2}-\eta_{1})^{2}+g\eta_{2}(1-\eta_{2})+G\eta_{1}(1-\eta_{1})}{ (G+g)(G\eta_{1}+g\eta_{2})+4Gg\sqrt{\eta_{1}\eta_{2}}\cos\varphi },$$
where $G_{1}$ is assumed to be equal to $G_{2}$ in the following discussions, thus Eq. (5) will be reduced to a simplified form of $1/[4Gg(1+\cos \varphi )+1]$ when no losses are present in Fig. 2(a). Following the similar procedures, the DS between the two output beams in Fig. 2(b) can be calculated as
$$DS_{b}=\frac{DS_{bn}}{DS_{bd}},$$
where the numerator DS$_{bn}$ and denominator DS$_{bd}$ of DS$_{b}$ are
$$\begin{aligned}DS_{bn} &=[2Gg\eta_{2}(1+\cos\varphi )-\eta_{1}(G^{2}+g^{2}+2Gg\cos\varphi)]^{2}+\\ &\quad\quad (\eta_{2}-\eta_{1})^{2}Gg(G+g)^{2}(1+\cos\varphi )^{2}+(\eta_{2}-\eta_{1})^{2}Gg\sin\varphi ^{2}+\\ &\quad\quad 2Gg\eta_{2}(1-\eta_{2})(1+\cos\varphi)+\eta_{1}(1-\eta_{1})(G^{2}+g^{2}+2Gg\cos\varphi ), \end{aligned}$$
and
$$DS_{bd}=2Gg\eta_{2}(1+\cos\varphi )+\eta_{1}(G^{2}+g^{2}+2Gg\cos\varphi ),$$
respectively. Similarly, Eq. (6) will also be reduced to the simplified form of $1/[4Gg(1+\cos \varphi )+1]$ when no losses are present in Fig. 2(b). In order to make a direct comparison with a quantum SPR sensor based on a single FWM process in the red dashed box in Fig. 2(a) [21–26], its DS can also be expressed as
$$DS_{s}=\frac{2G_{s}^{2}(\eta_{1}-\eta_{2})^{2}+G_{s}\eta_{1}(1-2\eta_{1} )+g_{s}\eta_{2}(1-2\eta_{2})+2G_{s}\eta_{2}(2\eta_{1}-\eta_{2})}{G_{s}\eta_{1}+g_{s}\eta_{2}},$$
where $G_{s}$ is the power gain in a single FWM process and $g_{s}=G_{s}-1$. To make a fair comparison, the intensities of the sensing beams involving the plasmonic interaction should be equal. Specifically, the intensities of the sensing beams from a quantum SPR sensor based on a single FWM process, I-type, and II-type nonlinear interferometric SPR sensors are $G_{s}\eta _{1}$, $G^{2}\eta _{1}+g^{2}\eta _{2}+2Gg\sqrt {\eta _{1}\eta _{2}}$ ($\varphi =0$), and $\eta _{1}(G^{2}+g^{2}+2Gg)$ ($\varphi =0$), respectively. For the sake of simplicity, firstly, $\eta _{1}=\eta _{2}$ is assumed and this assumption can also preserve the quantum correlations existed in the three different sensors; Secondly, if the seed beam is amplified by a single FWM process with power gain $G_{s}=3$, then the amplification factor $G^{2}+g^{2}+2Gg$ from nonlinear interferometer should also be equal to 3, thus the power gain $G$ of each single FWM process in nonlinear interferometer is equal to $(1+\sqrt {3})/2\approx 1.366$. Doing this can guarantee the intensities of the three sensing beams are always equal. In this sense, the values of Eq. (6) and Eq. (9) are equal, which means that quantum performance of a single FWM process with power gain $G_{s}=3$ is equivalent to the one of II-type nonlinear interferometer with $G=1.366$. Therefore, the quantum performance of a quantum SPR sensor based on a single FWM process with power gain $G_{s}=3$ can be fully replaced with the one of II-type nonlinear interferometric SPR sensor with $G=1.366$ in Fig. 2(b), and thus the sensing performance will be mainly compared between I-type nonlinear interferometric SPR sensor and II-type nonlinear interferometric SPR sensor.

Based on the above discussions, the sensing principle of DS values with respect to refractive index will be shown as follows. As is depicted in Fig. 3(a), the DS values of I-type and II-type nonlinear interferometric SPR sensors are always below SQL (the cyan dashed line), while the DS value of I-type nonlinear interferometric SPR sensor (the red curve) is much smaller. This is because the two beams generated from FWM$_{1}$ process still preserve quantum correlation after experiencing the identical attenuation, and its correlation degree will be enhanced by the next FWM$_{2}$ process. While for II-type nonlinear interferometric SPR sensor, the quantum correlation shared by the two output beams will only be degraded by the attenuation process. Similarly, the dependence of I-type and II-type nonlinear interferometric SPR sensors on the incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve) is shown in Fig. 3(b), it can be seen that both of the DS values obtain the maximum values near resonance angle due to the higher loss, and the minimum values far from resonance angle due to the lower loss. This phenomenon can be explained as follows. As shown in Figs. 1(b)–1(c), based on the operational principle of a conventional SPR sensor, the reflected beam will experience more attenuation near resonance angle than far from resonance angle, and more attenuation means the introduction of more loss to the reflected beam, this will inevitably lead to the disappearance of more quantum correlations. Meanwhile, quantum correlation is characterized by the DS value, a lower (higher) quantum correlation corresponds to a larger (smaller) DS value. Thus both of the DS values are totally different between near resonance angle and far from resonance angle. Note that the maximum DS values of I-type and II-type nonlinear interferometric SPR sensors are 0.3462 and 0.9989 respectively, this means that near resonance angle the correlation degree of I-type nonlinear interferometric SPR sensor is improved by a factor of 10Log(0.9989/0.3462)=4.6 dB compared with that obtained with II-type nonlinear interferometric SPR sensor.

 figure: Fig. 3.

Fig. 3. (a) The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of attenuation $\eta _{1}$. (b) The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: SQL. Other details can be seen in Fig. 1.

Download Full Size | PDF

The aforementioned sensing principle is based on angle demodulation, intensity demodulation as the other demodulation technique, i. e., the dependence of the DS value on refractive index will be shown as follows. As shown in Fig. 4, the DS values decrease with the increase of refractive index. Such behaviors can be explained as follows. As shown in Fig. 1(d), a low (high) refractive index corresponds to a low (high) reflection coefficient and a high (low) loss, and a high (low) loss indicates the decreasing (increasing) of the quantum correlations. To sum up, the correlation degree of I-type nonlinear interferometric SPR sensor near resonance angle is improved by a factor of 10Log(0.9989/0.3462)=4.6 dB compared with that obtained with II-type nonlinear interferometric SPR sensor.

 figure: Fig. 4.

Fig. 4. The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: SQL.

Download Full Size | PDF

4. Estimation precision ratio enhancement

The first sensing parameter, i. e., the DS values of the two output beams generated from I-type and II-type nonlinear interferometric SPR sensors as a function of refractive index has been analyzed. In this section, the estimation precision $\delta n_{s}$ as the second sensing parameter can also be used to characterize the sensing performance, which can be obtained through an error propagation analysis, such that [55]

$$\delta n_{s}=\frac{\langle\Delta\hat{S}\rangle}{\left\vert \frac{\partial\left\langle \hat{S}\right\rangle }{\partial n_{s} }\right\vert},$$
where $\hat {S}$ consists of some combination of measurements of the two output beams and $\langle \cdots \rangle$ denotes the mean value. In general, this parameter is used to study the estimation ability of refractive index via a sensor. A smaller value indicates a reduction in the uncertainty of the estimation and thus an increase in the sensitivity. For the two mode squeezed state, the signal $\hat {S}$ can be written as the form of intensity difference between the two output beams. So the denominator of Eq. (10) $\left \vert \frac {\partial \left \langle \hat {S}\right \rangle }{\partial n_{s}}\right \vert$ can be simplified as $N_{s}\left \vert \frac {\partial \left \vert r_{spr}\right \vert ^{2}}{\partial n_{s}}\right \vert$, where $N_{s}$ is the intensity of the sensing beam. Note that the intensities of all the sensing beams are set to be equal in the previous section, thus the sensing performance of the different SPR sensors only depends on the numerator, i. e., nonlinear features existed in the different SPR sensors. Therefore, to quantify the effect of nonclassical features, i. e., quantum correlations, on the estimation precision enhancement, it is necessary to define a ratio of the precision based on SQL to the precision from quantum correlations. The ratio is defined as
$$\Re=\frac{\delta n_{s(SQL)}}{\delta n_{s}}=\frac{\langle\Delta\hat {S}\rangle_{(SQL)}}{\langle\Delta\hat{S}\rangle},$$
where $\Re$ greater than 1 reveals an enhancement in the estimation precision $\delta n_{s}$, or equivalently a quantum noise reduction compared with the classical scenario. For the cases of I-type and II-type nonlinear interferometric SPR sensors, their estimation precision ratios $\Re$s are the reciprocal of the square root of the corresponding DSs, i. e., $\Re =\frac {1}{\sqrt {DS}}$. As shown in Fig. 5(a), the $\Re$ values of I-type nonlinear interferometric SPR sensor are always greater than that obtained with II-type nonlinear interferometric SPR sensor, resulting from a much stronger quantum noise reduction existed in I-type nonlinear interferometric SPR sensor mentioned before. Similarly, the $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve) are also shown in Fig. 5(b), a tiny change of the refractive index from 1.301 (black curve) to 1.302 (red curve) can be identified with the higher $\Re$ values by observing a shift of the resonance angle using I-type nonlinear interferometric SPR sensor compared with II-type nonlinear interferometric SPR sensor. However, I-type nonlinear interferometric SPR sensor show the estimation precision enhancement even in the high loss condition, i. e., near resonance angle, in which I-type nonlinear interferometric SPR sensor show the distinct differences compared with II-type nonlinear interferometric SPR sensor. This is mainly due to the fact that two beams involving plasmonic interaction in I-type nonlinear interferometric SPR sensor will still preserve quantum correlation after experiencing the identical attenuation, and will be further enhanced by the next FWM$_{2}$ process. While the quantum correlation shared by the two beams output from II-type nonlinear interferometric SPR sensor will only be degraded after the plasmonic interaction. This point can also be confirmed in Fig. 3(a), meanwhile estimation precision ratio is defined as the reciprocal of the square root of the corresponding DS, and thus I-type nonlinear interferometric SPR sensor will show estimation precision enhancement compared with II-type nonlinear interferometric SPR sensor in the high loss condition. In other words, the minimum values from I-type and II-type nonlinear interferometric sensors are 1.7 and 1.0 respectively, which means that the ability of the estimation precision enhancement of I-type nonlinear interferometric SPR sensor near resonance angle is enhanced by a factor of 10Log[1.7/1.0]=2.3 dB compared with the one of II-type nonlinear interferometric SPR sensor.

If the intensity demodulation instead of angle demodulation is involved in the discussions of the estimation precision enhancement, the results are shown in Fig. 6, the $\Re$ values from both I-type and II-type nonlinear interferometric SPR sensors increase with the increase of refractive index, while they show the distinct differences in the low refractive index range mentioned before and their values saturates at the value of 2.24 in the high refractive index limit. In a word, the estimation precision ratio of I-type nonlinear interferometric SPR sensor under the high loss condition can obtain a maximum 2.3 dB enhancement compared with II-type nonlinear interferometric SPR sensor.

5. SNR enhancement

After DS and estimation precision ratio from both I-type and II-type nonlinear interferometric SPR sensors are discussed, the third sensing parameter, i. e., SNR, will be studied in this section. For two mode squeezed states, its SNR can defined as the ratio of the intensity of the sensing beam to the intensity-difference noise between the two output beams. This definition has the following advantages: firstly, the intensity of the sensing beam rather than intensity-difference between the two beams as the signal can obtain a maximum value; Secondly, intensity-difference noise between the two output beams can obtain a minimum value compared with the noise of the sensing beam, because each beam from the twin beams has the noise level above SQL. Thus such cooperation between high signal and low noise level enable the SNR from nonlinear interferometric SPR sensors to be further enhanced than that obtained with classical scenario. Following this basic idea, the SNR values of I-type and II-type nonlinear interferometric SPR sensors can be given by

$$SNR_{a}=\frac{G^{2}\eta_{1}+g^{2}\eta_{2}+2Gg\sqrt{\eta_{1}\eta_{2}}\cos\varphi}{(G\eta_{1}-g\eta_{2})^{2}+Gg(\eta_{2}-\eta_{1})^{2}+g\eta_{2}(1-\eta_{2})+G\eta_{1}(1-\eta_{1})},$$
and
$$SNR_{b}=\frac{\eta_{1}(G^{2}+g^{2}+2Gg\cos\varphi)}{DS_{bn}},$$
respectively. Under the condition $\eta _{1}=\eta _{2}$, the SNR of the classical scenario is 0.5, which can be realized by splitting a beam in a coherent state with a power equal to the total power of the two correlated beams, one beam as the sensing beam to interact with the attenuated total reflection setup using the Kretschmann configuration, the other beam experiencing the identical attenuation as the reference beam, then they are both directed to a differential detector. Thus this balanced detection system makes it possible to cancel all the sources of classical noise and obtain a accurate measure of the SNR of classical scenario. As shown in Fig. 7, the SNR values of I-type and II-type nonlinear interferometric SPR sensors are both greater than that obtained with classical scenario, while the SNR value of I-type nonlinear interferometric SPR sensor is always greater than that obtained with II-type nonlinear interferometric SPR sensor. The maximum differences about SNR values can be seen from the high loss situation, similar to the discussion of estimation precision enhancement. Specifically, the minimum SNR values from I-type and II-type nonlinear interferometric SPR sensors are 1.73 and 0.60 respectively, this corresponds directly a 10Log10[1.73/0.60]=4.6 dB SNR improvement from I-type nonlinear interferometric SPR sensor under the high loss condition compared with II-type nonlinear interferometric SPR sensor.

The SNR values as a function of refractive index are shown in Fig. 8, similar to the dependence of estimation precision enhancement on refractive index in the previous section, here the SNR values of both I-type and II-type nonlinear interferometric SPR sensors also increase with the increase of refractive index, but their values saturates at the value of 3.0 in the high refractive index limit. In conclusion, the SNR value of I-type nonlinear interferometric SPR sensor under the high loss situation is enhanced by a factor of 4.6 dB compared with II-type nonlinear interferometric SPR sensor, while this advantage will disappear in the high refractive index limit.

 figure: Fig. 5.

Fig. 5. (a) The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of $\eta _{1}$. (b) The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: 1.

Download Full Size | PDF

 figure: Fig. 6.

Fig. 6. The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: 1.

Download Full Size | PDF

 figure: Fig. 7.

Fig. 7. (a) The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of $\eta _{1}$. (b) The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: 0.5.

Download Full Size | PDF

 figure: Fig. 8.

Fig. 8. The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: 0.5.

Download Full Size | PDF

6. Conclusions

We have theoretically characterized the sensing performance of I-type and II-type nonlinear interferometric SPR sensors and demonstrated the equivalence between II-type nonlinear interferometric SPR sensor with power gain $G=1.366$ and a quantum SPR sensor based on a single FWM process with power gain $G_{s}=3$. The distinct advantages of I-type nonlinear interferometric SPR sensor near resonance angle are presented as follows. Firstly, the DS as the first evaluation parameter, the DS value of I-type nonlinear interferometric SPR sensor is improved by a factor of 4.6 dB than that obtained with II-type nonlinear interferometric SPR sensor; Secondly, estimation precision ratio as the second evaluation parameter, the estimation precision ratio of I-type nonlinear interferometric SPR sensor is enhanced by a factor of 2.3 dB than that obtained with II-type nonlinear interferometric SPR sensor; Lastly, SNR as the third parameter, the SNR value of I-type nonlinear interferometric SPR sensor is improved by a factor of 4.6 dB than that obtained with II-type nonlinear interferometric SPR sensor. The results presented here may find other potential sensing applications based on a nonlinear interferometer, i. e., nonlinear interferometric bending, pressure, and temperature sensors.

Funding

National Natural Science Foundation of China (11804323, 11874155, 91436211, 11374104); Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-08-E00100); Basic Research Project of Shanghai Science and Technology Commission (20JC1416100); Natural Science Foundation of Shanghai (17ZR1442900); Minhang Leading Talents (201971); Program of Scientific and Technological Innovation of Shanghai (17JC1400401); National Basic Research Program of China (2016YFA0302103); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); the 111 project (B12024).

Disclosures

The author declares no conflicts of interest.

References

1. J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: Review,” Sens. Actuators, B 54(1-2), 3–15 (1999). [CrossRef]  

2. C. Li, S. Xu, J. Yu, Z. Li, W. Li, J. Wang, A. Liu, B. Man, S. Yang, and C. Zhang, “Local hot charge density regulation: Vibration-free pyroelectric nanogenerator for effectively enhancing catalysis and in-situ surface enhanced Raman scattering monitoring,” Nano Energy 81, 105585 (2021). [CrossRef]  

3. X. Zhao, C. Liu, J. Yu, Z. Li, L. Liu, C. Li, S. Xu, W. Li, B. Man, and C. Zhang, “Hydrophobic multiscale cavities for high-performance and self-cleaning surface-enhanced Raman spectroscopy (SERS) sensing,” Nanophotonics 9(16), 4761–4773 (2020). [CrossRef]  

4. J. Yu, M. Yang, Z. Li, C. Liu, Y. Wei, C. Zhang, B. Man, and F. Lei, “Hierarchical Particle-In-Quasicavity Architecture for Ultratrace In Situ Raman Sensing and Its Application in Real-Time Monitoring of Toxic Pollutants,” Anal. Chem. 92(21), 14754–14761 (2020). [CrossRef]  

5. H. S. Hashim, Y. W. Fen, N. A. S. Omar, J. Abdullah, W. M. E. Mustaqim Mohd Daniyal, and S. Saleviter, “Detection of phenol by incorporation of gold modified-enzyme based graphene oxide thin film with surface plasmon resonance technique,” Opt. Express 28(7), 9738–9752 (2020). [CrossRef]  

6. S. H. Al-Rekabi, Y. Mustapha Kamil, M. H. Abu Bakar, Y. W. Fen, H. N. Lim, S. Kanagesan, and M. A. Mahdi, “Hydrous ferric oxide-magnetite-reduced graphene oxide nanocomposite for optical detection of arsenic using surface plasmon resonance,” Opt. Laser Technol. 111, 417–423 (2019). [CrossRef]  

7. N. A. S. Omar, Y. W. Fen, J. Abdullah, M. Hazani Mat Zaid, W. M. E. Mustaqim Mohd Daniyal, and M. A. Mahdi, “Sensitive surface plasmon resonance performance of cadmium sulfide quantum dots-amine functionalized graphene oxide based thin film towards dengue virus E-protein,” Opt. Laser Technol. 114, 204–208 (2019). [CrossRef]  

8. S. Saleviter, Y. W. Fen, W. M. E. Mustaqim Mohd Daniyal, J. Abdullah, A. R. Sadrolhosseini, and N. A. S. Omar, “Design and analysis of surface plasmon resonance optical sensor for determining cobalt ion based on chitosan-graphene oxide decorated quantum dots-modified gold active layer,” Opt. Express 27(22), 32294–32307 (2019). [CrossRef]  

9. W. M. E. Mustaqim Mohd Daniyal, Y. W. Fen, J. Abdullah, A. R. Sadrolhosseini, S. Saleviter, and N. A. S. Omar, “Exploration of surface plasmon resonance for sensing copper ion based on nanocrystalline cellulose-modified thin film,” Opt. Express 26(26), 34880–34893 (2018). [CrossRef]  

10. A. A. Zainudin, Y. W. Fen, N. A. Yusof, S. H. Al-Rekabi, M. A. Mahdi, and N. A. S. Omar, “Incorporation of surface plasmon resonance with novel valinomycin doped chitosan-graphene oxide thin film for sensing potassium ion,” Spectrochim. Acta, Part A 191(15), 111–115 (2018). [CrossRef]  

11. C. Zhang, C. Li, J. Yu, S. Jiang, S. Xu, C. Yang, Y. Liu, X. Gao, A. Liu, and B. Man, “SERS activated platform with three-dimensional hot spots and tunable nanometer gap,” Sens. Actuators, B 258, 163–171 (2018). [CrossRef]  

12. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nat. Photonics 7(3), 229–233 (2013). [CrossRef]  

13. M. Piliarik and J. Homola, “Surface plasmon resonance (SPR) sensors: approaching their limits?” Opt. Express 17(19), 16505–16517 (2009). [CrossRef]  

14. X. Wang, M. Jefferson, P. C. Hobbs, W. P. Risk, B. E. Feller, R. D. Miller, and A. Knoesen, “Shot-noise limited detection for surface plasmon sensing,” Opt. Express 19(1), 107–117 (2011). [CrossRef]  

15. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004). [CrossRef]  

16. K. C. Neuman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical trops,” Biophys. J. 77(5), 2856–2863 (1999). [CrossRef]  

17. E. J. Peterman, F. Gittes, and C. F. Schmidt, “Laser-induced heating in optical traps,” Biophys. J. 84(2), 1308–1316 (2003). [CrossRef]  

18. M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016). [CrossRef]  

19. S. Kaya, J. C. Weeber, F. Zacharatos, K. Hassan, T. Bernardin, B. Cluzel, J. Fatome, and C. Finot, “Photo-thermal modulation of surface plasmon polariton propagation at telecommunication wavelengths,” Opt. Express 21(19), 22269–22284 (2013). [CrossRef]  

20. J.-S. Lee, S.-J. Yoon, H. Rah, M. Tame, C. Rockstuhl, S. H. Song, C. Lee, and K.-G. Lee, “Quantum plasmonic sensing using single photons,” Opt. Express 26(22), 29272–29282 (2018). [CrossRef]  

21. R. C. Pooser and B. J. Lawrie, “Plasmonic trace sensing below the photon shot noise limit,” ACS Photonics 3(1), 8–13 (2016). [CrossRef]  

22. R. C. Pooser and B. J. Lawrie, “Ultrasensitive measurement of microcantilever displacement below the shot-noise limit,” Optica 2(5), 393–399 (2015). [CrossRef]  

23. B. J. Lawrie, P. G. Evans, and R. C. Pooser, “Extraordinary Optical Transmission of Multimode Quantum Correlations via Localized Surface Plasmons,” Phys. Rev. Lett. 110(15), 156802 (2013). [CrossRef]  

24. W. Fan, B. J. Lawrie, and R. C. Pooser, “Quantum Plasmonic Sensing,” Phys. Rev. A 92(5), 053812 (2015). [CrossRef]  

25. M. W. Holtfrerich, M. Dowran, R. Davidson, B. J. Lawrie, R. C. Pooser, and A. M. Marino, “Toward quantum plasmonic networks,” Optica 3(9), 985–988 (2016). [CrossRef]  

26. M. Dowran, A. Kumar, B. J. Lawrie, R. C. Pooser, and A. M. Marino, “Quantum-enhanced plasmonic sensing,” Optica 5(5), 628–633 (2018). [CrossRef]  

27. B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, “Quantum Sensing with Squeezed Light,” ACS Photonics 6(6), 1307–1318 (2019). [CrossRef]  

28. C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32(2), 178–180 (2007). [CrossRef]  

29. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science 321(5888), 544–547 (2008). [CrossRef]  

30. R. C. Pooser, A. M. Marino, V. Boyer, K. M. Jones, and P. D. Lett, “Low-Noise Amplification of a Continuous-Variable Quantum State,” Phys. Rev. Lett. 103(1), 010501 (2009). [CrossRef]  

31. A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable Delay of Einstein-Podolsky-Rosen Entanglement,” Nature 457(7231), 859–862 (2009). [CrossRef]  

32. Q. Glorieux, L. Guidoni, S. Guibal, J. P. Likforman, and T. Coudreau, “Quantum correlations by four-wave mixing in an atomic vapor in a nonamplifying regime: Quantum beam splitter for photons,” Phys. Rev. A 84(5), 053826 (2011). [CrossRef]  

33. M. Jasperse, L. D. Turner, and R. E. Scholten, “Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic,” Opt. Express 19(4), 3765–3774 (2011). [CrossRef]  

34. M. Guo, H. Zhou, D. Wang, J. Gao, J. Zhang, and S. Zhu, “Experimental investigation of high-frequency-difference twin beams in hot cesium atoms,” Phys. Rev. A 89(3), 033813 (2014). [CrossRef]  

35. Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental Generation of Multiple Quantum Correlated Beams from Hot Rubidium Vapor,” Phys. Rev. Lett. 113(2), 023602 (2014). [CrossRef]  

36. H. Wang, C. Fabre, and J. Jing, “Single-step fabrication of scalable multimode quantum resources using four-wave mixing with a spatially structured pump,” Phys. Rev. A 95(5), 051802 (2017). [CrossRef]  

37. X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-Angular-Momentum Multiplexed Continuous-Variable Entanglement from Four-Wave Mixing in Hot Atomic Vapor,” Phys. Rev. Lett. 123(7), 070506 (2019). [CrossRef]  

38. S. Liu, Y. Lou, and J. Jing, “Interference-Induced Quantum Squeezing Enhancement in a Two-beam Phase-Sensitive Amplifier,” Phys. Rev. Lett. 123(11), 113602 (2019). [CrossRef]  

39. K. Zhang, W. Wang, S. Liu, X. Pan, J. Du, Y. Lou, S. Yu, S. Lv, N. Treps, C. Fabre, and J. Jing, “Reconfigurable Hexapartite Entanglement by Spatially Multiplexed Four-Wave Mixing Processes,” Phys. Rev. Lett. 124(9), 090501 (2020). [CrossRef]  

40. S. Li, X. Pan, Y. Ren, H. Liu, S. Yu, and J. Jing, “Deterministic Generation of Orbital-Angular-Momentum Multiplexed Tripartite Entanglement,” Phys. Rev. Lett. 124(8), 083605 (2020). [CrossRef]  

41. H. Wang, K. Zhang, N. Treps, C. Fabre, J. Zhang, and J. Jing, “Generation of hexapartite entanglement in a four-wave-mixing process with a spatially structured pump: Theoretical study,” Phys. Rev. A 102(2), 022417 (2020). [CrossRef]  

42. S. Liu, Y. Lou, and J. Jing, “Orbital angular momentum multiplexed deterministic all-optical quantum teleportation,” Nat. Commun. 11(1), 3875 (2020). [CrossRef]  

43. J. Jing, C. Liu, Z. Zhou, Z. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011). [CrossRef]  

44. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1, 1) interferometer,” Phys. Rev. A 86(2), 023844 (2012). [CrossRef]  

45. Z. Yan, X. Jia, X. Su, Z. Duan, C. Xie, and K. Peng, “Cascaded entanglement enhancement,” Phys. Rev. A 85(4), 040305 (2012). [CrossRef]  

46. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5(1), 3049 (2014). [CrossRef]  

47. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010). [CrossRef]  

48. D. Li, B. T. Gard, Y. Gao, C.-H. Yuan, W. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU(1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016). [CrossRef]  

49. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-Up SU(1, 1) Interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017). [CrossRef]  

50. M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1, 1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017). [CrossRef]  

51. B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with brightand vacuum-seeded SU(1, 1) interferometers,” Phys. Rev. A 95(6), 063843 (2017). [CrossRef]  

52. B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the SU(1, 1) interferometer,” Optica 4(7), 752–756 (2017). [CrossRef]  

53. J. Xin, J. Qi, and J. Jing, “Enhancement of entanglement using cascaded four-wave mixing processes,” Opt. Lett. 42(2), 366–369 (2017). [CrossRef]  

54. H. Wang, L. Cao, and J. Jing, “Characterization of Pairwise Correlations from Multiple Quantum Correlated Beams Generated from Cascaded Four-Wave Mixing Processes,” Sci. Rep. 7(1), 40410 (2017). [CrossRef]  

55. J.-S. Lee, T. Huynh, S.-Y. Lee, K.-G. Lee, J. Lee, M. Tame, C. Rockstuhl, and C. Lee, “Quantum noise reduction in intensity-sensitive surface-plasmon-resonance sensors,” Phys. Rev. A 96(3), 033833 (2017). [CrossRef]  

56. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of Quantum Noise Reduction on Twin Laser Beams,” Phys. Rev. Lett. 59(22), 2555–2557 (1987). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)
Fig. 1.
Fig. 1. (a) An attenuated total reflection prism setup using the Kretschmann configuration for the conventional SPR sensors. (b) The intensity of the reflected light as a function of the incident angle $\theta$ for the three different thicknesses of gold film: $d$=40 nm (red curve), $d$=50 nm (cyan curve), and $d$=60 nm (blue curve). Here the refractive index is set to equal to 1.301. (c) The intensity of the reflected light as a function of the incident angle $\theta$ for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). Here the thickness of gold film is set to equal to 40 nm. (d) The intensity of the reflected light as a function of refractive index of the sample in the range of $1.288\leq n_{s}\leq 1.319$. Here the incident angle and thickness of gold film are set to equal to 62$^{\circ }$ and 40 nm respectively. Other details can be seen in the main text.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Nonlinear interferometric SPR sensors. (a) I-type nonlinear interferometric SPR sensor; (b) II-type nonlinear interferometric SPR sensor. $\hat {a}_{in}$ is coherent input beam, $\hat {v}_{0}$ is vacuum input, $\hat {a}_{1}$ and $\hat {a}_{2}$ are the two output beams. $G_{i}$ is the power gain for FWM$_{i}$ process, $\eta _{i}$ is the attenuation for the beams $\hat {a}_{i}$ ($i=1, 2$). For the sake of comparison, a quantum SPR sensor based on a single FWM process with the power gain $G_{s}$ is also shown in the red dashed box in Fig. 2(a).

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. (a) The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of attenuation $\eta _{1}$. (b) The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: SQL. Other details can be seen in Fig. 1.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. The DS values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: SQL.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. (a) The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of $\eta _{1}$. (b) The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: 1.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. The $\Re$ values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: 1.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. (a) The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of $\eta _{1}$. (b) The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of incident angle for the two different samples: $n_{s}=1.301$ (black curve) and $n_{s}=1.302$ (red curve). The cyan dashed line: 0.5.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The SNR values of I-type (A) and II-type (B) nonlinear interferometric SPR sensors as a function of refractive index. The cyan dashed line: 0.5.

View in Article | Download Full Size | PDF

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

r s p r = r p g + r g s e 2 i k g z d 1 + r p g r g s e 2 i k g z d ,
r l m = k l z ε m − k m z ε l k l z ε m + k m z ε l ,
k l z = 2 π λ ε l − ε p sin ⁡ θ 2 ,
a ^ 1 = ( G 2 g 1 η 2 e i φ + G 1 g 2 η 1 ) a ^ i n † + ( G 1 G 2 η 2 e i φ + g 1 g 2 η 1 ) v ^ 0 + G 2 ( 1 − η 2 ) e i φ v ^ 1 + g 2 ( 1 − η 1 ) v ^ 2 † , a ^ 2 = ( G 1 G 2 η 1 e i φ + g 1 g 2 η 2 ) a ^ i n + ( G 2 g 1 η 1 e i φ + G 1 g 2 η 2 ) v ^ 0 † + g 2 ( 1 − η 2 ) v ^ 1 † + G 2 ( 1 − η 1 ) e i φ v ^ 2 ,
D S a = ( G η 1 − g η 2 ) 2 + G g ( η 2 − η 1 ) 2 + g η 2 ( 1 − η 2 ) + G η 1 ( 1 − η 1 ) ( G + g ) ( G η 1 + g η 2 ) + 4 G g η 1 η 2 cos ⁡ φ ,
D S b = D S b n D S b d ,
D S b n = [ 2 G g η 2 ( 1 + cos ⁡ φ ) − η 1 ( G 2 + g 2 + 2 G g cos ⁡ φ ) ] 2 + ( η 2 − η 1 ) 2 G g ( G + g ) 2 ( 1 + cos ⁡ φ ) 2 + ( η 2 − η 1 ) 2 G g sin ⁡ φ 2 + 2 G g η 2 ( 1 − η 2 ) ( 1 + cos ⁡ φ ) + η 1 ( 1 − η 1 ) ( G 2 + g 2 + 2 G g cos ⁡ φ ) ,
D S b d = 2 G g η 2 ( 1 + cos ⁡ φ ) + η 1 ( G 2 + g 2 + 2 G g cos ⁡ φ ) ,
D S s = 2 G s 2 ( η 1 − η 2 ) 2 + G s η 1 ( 1 − 2 η 1 ) + g s η 2 ( 1 − 2 η 2 ) + 2 G s η 2 ( 2 η 1 − η 2 ) G s η 1 + g s η 2 ,
δ n s = ⟨ Δ S ^ ⟩ | ∂ ⟨ S ^ ⟩ ∂ n s | ,
ℜ = δ n s ( S Q L ) δ n s = ⟨ Δ S ^ ⟩ ( S Q L ) ⟨ Δ S ^ ⟩ ,
S N R a = G 2 η 1 + g 2 η 2 + 2 G g η 1 η 2 cos ⁡ φ ( G η 1 − g η 2 ) 2 + G g ( η 2 − η 1 ) 2 + g η 2 ( 1 − η 2 ) + G η 1 ( 1 − η 1 ) ,
S N R b = η 1 ( G 2 + g 2 + 2 G g cos ⁡ φ ) D S b n ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use