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
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 by3. 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
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
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.
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.
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]
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
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.
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.
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