1 Introduction

When a monochromatic beam is propagated through a periodic structure such as grating platforms, the pattern of that configuration is provided to repeat itself occasionally with increasing distance of the pattern from the platform which known as Fresnel regime (Salama et al. 1999; Siegel et al. 2001; Wang et al. 2009). This distance is known as the Talbot distance, and fantastic effects may also be defined at this distances, where the multiple frequency self-imaging pattern can occur. This phenomena as a self-imaging pattern of the grating structure was introduced by H.F. Talbot in 1836, and has been the received remarkable attention from research groups in the field of atom optics and plasmonics (Podanchuk et al. 2014; Kovalenko et al. 2013; Zhang et al. 2009; Dennis et al. 2007). Therefore, various research groups have used talbot effect form practical applications in gratings and diffractive structure. Also, several optimized algorithms are proposed (Chen et al. 2020; Wang and Chen 2020; Xu et al. 2019; Zhao et al. 2019). As an example Li et al proposed intelligence platform (Li et al. 2018). Wang et al proposed machine learning algorithm (Wang et al. 2017). And several related works are presented recently (Xia et al. 2017; Chen et al. 2016; Shen et al. 2016; Hu et al. 2015). This optimized algorithms have advanced applications (Xu and Chen 2014; Zhang et al. 2020, 2021; Zhao et al. 2021). Chen group work on bio-science with optimized algorithm (Tu et al. 2021; Shan et al. 2021; Yu et al. 2021; Hu et al. 2021; Zhao et al. 2014; Yu et al. 2020). The interest in this phenomena is not only theoretical aspect (Koriakovskii and Marchenko 1981; Liu et al. 2015). The Talbot effects as a remarkable phenomena of periodic structure have a advanced applications, such as switch, sensor, optical metrology, laser array illumination, detector and so on Feng et al. 2020; Fu and Yang 2020; Jiang et al. 2020,?. For example protein is a main components which is considered by research groups (Zhang and Liu 2019; Xu et al. 2020). Alzheimer detection is also so important (Zhu et al. 2020) and several sensing platforms (Zhu et al. 2020; Hu et al. 2020; Qu et al. 2019; Jiang et al. 2013; Wang et al. 2020; Zou et al. 2019). However, the self-imaging pattern is seen as the period of the structure is highly larger than the incident light beam, for the condition of resonant diffraction periodic structure whose period is relatively comparable to the incident frequency, the electric field profile at the repeated distance of the periodic structure will also revive periodically, called as the quasi-Talbot phenomena (Podanchuk et al. 2013, 2011, 2013, 2015; Iwata et al. 2011). In the year 2012, Hua research group developed Talbot effect beyond the paraxial limit at optical frequencies (Hua et al. 2012). In the year 2018, JINWOO group experimentally demonstrate a new design for passive Talbot amplification of repetitive optical waveforms (Jeon et al. 2018). In the year 2020, Aviad and co-worker focused on use of talbot effect in label free biosensors for therapeutic purpose. They proposed a label-free sensor on a chip, operating in near-infrared for monitoring of absorption line signatures based on molecular vibrations (Katiyi and Karabchevsky 2020). In the same year, X-ray measurements is introduced by Talbot effect by Brazhnikov research group. These proposed sensors able to detect a very small quantity of molecules (Brazhnikov et al. 2020).

In several practical applications, long grating structures are applied because of its good behavior, with periods ranging from several centimeter to higher than hundred of centimetres (Lin et al. 2020; Zuo et al. 2015, 2017; Zhang et al. 2020). However, there exist some advanced nano-applications in which small footprint displacements and exact contrast need to be calculated (Li et al. 2018, 2017; Wang et al. 2017; Xia et al. 2017). In typical, several applications based on the Talbot effect, cannot be used to nano-scale configuration with grating platforms comparable to the incident beam wavelength. Some numerical researches have focused on the self-image patterns in one-dimenssional structure the paraxial limit, and many of them differences in the self-image patterns were predicted. Also, these numerical results, always, have not been validated by experimental ones owing to their restriction in fabrication process.

Fortunately, in this years, by introducing the plasmonics fields, the Talbot effect was seen in naoscale structure (Wang et al. 2010; Li et al. 2011; Shi et al. 2015; Kim et al. 2020). However, the fast damping of surface waves in nobel metals restricted the investigate to only the first Talbot distance. Therefore, the new material such as polymers with high mobility features is needed (Kim et al. 2020). Despite remarkable practical investigations on biosensors based on talbot, there is still much researches to improve the overall performance of such biosensors.

In this work, we have analyzed the Talbot effect of polymer grating in the self-imaging pattern of Cytop polymer for biomaterial detections. We have numerically modeled the periodic structure with finite-difference time-domain (FDTD) as an Ether, Ethyleneglycol, Chlorobenzene and Quinoline biosensor. As expected, the Talbot effect appears. We have numerically found that the contrast of the self-images pattern changes as we harness the biomaterials. Also, a comparison of the previous results with the proposed model provided and shows that a changes of the contrast of the self-images pattern is due to the change of refractive index of biomaterial. Therefore, the proposed structure can be used as a highly sensitive biosensor.

The rest of this paper is organized as follows. In Sect. 2, the numerical structure of the Talbot biosensor is presented. Then, pivotal parameters of sensors are introduced. In the same Section, the main operation mechanism of proposed model is provided. In Sect. 3, by utilizing the different biomaterials, the Talbot effect for monitoring of the materials is applied. Moreover, obtained result compared with some previous works. Finally, the main conclusions in Sect. 4 is presented.

2 Proposed Talbot bioensor

2.1 Structure of the Talbot bioensor

Figure 1 presented the 3D-view of the proposed biosensor. As can be seen, the proposed plasmonics talbot biosensor is composed of a Cytop polymer grating layer for generating of talbot wave and air medium. The talbot structure is assumed to be illuminated by a tunable semiconductor laser in the range of 550 nm to 650 nm, and incident angle of \(\theta\) is injected from left edge side. As a talbot biosensor, several biomaterials including Ether, Ethyleneglycol, Chlorobenzene and Quinoline are inspected. Schematic configuration of proposed sensor is depicted in Fig. 1. It consists of Cytop as substrate with grating period of \(\Lambda\) and duty cycle of 50 percent. The refractive index of polymer Cytop in range of 0.2 μm to 1.2 μm is shown in Fig. 2. The grooves are etched to depth of 0.5 μm. Other structural parameters of the device are tabulated in Table 1.

The main reason for choosing the talbot biosensor is that its simple and practical configuration with relatively compact footprint. The total footprint of the biosensor is 3.9 μm \(\times\)2.4 μm \(\times\)1.9 μm which is a very good candidate for portable platforms. It is worth to mentioning that the surface of injected biomaterial in the talbot biosensor has a grating configuration; therefore, it can be used for simultaneous biomaterials detection. For selecting the appropriate parameters such as previous works, optimized algorithm is considered (Zhang et al. 2019, 2020a, b, c, d, 2021). For example Yang group used classification for optimization of the structures (Yang et al. 2019; Zhang et al. 2021; Gong et al. 2019).

Fig. 1
figure 1

The 3D-schematic of the talbot biosensor

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Table 1 The layers feature of the proposed Talbot biosensor
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Fig. 2
figure 2

The calculated of refractive index of polymer Cytop

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Incident light of plane wave has been applied along the z axis. The pattern of electric field interference of the structure with air background is shown in Fig. 3. For a diffraction grating the self-imaging pattern is repeated at a distance known Talbot plane, c:

$$\begin{aligned} c=(\lambda /n_r)(1-\sqrt{1-(\lambda /n_r\Lambda )_2}) \end{aligned}$$
(1)

Where \(\lambda\) and n\(_r\) are the free space wavelength and background refractive index, respectively. To examine sensing operation of device, grooves were filled with different material as listed in Table 2. From Table 2 it can been seen that for different wavelength, refractive index of biomaterials change, and the self-imaging intensity pattern changes but the Talbot distance remains the same.

Table 2 The refractive index of tested biomaterial as a function of wavelength
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In next section, firstly, the crucial parameters of the biosensor is reviewed. Then, to gain a deep point of view about the mechanism of the structure several parameters are adjusted to find the appropriate results.

2.2 Talbot biosensor characteristics

To obtain deep point of view about talbot biosensor, potential results should be considered to detection of biomaterials including FoM, and sensitivity. In this regard, the sensitivity is expressed as:

$$\begin{aligned} S=\frac{\Delta \lambda }{\Delta n} \end{aligned}$$
(2)

where \(\Delta \lambda\) and \(\Delta n\) are respectively, reflection red-blue shifts and the biomaterial refractive index changes.

Also, figure of merit (FoM), as another main feature of the talbot biosensor, is calculated from:

$$\begin{aligned} FoM=[\frac{S}{FWHM}\ (1/RIU)] \end{aligned}$$
(3)

where FWHM refers to the full width at half of the maximum parameter at the central wavelength.

In next section, firstly, it assume air condition for talbot biosensor with room temperature condition \(T=300 ^{\circ }K\), and the incident wavelength of laser is tuned from \(\lambda _{0}\) = 550 nm to \(\lambda _{0}\)=650 nm, and calculation is done by using FDTD package. Then, the effect of the different biomaterials including Ether, Ethyleneglycol, Chlorobenzene and Quinoline on the talbot reuslts are considered. Finally, to improve the calculated result, by considering best geometry of the talbot biosensor, highest sensitivity is calculated.

Fig. 3
figure 3

The calculated of self-imaging pattern as the biomaterial is air

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3 Result and discussion

To deep study the performance of the talbot biosensor, the FDTD method for detection of air is used. The electric field profile in this case is obtained in Fig. 3. In this case the temperature is 300 K. Here, the geometrical parameters are set as L\(_g\)= 1200 nm, w\(_g\)= 500 nm, and n\(_{air}\)= 1 so that the talbot resonance wavelength is occurred around 550 nm. As can be seen, the self-imaging pattern is repeated with highly sensitivity and high contrast.

Fig. 4
figure 4

The transmission spectra of structure for different \(\Lambda _s\)

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To further study the results of the proposed configuration, we have also investigated the effect of the \(\Lambda _s\) on the transmittance curve. As can be seen in Fig. 4, at resonance wavelength, the transmission variation by \(\Lambda _s\) is less than 0.1 and the throughput is greater than 0.9.

Fig. 5
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The transmission spectra of structure for different D.Cs

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Fig. 6
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The transmission spectra of structure for different Lg

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As D.C (Wg/\(\Lambda _g\)) is the another important parameter for evaluating the structure, this parameters is provided in Fig. 5. To discuss how the strong light-matter interaction between incident light and layer of Cytop and its effects on the sensing characteristics, we used different D.C of 30%, 40%, 50%, 60%, and 70% for transmission spectra, while keeping other parameters fixed. Figure 5 shows the transmission versus the resonance wavelength for different D.C. As can be seen, the variation of D.C leads to red and blue-shift of the transmission spectrum which is used for sensing mechanism.

To investigate how the length Lg affects the sensing properties, we tuned the Lg from 1.8 to 2 μm in steps of 0.05 μm, while keeping the other parameters fixed. Figure 6 illustrates the relationship between the transmission spectrum and the wavelength for different Lg. Figure 6 shows the red shift of the transmission can be used for sensing mechanism.

Fig. 7
figure 7

The calculated of self-imaging pattern as the biomaterials are Ether, Ethyleneglycol, Chlorobenzene and Quinoline

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To exact study the effect of different biomaterials on the electric field profile, biomaterials are changed, whereas the other parameters are fixed. As illustrated in Fig. 7, different self-imaging pattern are produced for various biomaterials. In this case, easily several biomaterials can be detected.

Fig. 8
figure 8

The calculated of self-imaging pattern as the biomaterials are Ether, Ethyleneglycol, Chlorobenzene and Quinoline

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The transmission spectra of sensor for different sensing materials is depicted in Fig. 8. The dip resonance frequency of device for air is about 525.2 nm. As it seen, by increasing the \(\Delta\)n the resonance frequency is red shifted.

Table 3 The calculated sensing parameters of the proposed structure
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Finally, we analyze the performance of parameters for different biomaterials including Ether, Ethyleneglycol, Chlorobenzene and Quinoline. The resonance frequency and relative sensitivity of talbot biosensor for each material is listed in Table 3.

As can be observed in Table 4, highest sensitivity of 324 nm/RIU by considering Chlorobenzene can be provided. Also, the highest FoM of 20.99 for Quinoline is obtained. The results are highly improved compared with previous works Farmani et al. 2018; Farmani 2019; Farmani and Mir 2019; Farmani et al. 2020; Hamzavi-Zarghani et al. 2019; Amoosoltani et al. 2019; Mozaffari and Farmani 2019; Farmani et al. 2020. Finally, by considering FDTD algorithm and the excitation of talbot waves, the performance of the talbot biosensor remarkably enhanced compared to the obtained results of the previous works provided in Table 4. As a result, the obtained results can be used in recent advanced applications Wang et al. 2017; Zhang et al. 2020, 2019; Sun et al. 2019.

Table 4 Comparison of the sensing parameters of the present and previous works
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4 Conclusion

Here, we have reported the model for a high sensitivity surface plasmon resonance biosensor for biomaterials detection, based on plasmonics Talbot effects. The performance of the biosensor was numerically studied with FDTD method. To evaluate of the biosensor different biomaterials including Ether, Ethyleneglycol, Chlorobenzene and Quinoline were also studied. It was observed that, for small variation of \(\Delta n\)= 0.4, in the biomaterials refractive index, FoM and sensitivity as high as 20.99 and 324nm/RIU are achievable in the biosensor, respectively. We envision that the proposed biosensor based on plasmonics Talbot effect can be used as a potential platform for on-chip biosensors.