Introduction

Surface plasmon resonance occurs when a photon hits the interface between two materials with dielectric function of opposite signs (typically, a metal film and a dielectric). At a certain angle of incidence, electrons in the metal surface layer absorb a portion of the p-polarized incident light and oscillates in response to light excitation. These electron excitations are called plasmons. They propagate parallel to the metal surface generating a surface plasmon wave (SPW) in the interface. SPW frequency depends on the parameters of the sample solution near the interface, namely, on the average refractive index near the sensor’s surface [1,2,3].

Because of its prominent features such as its sensitivity, fast response, great accuracy, radio frequency interference (RFI), and electromagnetic interference (EMI) immunity and its relatively small size and low cost, surface plasmon resonance (SPR) sensors have been a major subject of interest during the last two decade [4, 5].

The first observed SPR phenomena were by Wood [6], but Liedberg et al. [7] introduced the principle for gas sensing and bio-sensing. Since then, a numerous SPR sensing structures have been reported. Many introduced different sensor configurations like removing the fiber cladding [8,9,10], tapering the fiber [11, 12], U-bent fiber [13,14,15], and the use of fiber Bragg grating [16,17,18].

On the other hand, investigating the optimal plasmonic materials was never settle, starting with metals gold [11, 14, 16,17,18], silver [9,10,11, 13], aluminum [19,20,21], copper [22, 23], and platinum [24, 25] to semiconductors [26, 27] and graphene [8, 13, 15, 16, 21, 23]. The plasmonic material performance depends on how it interacts with light: transmission, absorption, and reflection. This behavior is described by the material complex dielectric function [3]. The complex dielectric function consists of a real part and an imaginary part that describe the deflection of light and its absorption, respectively.

There are different calculation methods for determination of metal complex dielectric function (permittivity); the most common are Drude, Lorentz-Drude (L-D), and Brendel-Bormann (B-B). Drude model is the basic one, which describes the free electron effects and proves to be accurate for the low-energy part of the dielectric function [3, 28, 29]. Lorentz reduced the error in Drude model by adding a compensation term for the inter-band electron transitions. Although L-D model extends the validity of dielectric function calculation, it could not describe the sharp absorption, especially in noble metals [28]. The B-B model further improves the accuracy by replacing Lorentz oscillators with a superposition of infinite number of oscillators.

Many studies compared the three models’ performance relevant to measured data; the B-B model showed the best accuracy with more than 94% overlap with the measured data; L-D model was less accurate, and Drude model was the least accurate [28,29,30,31]. Although the Drude model performance was not the best, many SPR sensor studies adopted it as a simple calculation method for metal complex dielectric function [32,33,34,35], and few papers used the more accurate L-D model [36, 37].

In this paper, we investigated how the selection of the used dielectric function model affects the gold SPR sensor’s characteristics, mainly the sensitivity, full width at half maximum (FWHM), and the signal-to-noise ratio (SNR), using the three dielectric function models Drude, L-D, and B-B, and the experimentally measured dielectric function introduced in [38].

Method

We created a model of gold SPR fiber optic sensor; the sensor consisted of a fused silica multimode fiber with a part of its cladding removed and a thin layer of gold deposited on its core as shown in Fig. 1. The fiber optic parameters are shown in Table 1.

Fig. 1
figure 1

Sensor’s configuration

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Table 1 Fiber optic parameters
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The fiber core refractive index has been calculated using Sellmeier formula Eq. (1) of fused silica dispersion in visible and near-infrared range [39]:

$$ {n}_c^2\left(\uplambda \right)=1+\sum \limits_{i=1}^M\ \frac{B_i\ {\uplambda}^2}{\uplambda^2-{\uplambda}_i^2} $$
(1)

where Bi is the ith oscillator strength for the resonance wavelength λi (Table 2)and λ indicates the wavelength of incident light in (μm).

Table 2 Sellmeier constants for fused silica [39]
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Dielectric Function Models

We created three sensor models adopting each of Drude, L-D, and B-B models as the dielectric function’s calculation model using parameters provided in [28] and another reference model using the experimentally measured data of the dielectric function obtained from [38].

Drude model:

$$ \epsilon \left(\omega \right)=1-\frac{\varOmega_p^2}{\omega \left(\omega -i{\varGamma}_0\right)} $$
(2)

Lorentz-Drude model:

$$ \epsilon \left(\omega \right)=1-\frac{\varOmega_p^2}{\omega \left(\omega -i{\varGamma}_0\right)}+\sum \limits_{j=1}^k\frac{f_j{\omega}_p^2}{\left({\omega}_j^2-{\omega}^2\right)+ i\omega {\varGamma}_j} $$
(3)

where ωp is the plasma frequency, k is the number of oscillators with frequency ωj, strength fj, and lifetime 1/Γj, while \( {\Omega}_p=\sqrt{f_o}\ {\omega}_p \) is the plasma frequency associated with intraband transitions with oscillator strength fo and damping constant Γo.

Brendel-Bormann model:

$$ \epsilon \left(\omega \right)=1-\frac{\varOmega_p^2}{\omega \left(\omega -i{\varGamma}_0\right)}+\sum \limits_{j=1}^k{\chi}_j\left(\omega \right) $$
(4)

where

$$ {\chi}_j\left(\omega \right)=\frac{1}{\sqrt{2\pi }{\sigma}_j\ }\ \underset{-\infty }{\overset{\infty }{\int }}\mathit{\exp}\left[-\frac{{\left(x-{\omega}_j\right)}^2}{2{\sigma}_j^2}\right]\times \frac{f_j{\omega}_p^2}{\left({x}^2-{\omega}^2\right)+ i\omega {\varGamma}_j}\ dx $$
(5)

Since the B-B model’s integral calculation solution contains the computation of the complex complementary error function: erfc(z) [28], we used the rational expansion method introduced by Weideman [40].

For a surface plasmon wave to occur, the resonance condition is:

$$ \frac{2\pi }{\lambda }\ {n}_{\mathrm{c}}\ \sin \theta =\mathit{\operatorname{Re}}\left[\frac{2\pi }{\lambda }{\left(\frac{\varepsilon_m{n}_s^2}{\varepsilon_m+{n}_s^2}\right)}^{1/2}\right] $$
(6)

where nc, ns, and εm are the fiber core refractive index, sensing layer refractive index, and metal layer dielectric constant, respectively [41]. For wavelength interrogation fiber optic SPR sensor, the condition is met at a specific wavelength (resonance wavelength), at which the intensity of the transmitted light is the lowest [42]. The resonance wavelength can be obtained from the SPR curve, which shows the detected transmitted light at the fiber’s end, after passing through the sensing area, as a function of the wavelength.

The light was launched into the fiber from a collimated light source through a microscopic lens. For a given incidence angle θ, and a fiber core refractive index nc, the corresponding optical power P(θ) is [43]:

$$ P\left(\theta \right)\propto \frac{n_c^2\ \sin \theta \cos \theta }{{\left(1-{n}_c^2{\cos}^2\theta\ \right)}^2} $$
(7)

To calculate the transmitted power, the reflection intensity coefficient (RP) of each ray is raised to the power of its corresponding number of reflections (Nref). Accordingly, the formula of the normalized transmitted power (Pt) for a p-polarized light is [44]:

$$ {P}_t=\frac{\int_{\theta_{cr}}^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{R}_P^{N_{ref}\left(\theta \right)}P\left(\theta \right) d\theta}{\int_{\theta_{cr}}^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}P\left(\theta \right) d\theta} $$
(8)

where RP is the reflection intensity coefficient for a p-polarized incident wave and is defined by:

$$ {R}_P={\left|{r}_p\right|}^2 $$
(9)

rp represents the reflection coefficient for p-polarized light and was obtained by applying the N-layer model [45] and creating our 3-layer transfer matrix as described in [44].

Nref(θ) is the total number of reflections of the light in the sensing region and is defined by:

$$ {N}_{ref}\left(\theta \right)=\frac{L}{D\ \tan \left(\theta \right)} $$
(10)

where L is the length of the sensing area, D is the fiber core diameter, and θ is the incidence angle.

θcr is the fiber’s critical angle and is described by:

$$ {\theta}_{cr}={\sin}^{-1}\left(\frac{n_{cl}}{n_1}\right) $$
(11)

n1 and ncl are the refractive indices of the core and cladding, respectively.

Results

First, we calculated the Au dielectric function over our frequency range of interest from 0.8 to three (eV), i.e., wavelength range from 1400 down to 400 (nm) using Drude, L-D, and B-B models. The real and imaginary parts of the models-obtained dielectric functions compared to the measured data were illustrated in Fig. 2b.

Fig. 2
figure 2

a Real part of Au dielectric function using Drude, L-D, and B-B models compared with the measured data. b Imaginary part of Au dielectric function using Drude, L-D, and B-B models compared with the measured data

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As we can see from Fig. 2, the B-B model gives the best fit of the measured Au dielectric function in both its real and imaginary parts. On the contrary, Drude model has the worst fit of the three models. In spite of having the best fit for lower frequencies, its deviation increases rabidly above 1.8 eV (eV), i.e., under 688 nm especially for the imaginary part. This matched the previously reported results in [3, 28, 29, 31].

Then, we calculated the SPR curve for the visible and near-infrared (NIR) range at a medium refractive index (RI) of 1.33, using each of the discussed dielectric function models’ sensors and the measured dielectric function’s sensor as shown in Fig. 3.

Fig. 3
figure 3

Fiber optic SPR sensor normalized transmitted p-polarized light of Drude, L-D, and B-B models’ sensors compared with the measured dielectric function model’s sensor at RI 1.33

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As we can see from Fig. 3, the SPR curve of the measured dielectric function sensor has its resonance wavelength at 597 nm. The nearest to it is the B-B model’s curve at 625 nm, while the L-D model’s is a bit higher than the B-B’s at 639 nm. Moreover, the Drude’s is the farthest at 533 nm. Note that Drude model SPR curve nearly matches the measured dielectric function above 700 nm and deviates under that as expected.

To evaluate the percentage relative error in determining the sensor’s resonance wavelength using the three models with reference to the sensor’s resonance wavelength using the measured dielectric function, we used Eq. (12):

$$ \% Error=\left|\frac{Theoritical\ value- Measured\ value}{Measured\ value}\right|\times 100 $$
(12)

As we can see from Table 3, the Drude model’s sensor has the highest error regarding the resonance wavelength at a refractive index of 1.33 compared with L-D model’s which has lower error and B-B model’s which is the nearest to the measured dielectric function model’s value. The resonance wavelength (λres) values for each of the models through the RI range from 1.33 to 1.41 is indicated in Table 4.

Table 3 Error in sensor’s resonance wavelengths obtained using the three dielectric function models with reference to the one obtained using the measured dielectric function’s model at RI of 1.33
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Table 4 Resonance wavelength (λres) values for each of the models through the RI range from 1.33 to 1.41
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From Table 4, it is worth noting that, while the resonance wavelength values of L-D and B-B models are always higher than that of the measured dielectric function’s model, the Drude model’s values are lower than that of the measured dielectric function’s model only at the beginning of the RI range up to 1.37. From 1.38 up to 1.41, the Drude model’s resonance wavelength becomes higher than that of the measured dielectric function’s model.

We calculated the SPR curves for refractive indices from 1.33 to 1.41 for the discussed dielectric function models’ sensors and the measured dielectric function model’s sensor as shown in Fig. 4b, c, and d.

Fig. 4
figure 4

a Normalized transmitted p-polarized light of Drude model’s sensor for RI from 1.33 to 1.41. b Normalized transmitted p-polarized light of L-D model’s sensor for RI from 1.33 to 1.41. c Normalized transmitted p-polarized light of B-B model’s sensor for RI from 1.33 to 1.41. d Normalized transmitted p-polarized light of the measured dielectric function model’s sensor for RI from 1.33 to 1.41

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Figure 4b, c, and d show the difference in the obtained SPR curves using the three models’ sensors and the measured dielectric function’s model. At 400 nm, the normalized transmitted power equals 0.3 for the measured dielectric function’s model and B-B model. L-D’s value is between 0.3 and 0.4, and Drude’s value equals almost one. The minimum power level over the RI range is around 0.2 for Drude model; on the contrary, it reached 0.1 for the other models. For the higher frequencies, while L-D and B-B models power level stayed under 0.8, Drude model and the measured dielectric function’s model reached higher level around 0.9. We can also see that the Drude and the measured dielectric function models’ SPR curves have narrower dips than the other two models, especially the L-D model, which has the widest dips.

As the sensitivity of a sensor is its prominent feature, we calculated the sensitivity of each model’s sensor for the different RI values as shown in Table 5. The sensitivity of an SPR sensor with spectral interrogation equals the shift in the resonance wavelength per unit change in the medium’s refractive index and is given by:

$$ {S}_n\left(\upmu \mathrm{m}/\mathrm{RIU}\right)=\frac{\delta {\lambda}_{res}}{\delta {n}_s} $$
(13)

where RIU is the refractive index unit [46]. The calculated sensitivity valves of each sensor model over the RI range from 1.34 to 1.4 are indicated in Table 5.

Table 5 SPR sensor’s sensitivity using the three models and the measured dielectric function
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As we can see from Table 5, the measured dielectric function’s sensor has the lowest sensitivity values for all the refractive indices. The sensitivity values of B-B model’s sensor are the closest to the measured dielectric function model’s and Drude’s are the furthest while L-D’s are in between and close to the B-B model’s having the same sensitivity values for RI values 1.39 and 1.40. The percentage relative error in sensitivity values of each model’s sensor and the measured dielectric function’s sensor as a reference is calculated and illustrated in Fig. 5.

Fig. 5
figure 5

Percentage relative error in sensitivity of SPR sensor using the three models with reference to the sensor’s sensitivity using the measured dielectric function’s model

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As we can see from Fig. 5, the error in sensitivity calculation for Drude model’s sensor reached up to 80% between 1.33 and 1.34 RI with a mean relative error of 59.2%. As the RI reached higher values, the Drude model’s error gradually decreases. On the other hand, using B-B model greatly decreased the relative error to a maximum of 22% between 1.34 and 1.35 RI, which is four times less than Drude model’s max error and almost constant through the entire RI range with a mean relative error of 17.4%. L-D model gives higher error than B-B model but still lower than Drude model’s error with a maximum of 27.8% between 1.34 and 1.35 RI and a mean of 22.3%.

The second investigated sensor parameter is the FWHM, which is defined as the spectral width of the sensor’s SPR curve at half maximum [47]. The lower the SPR curve spectral width the better, because it will be easier to distinguish smaller wavelength shifts. The values of FWHM of each sensor model for RI range from 1.33 to 1.4 are indicated in Table 6.

Table 6 FWHM of the sensor’s SPR curves using the three models and the measured dielectric function
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As we can see from Table 6, the measured dielectric function model has the lowest FWHM values over the entire range, and all the other models have a relatively higher FWHM values. Figure 6 depicts the percentage relative error in FWHM values of each model’s sensor and the measured dielectric function model’s sensor as a reference.

Fig. 6
figure 6

Percentage relative error in FWHM of SPR sensor curves using the three models with reference to the measured dielectric function’s model

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Figure 6 indicates that Drude model unexpectedly gives better performance in terms of FWHM calculation accuracy than the other two models with respect to the measured dielectric function model’s values over the RI range from 1.33 to 1.35 with a max relative error of 46.2% at 1.37 RI and a mean relative error of 31.6%. B-B model’s performance is not as good as Drude’s, but not far from it, and precedes it above 1.36 with a max relative error of 40.6% at 1.36 RI and a mean relative error of 25.6%. L-D model is the least accurate among the three models with a max relative error of 96.9% at 1.36 RI and a mean relative error of 66.8%. It beats the other two models only at 1.4 RI.

The SNR of an SPR sensor is another important aspect; it indicates how accurately and precisely the resonance wavelength can be detected and, hence, the refractive index of the sensing layer. For a spectral interrogation SPR sensor, the SNR is a dimensionless parameter defined by:

$$ SNR=\frac{\delta {\lambda}_{res}}{FWHM} $$
(14)

where δλres is the shift in the resonance wavelength of the SPR curve induced by an amount of change in the sensed medium refractive index [47]. The SNR values of each sensor model for RI range from 1.33 to 1.4 are stated in Table 7.

Table 7 SNR values of each sensor model for RI range from 1.33 to 1.4
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In Table 7, the Drude model’s SNR values are always higher than that of the measured dielectric function’s model. On the other hand, L-D and B-B models give lower SNR values than that of the measured dielectric function’s model for all the RI range except at 1.4 RI. The percentage relative error in SNR values of each model’s sensor and the measured dielectric function model’s sensor as a reference was calculated and illustrated in Fig. 7.

Fig. 7
figure 7

Percentage relative error in SNR of SPR sensor models with reference to the measured dielectric function’s model

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From Fig. 7, it is obvious that B-B model gives very accurate SNR values with almost zero error from 1.33 to 1.34 RI, a max relative error of 31.2% at 1.4 RI, and a mean relative error of 10.5%. Drude model has a very high error at the start of RI range with a max of 61.1% at 1.33 RI and a mean relative error of 23.8%. The error decreases gradually as the RI increases until it becomes the dominant model in terms of accuracy from 1.36 up to 1.39 RI. L-D model has an intermediate relative error from 1.33 to 1.34 RI and gives the highest error among the three models over the rest of the RI range with a max of 49% at 1.4 RI and a mean relative error of 31.7%.

Discussion and Conclusion

In this paper, at first we confirmed that Brendel-Bormann model gives the most accurate representation of Au dielectric function among the three most used dielectric function models (Drude, Lorentz-Drude, and Brendel-Bormann) compared with the Au measured dielectric function in the visible and NIR range.

Then, we investigated the performance of each of the three models along with the measured data as a reference, when adopted in an SPR sensor model. Based on our results, Brendel-Bormann model preceded the other two models in determining the SPR sensor’s main parameters, sensitivity, FWHM, and SNR with a mean relative error of 17.5%, 25.6%, and 10.5%, respectively. Drude model gave the second best result for FWHM and SNR with a mean relative error of 31.6% and 23.8%, respectively, and the least accurate sensitivity values with a mean relative error of 59.2%. L-D model performed better than Drude model in case of sensitivity with a mean relative error of 22.3% but was the least accurate in determining FWHM and SNR with a mean relative error of 66.8% and 31.7%, respectively.

Using Drude model to describe the gold complex dielectric function of an SPR sensor severely degrades the accuracy of the sensor’s sensitivity and FWHM and SNR calculations. Using L-D model instead of Drude’s improves the accuracy of the sensitivity calculation but degrades the FWHM and SNR calculation accuracy. The best choice is B-B model, which has the best accuracy for all the sensor’s parameter calculations leading to a more realistic representation of the SPR sensor and reducing the gap between theoretical sensor’s models and the experimental results.