Classical and Quantum Surface Plasmon Resonance Biosensing

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Introduction
Over the past few decades, we have seen several new viral strains emerge around the world [1].In the late 1990s, there was the HIV pandemic; then, in the early 2000s, that is, 2009-2010, we had the swine fu; in the recent 2000s (2014-2016), we have seen the Ebola outbreak; and most recently (2019), we had a COVID outbreak [2].Tis is a very short list in a long list of outbreaks of diseases.Many are expected to occur over the next few years.One major problem is that viral strains are rapidly evolving, often faster than technology is capable of keeping up, leading to epidemics and pandemics.Te occurrence of these disease outbreaks has heightened the importance of having precise diagnostics more than ever before, and optical biosensors, such as surface plasmon resonance (SPR)-based biosensors, are proving to be a strong contender in the fght against disease outbreaks [3][4][5][6].SPR-based biosensors are not only good for providing accurate diagnostics but are also useful in drug development as well; rapid treatment development is also critical to stop the progression of associated diseases.SPR-based optical biosensors are reliable diagnostic devices that can make good point of care (PoC) systems due to their features such as portability, reproducibility, sensitivity, and specifcity [7,8].SPR biosensors enable for the study of the binding kinetics of biochemical reactions that occur in a solution environment.In addition, they assess dissociation constants, binding and association rate constants, and stoichiometry for the kinetics of biochemical binding interactions.SPR is utilized for equilibrium analysis, concentration analysis, and kinetic analysis, all of which are crucial in detecting various phases of diseases like Epstein-Barr virus [9].Identifying antibodies that bind to diverse antigens on a virus's surface can be utilized to determine the presence of specifc illnesses [9].SPR biosensors are an excellent tool for detecting the presence of these antigens.Beyond the realm of biosensing, SPR techniques also fnd utility in food monitoring [10], such as detecting antibiotics which are found exclusively in milk samples [11] and identifying the mycotoxin patulin [12].SPR technologies have been used successfully in real time monitoring of the binding kinetics in a reaction between dengue antibodies with its associated antigen [13].Te fexibility in terms of applications of SPR technologies and biosensors makes them very useful.SPR-based devices also have applications that include the detection and quantifcation of viruses such as the enterovirus 71 [14], membrane interaction studies which are applicable/applied to virus detection [15], and design of diseases such as COVID-19 [16].SPR biosensors have surpassed traditional viral detection and quantifcation techniques in utility, and this is particularly true in the felds of medical diagnosis and healthcare [16].A multibillion dollar global market exists for SPR-based biosensors, and major manufacturers include Biacore and Dynamic Biosensors.
In 1957, General Electric physicist Rufus Ritchie made a theoretical prediction of the existence of surface plasmons [17].However, the SPR phenomenon had been observed earlier by Robert Wood, a physicist at Johns Hopkins University, in 1902.When Wood directed polarized light at a difraction grating with a metal backing, he noticed a series of alternating dark and bright stripes in the light that was refected [18].Tis was the frst recorded instance of SPR, but, at the time, there was no explanation for it.In 1907, Lord Rayleigh, a British physicist, was the frst to develop a theoretical treatment of the optical observation of SPR in 1907.He hypothesized that the phenomenon was caused by the interaction between light and electrons on the surface of the metal.Dieter Otto demonstrated in 1968 that surface plasmon excitation arises as a result of attenuated total internal refection (ATR), a technique that involves directing light at a particular angle through a prism or other transparent material to create an evanescent wave that interacts with the metal surface [19].Otto Max Kretschmann and Heinz Friedrich Wilhelm Raether both obtained comparable results from a diferent ATR confguration in the same year (1968) [20,21].It was not until 1983 that Liedberg et al. demonstrated the application of SPR to biosensing [22].Liedberg and colleagues showed that by coating a gold surface with a thin layer of biomolecules, they could use SPR to detect changes in the refractive index of the surrounding medium caused by the binding of complementary biomolecules.Since then, SPR technologies have developed into standard tools to study kinetic interactions between target molecules and biorecognition molecules.Surface plasmon resonance (SPR) became the core technology in afnity biosensors for the study of biomolecular interaction analysis (BIA) in the early 1990s [23].Tese biosensors use SPR to detect and quantify the interactions between biomolecules, such as proteins and DNA, in real time and without the need for labeling or other modifcations to the molecules.Te fundamental/ core principle of SPR-based BIA is to immobilize one of the biomolecules of interest on the sensor chip and then pass the other molecule over the surface.As the second molecule binds to the immobilized molecule, the refractive index at the surface changes, causing a shift in the SPR signal.By monitoring this shift, researchers can determine the strength of the interaction and other properties, such as the kinetics and afnity constants [23].SPR technologies are prevalent to the point where they have been used to study diseases (and the viruses, organisms, or bacteria that cause them), such as AIDS (HIV) [24], COVID [25], and malaria [26].It should be noted that there are other approaches regarding the properties and applications of plasmonicbased sensor structures which include MDM (metal-dielectric-metal) waveguide-based resonator [27][28][29], rodshaped plasmonic sensor [30], and hybrid MDM-based plasmonic nanostructure [31,32].All of the simulations described in this work were generated using Mathematica code which was written by the authors.Any other programming language can be used for similar work such as Python and MATLAB.
Although SPR-based plasmonic nanosensors are commercially available, the precision of current SPR instrumentation is limited when it comes to measuring light refectivity from the sensor.Tis limitation can lead to problems in various applications.For example, examining the movement and behavior of medications targeting various HIV-1 virus variants necessitates a level of accuracy beyond what is presently ofered by commercially accessible SPR-based nanosensors [33].In order to tackle this problem, scientists need to explore the underlying principles of the sensor's physics and take into account the foundational elements of optics, including resources rooted in quantum mechanics [34].
One very important parameter that is used to characterize the performance of a biosensor is its limit of detection (LoD).LoD is the minimal detectable change in a parameter of interest, x, where x can be the wavelength, refractive index, phase, or incident angle in the case of an SPR experiment.For a signal, y, the LoD of an SPR-based biosensor is dependent based on the transduction signal's measured noise, Δy, that is, the minimum resolvable signal.
where S y is the sensitivity [26].In order to improve the limit of detection (LoD) of SPR-based biosensors, researchers can either increase the sensitivity of the system or reduce the noise level.However, there are inherent limitations to how much the sensitivity can be improved due to the statistical nature of the light used in the system.Te sensitivity of an SPR-based biosensor is determined by the magnitude of the change in the refractive index at the surface of the sensor, which is proportional to the amount of analyte bound to the sensor surface.Te refractive index change causes a shift in the SPR signal, which can be measured with high precision using a detector.However, the precision of the detector is ultimately limited by the statistical fuctuations in the intensity of the light used to excite the plasmons at the surface of the sensor.Tis is known as shot noise, and it sets a fundamental limit to the sensitivity of the system [35].
According to research, the use of quantum states of light (such as the NOON, squeezed, and Fock) has great potential in reducing measurement noise, Δy, thus improves the LoD of plasmonic-based biosensors, going below the restrictive and classically limiting shot-noise limit (SNL) bound [35][36][37][38][39][40].

International Journal of Optics
Te formula for the sensitivity of the biosensor is given as follows: where p is a parameter to be estimated in the measurement of the signal y.Te precision (LoD) with respect to the estimated p value is defned as follows: To compare the performance of various types of SPR sensors in detecting the refractive index, for example, we can rewrite the LoD formula with respect to the refractive index as follows: where n is the refractive index of the sample that the detector is monitoring.In general, for SPR biosensing, y can be either of the analysis quantities (λ spr , Θ spr , n spr , ϕ spr ) where λ spr is the wavelength of light incident on the biosensor, Θ spr is the incident angle of light on the biosensor, n spr is the index of refraction of the sample monitored by the biosensor, and ϕ spr is the phase.Te aim of this research is to reduce the noise in the transduction signal, Δy, which will lead to an improvement in the precision of the biosensor measurement, Δx.By applying quantum states of light, researchers hope to go beyond the limitations of classical metrology and achieve greater precision in biosensor measurements [34].Quantum-enhanced SPR-based biosensors have already been shown to provide enhancement over their classical counterparts.In the later sections of the paper, we will discuss some quantum plasmonic resonance sensing and the potential of quantum biosensing to enhance the performance of SPR-based biosensors.Tis review provides an overview of the surface plasmon resonance mechanism, applications of SPR, and some limitations, as well as report on recent research to address certain limitations and quantum-based SPR sensing.

Introduction to Surface Plasmon Resonance
Surface plasmons are described or defned as collective oscillations of free electrons that occur at the interface between a metal and a dielectric material, such as air or water.
Tese oscillations are evanescently confned to the interface, meaning that they decay exponentially in the direction perpendicular to the interface [41].Te ability of a metal to excite surface plasmons is determined by its electromagnetic properties, specifcally its dielectric function or complex refractive index.Metals with a high density of free electrons, such as gold, silver, and copper, are often used for surface plasmon applications because they have a large negative real part of the dielectric function at optical frequencies, which allows for strong coupling between light and the collective oscillations of the electrons on the metal surface, known as surface plasmons.In contrast, metals with a low density of free electrons, such as aluminum and titanium, do not support surface plasmons at optical frequencies due to their weak coupling with light.Other factors, such as the size and shape of the metal nanostructures, can also afect the ability to excite surface plasmons.Gold or silver (for biological experiments, silver may not be used because of its antibacterial and antiviral properties) is typically used to excite surface plasmons because their electromagnetic properties (charge arrangement and conductivity) are the most suitable for getting an excitation response.Surface plasmon waves are evanescent waves whose propagation is confned to a metal-dielectric interface, as shown in Figure 1.Te surface plasmon wave originates as a result of the coupling between an incident electromagnetic wave and oscillating electrons on a thin metal surface (at a specifc angle of incidence of the electromagnetic wave).Te conditions for this coupling to occur are discussed in the next section, where we derive the propagation coefcient.Figure 1 shows the excitation of surface plasmons by sending an electromagnetic wave towards a thin metal sheet through some dielectric medium of refractive index n 1 .Te surface plasmon is only few nanometers high.Tere are two main sets of confgurations used to excite surface plasmons, called the Krestchmann confguration (Figure 1(a)) and the Ottoman confguration (Figure 1(b)) as shown in Figure 1.Te diference being that, for the Krestchmann confguration, the analyte of interest sits on the surface of the gold and the plasmons are excited on the top part of the gold layer, while for the Otto confguration, the analyte of interest is between the gold layer and the prism and the plasmons are excited at the base of the gold.When SPR occurs, the intensity R (also known as refectance) of the refected electromagnetic wave is measured by the detector as shown in Figure 1 and decreases sharply (this happens as the angle of incidence of the incoming beam changes).Te refection coefcient or intensity (R) is determined by the angle of incidence of the incoming light and the refractive index of the analyte being examined.Te refection coefcient is defned as the ratio of the amplitude of the refected wave to the amplitude of the incident wave [19].Te relationship between the refection coefcient and the incident angle and refractive index can be described by Fresnel's equations, which are used in optics to calculate the refection and transmission of electromagnetic waves at the interface between two media with diferent refractive indices.Te minimum point of the intensity decline is sometimes referred to as an SPR dip, and this drop in energy is a result of the coupling of incident light with electrons in the metal layer.Te profle generally resembles the plot in Figures 2(a) and 2(b), depending on whether the angle or index of refraction is changing.Te intensity R of the electromagnetic wave refected depends on the angle of incidence θ i and the index of refraction of the dielectric medium sitting on the metal surface.Te resonance dip measurement can also be obtained from the change in the index of refraction, that is, by keeping the incident angle constant and changing the index of refraction of the dielectric on the biosensor surface (that is the metal surface), as shown in Figure 2(b).Changing the refractive index of the dielectric medium while keeping the International Journal of Optics incident angle fxed above the resonance angle can be used to tune the SPR signal, and by altering the refractive index of the dielectric material, we can observe a decrease in the SPR signal, as shown in Figure 2(a).Because refectance, R, is a collective function of both, angle of incidence and the index of refraction, we can study its relationship with either parameter by holding the other constant; however, we can construct the three-dimensional plot as shown in Figure 2(b).To fnd the angle of incidence and the index of refraction parameters that give the lowest resonance dip, Figure 2(c) shows a 3D plot of the dependence of R, both on the index of refraction and on the incident angle.Tis can be useful in identifying the true minimum of R.

Propagation Coefcient Derivation from the Wave Vector.
Te dispersion relation is a mathematical expression that describes the relationship between the frequency and wavelength of a wave in a given medium.Te propagation coefcient is vital for understanding the excitation of evanescent waves.It is the key result that tells us the conditions under which the coupling between the incident light and the metal plasmons can occur.In this section, we derive the propagation coefcient from the wave vector equation.To get a derivation of the propagation constant, it is critical to follow the calculation from the textbook of Maier [41].To obtain classical feld expressions for surface plasmons sustained by a single interface, Maxwell's equations are employed.Beginning with the wave equation, the dispersion relation in equation ( 5) can be obtained.
where k 0 is the free space permittivity of light, ϵ(ω) is the dielectric function of the metal layer, and ω is a frequency.

Discussion of Sensing Techniques and Coupling Strategies.
Te propagation-dispersion relation is an important concept in the study of surface plasmon excitation.It describes the relationship between the momentum and frequency of the surface plasmon waves, which can be excited at the interface between a metal and a dielectric material.It also gives information on whether or not SPR can occur in a setup.Propagation relation studies for diferent media help us to understand the diferent combinations of materials that can excite surface plasmons.Figures 3(a) and 3(b) show the propagation relations in diferent media, and we can see the combination of media that allows the excitation of plasmon resonance.
From Figure 3(b), we can see that surface plasmons can only be excited at the metal-dielectric interface, where the mode matching condition is satisfed, that is, when k ‖SiO 2 � k ‖sp .It is visible from Figure 3 that getting the coupling conditions is not an easy exercise in practice.It is also clear from Figure 3 that in the air-metal interface, the coupling conditions are never satisfed, that is, k ‖air ≠ k ‖sp meaning there cannot be any coupling without the introduction of some other medium.Tis is due to the fact that surface plasmon modes cannot directly interact with the far feld (i.e., free space electromagnetic radiation) and vice Both confgurations are practically useful, but the Krestchmann confguration tends to be easier to confgure and is more commonly used.In these setups, light is sent at a varied angle of incidence towards a glass prism (BK7) that has a gold layer on its surface (or has a gold-coated slide sitting above it).Tis gold layer is only a few nanometers thick.At a certain angle, the incident light couples with oscillating electrons in the thin gold layer, resulting in an evanescent wave propagating on the gold surface, which is called the surface plasmon wave.When this surface plasmon wave forms, we observe a dip in the intensity of the refected wave, which is collected by the detector.Te surface plasmon wave is sensitive to changes in the dielectric properties of the analyte (with a refractive index n 2 ) on the surface of the gold.Tis sensitivity changing dielectric conditions on the surface of the gold is used for biosensing applications.International Journal of Optics versa; this is known as the reciprocity theorem (which states that the coupling between two systems is the same, regardless of which one is the source and which one is the receiver).To solve this problem, we need to excite surface plasmons below the light line (i.e., at frequencies and wavevectors that are not accessible through conventional optical methods), and one common approach is to use a high refractive index medium, such as a BK7 prism or a grating, to couple light into the metal-dielectric interface at a specifc angle of incidence.From Figure 3, light in the glass will not excite the plasmons bound at a metal/glass boundary, but they will be excited at the metal-dielectric interface if the mode matching condition is met, that is, where k ‖SiO 2 � k ‖sp .From Figures 3(a) and 3(b), the mode matching conditions are satisfed at the crossing point between the dispersion curves.
Tere are many mechanisms reported to excite surface plasmons [43].Here, we will briefy discuss four fundamental mechanisms for excitation of the surface plasmon, that is, through optic coupling, waveguide coupling, grating prism coupling, and fber coupling.It also be noted that the SPR efect can also arise from SPR-based nanostructures [44][45][46][47].Tis is due to a phenomenon known as localized surface plasmon resonance (LSPR) that occurs when metallic nanoparticles, typically gold or silver, interact with light [48,49].It is a type of SPR that happens on a nanometer scale, resulting in enhanced electromagnetic felds near the surface of the nanoparticles; however, this is not in the scope of this work.Prism couplers have been reported to be a convenient SPR confguration with a very small LoD; however, the size of the prism can be cumbersome and poses challenges in terms of incorporating it into other optical and electrical components [43].Tere are a few strategies that can be used to mitigate these challenges.One approach is to use microprisms or nanoprism arrays, which can be much smaller than conventional prisms and can be fabricated using advanced lithography techniques.Tis can help to As the angle of incidence of light hitting the prism changes, the intensity of the refected beam R changes.Tis is because the incident beam begins to couple with the plasmons in the gold metal exciting them and generating a surface plasmon wave in the process; the energy of the beam is thus converted to this wave, and we start to get a drop in power.As the coupling becomes stronger and the evanescent waves become more intense, the refected beam becomes weaker until it reaches the minimum point, that is, the resonance dip.(a) SPR dip with resonance angle.Te same resonance dip efect can be observed when the incidence angle is kept constant and the refractive index changes instead.Tis is shown in (b) with an SPR dip plotted against the dielectric material's refractive index.(c) A 3D representation of the SPR dip is shown, considering the refractive index of the dielectric material and the angle of incidence.In this example, the coupling occurs at an incident angle of θ � 73 °and a dielectric material of refractive index n 2 � 1.39 and the refractive index of the prism n 1 � 1.52.Tis was taken from reference [40].
International Journal of Optics reduce the overall size and complexity of the SPR setup.Waveguide coupling is also an alternative to the prism setup because the waveguide confguration has the advantage over the prism confguration that it is as robust and easier to incorporate with other electrical and optical components.
Although the prism and waveguide mechanisms ofer a low LoD, they have drawbacks that restrict their use.Te inability of the evanescent feld in SPR structures to permeate the surrounding medium beyond 100 nm makes it difcult to detect large target molecules.Examples of such molecules can be cells or bacteria, and this is one of these limitations [43].Additionally, these mechanisms only have one surface plasmon for detecting changes in refractive index (RI).Tis makes it impossible to distinguish changes in the surface refractive index from bulk solution refractive index [50].Some models have been proposed to minimize these disadvantages, that is, allow the sensor to diferentiate the change in the background index of refraction and the surface-bound refractive change [7], and increase the penetration depth of the surface plasmon into the medium surrounding.Grating prism coupling is a versatile and effcient method for exciting surface plasmons on metaldielectric interfaces [51][52][53][54].Tis technique involves the use of a difraction grating placed in contact with a metal surface, followed by coupling of light through a prism.Te grating provides a periodic modulation of the refractive index, causing the incident light to couple into specifc angles corresponding to the plasmon resonance conditions.Tis coupling mechanism is highly dependent on the grating parameters such as period, depth, and the refractive indices of the involved materials.By adjusting these parameters, researchers can control the excitation efciency and the resonance angle of the surface plasmons.Grating prism coupling is particularly useful for studying localized and propagating surface plasmons and is commonly employed in various sensing and spectroscopy applications.Fiber coupling is another essential method for exciting surface plasmons, ofering a fexible and remote approach [51,[55][56][57][58]. Tis technique involves the use of a tapered optical fber that is brought into close proximity with a metal-dielectric interface.Light is launched into the fber, and due to the strong evanescent feld at the taper, energy is transferred to the surface plasmon modes.Fiber coupling is advantageous as it enables easy manipulation and positioning of the excitation point, allowing researchers to explore diferent regions of a sample [59].Moreover, it ofers compatibility with other optical components, facilitating integrated systems for sensing and imaging applications.However, it is important to consider the taper dimensions, polarization matching, and alignment precision to achieve efcient and controlled excitation.
Although four primary SPR sensing techniques exist, this paper will concentrate predominantly on the prism mechanism.Te main reason being that the prism mechanism is a well-used mechanism, especially by companies such as Biacore, which manufacture biosensing devices, and it also easily translates into the quantum sensing regime [60].Te popular prism confguration is the Krestchmann confguration discussed previously (as opposed to the Otto confguration).In the prism mechanism, we can also study sensing submechanisms; for example, we can consider an intensity-sensing-based approach, a wavelength-sensingbased approach, an angular-sensing-based approach, and a phase-sensing-based approach in a prism-based approach such as the Kretschmann confguration.We will examine the intensity-based approach for this review because of the ease .ω is the surface plasmon frequency.Te high-index medium is indicated by the letter "h" in the wave number k's subscript.Tis shows that if the electromagnetic wave does not propagate through some high refractive index medium frst before hitting the gold layer in the setup, then there would be no coupling as the propagation coefcients never meet; however, if the light goes through some high-index medium, then there will be coupling.Tis fgure was drawn from an example in reference [42].(b) Tis graphic displays the propagation relations in a Kretschmann prism confguration as well as the reasons surface plasmons at the metal-glass contact cannot be activated (coupling conditions are not satisfed).If a prism (high refractive index medium, such as a BK7 SiO 2 ) is incorporated into the setup and the light passes through it frst, surface plasmons are excited at the metal-air interface.Tis is demonstrated by the intersection of the blue prism dispersion line and the red surface plasmon dispersion curve at the metal/air contact.Surface plasmons at the metal/glass contact cannot be stimulated, according to the green curve.Tis fgure was drawn from an example fgure in reference [42].6 International Journal of Optics with which it translates into the quantum sensing regime; however, in principle, we could also look at a phase-sensing approach [38].Te angular and wavelength-sensing approaches do not easily translate to the quantum regime but will be discussed briefy.

Angular
Sensing.An angular interrogation SPR sensor measures the change in refection intensity over a range of incident angles at a fxed wavelength (see Figure 4(a)) (λ, it is assumed that the wavelength of incident light here is fxed).Te incident angle that corresponds to the resonance dip shown in the previous section is called the SPR dip angle and corresponds to the point where the coupling efect is strongest.Te shift observed in the SPR resonance angle (Δθ) corresponds to a change in the refractive index of the dielectric medium located on the gold surface (Δn s ).Te relationship between the resonance dip angle to the SPR biosensor and the refractive index is shown in the following equation: where θ SPR is the resonance angle of the SPR (the change in the resonance angle can be plotted as a time-dependent function, but this will be discussed when we look at interaction kinetics in the next section), λ is the wavelength of the electromagnetic wave incident on the prism, ϵ m (λ) is the dielectric constant of the metal flm, which is a function of the wavelength of the incident beam, λ, ϵ s is the dielectric constant of the dielectric medium, and n p is the index of refraction of the prism.Here, ϵ 1/2 s � n s , where n s is the refractive index of the dielectric medium sitting on the surface of gold.When the index of refraction is fxed and the angle angle of resonance is changed, we obtain a drop in the resonance profle, as shown in Figure (2(a)).An interesting phenomenon occurs when the index of refraction of the analyte on the surface of the gold changes (this can be because of binding reactions on the prism surface), and the position of the dip shifts to the left or right, depending on whether the index of refraction value increases or decreases due to the dielectric medium on the surface of the metal plate.Figure 5 shows this change.In Figure 5, the index of refraction of the analyte on the surface of the prism changed from a value of 1.39 to 1.4, where the position of the dip changed to the right.In kinetic studies, researchers study this angle position drift (or refectivity intensity drift) which is time dependent.As a result, we can obtain sensorgram plots that show dip angle drift over time.Tese drifts are essential in sensing experiments as they give information about binding parameters, such as binding constants, in kinetic binding studies.Te sensitivity of the resonance angle to changes in nanoseconds (n s ) can be obtained from the coupling condition and can be written mathematically as follows: where Given the aforementioned relationship, it is clear that the refractive indices of the dielectric (analyte), the prism material, and the metal all afect the change in resonance angle.In principle, one could plot the sensitivity equation for diferent metals for comparison.

Wavelength Sensing.
A wavelength-based SPR sensor measures the change in refection intensity over a range of wavelengths at a fxed incident angle.Typically, with this biosensor, a broadband light source covers the target spectral range.Utilizing a spectrometer, the SPR spectrum is recorded.Here, a specifc wavelength can also induce a SPR dip; this is the dip wavelength λ SPR .Te refectance curve can also be calculated by measuring the incident light as a function of wavelength (or frequency) that is injected into the prism setup with a fxed incidence angle.Te sensitivity to refractive index changes would be calculated as |dλ/dn|.Sensitivity can also be calculated as a function of wavelength, as shown in the following equation: where (10)

Intensity Sensing.
When utilizing an intensity-based SPR sensor, the refected intensity is measured at a set incident angle and wavelength.If the coupling conditions for light surface plasmon have been fulflled, there is a reduction in the refectivity of the SPR dip, known as ATR.Tis leads to a near zero drop in the measured refectance.To demonstrate that incident light is converted to a surface plasmon mode, we can solve the Maxwell equations.We calculate the refection coefcient using three-layer system equations, which is expressed mathematically as follows: where Here, . ϵ l is the respective permittivity, and d is the thickness of the metal layer.Refectance, R, is proportional to the squared refectivity coefcient (R ∝ |r sp | 2 ).Te sensitivity to changes in the refractive index would be calculated as follows: Here, R is the refectance and S R is the intensity sensitivity.

Phase Sensing.
A phase-based SPR sensor measures the phase response with a change in angle of incidence and at a fxed wavelength.Tere is a steep slope of the phase response over a very small range of refractive index values.
Phase modulation has previously been shown to give higher resolution compared to conventional SPR sensors based on angle and wavelength modulations [62].From the Fresnel model [20,21], the refection coefcients of the p-polarization (which excites surface plasmons) can be expressed as follows: where r is the refection coefcient described earlier and ϕ is the phase.Sensitivity to changes in the refractive index would be calculated as follows: Here, θ is the phase and S θ is the phase sensitivity.

Analysis of Binding Reaction Kinetics.
Interaction kinetics describe the binding and unbinding processes between ligands and receptors, where receptors are described as chemical structures made up of proteins, and ligands are molecules that chemically bind to a receptor to create a biochemical complex.Te interaction kinetics between the ligands and receptors is divided into three phases, namely, association, steady state, and dissociation.Te three phases are shown in Figure 6.When a ligand binds to its receptor, the process is called association, and when equilibrium is reached and the quantity of binding and unbinding molecules is equal, the process is called steady state.Dissociation, on the other hand, refers to the process where the bonds between the complex molecules start to break or unbind more quickly than they can bind; [63] this can be because the ligands are being fushed out.Te complex concentration receptor ligand in the system at any time is mathematically written as k a [L][R].Here, k a is a parameter called the association constant (which measures the rate at which a ligand and its receptor bind to form Refractive index, n 2 = 1.39 Figure 5: We see a shift/drift to the right in the position of the angle dip as the index of refraction of the analyte on the gold surface changes.Te orange curve represents the curve obtained when the index of refraction of the analyte is 1.39 but as it changes to 1.4 a new curve is obtained, that is, the red curve is obtained and a new resonance dip position is established.Te dip position is altered by changing the index of refraction of the dielectric on the gold surface.Tis shift is called the angular shift and corresponds to the change in refractive index.Tis fgure was generated in Mathematica programming language using details from an example fgure in reference [42].8 International Journal of Optics a complex) and is measured in per molar per second ). Te collision rates between the ligand and the receptor molecules are used to determine the association constant.At time t � 0, the concentration of the receptor is expressed as [R 0 ] and while the ligand concentration is expressed as [L 0 ].Due to the employment of ligands and receptors, the concentration of receptor molecules decreases as the concentration of the receptor-ligand complex increases.At time t ≥ 0, we have the concentration of ligands, where [C] is the complex concentration and the concentration of receptors, Reverse reactions in the equilibrium state can also be considered, where a decrease in the concentration of the receptor-ligand complex is directly proportional to an increase in the concentrations of the ligand and receptors.Here, the concentration of the ligand receptor is where k d is a parameter called the rate of dissociation constant and is measured in per second units ([s − 1 ]).Dissociation is a frst-order reaction meaning the rate of dissociation is proportional to the concentration of the complex.In other words, as the concentration of the complex decreases, the rate of dissociation also decreases proportionally.
According to the law of mass action, the rate of a chemical reaction is proportional to the product of the concentrations of the reactants of the reactants raised to the power of the stoichiometric coefcients of the reactants.In the context of biomolecular interactions, this means that the rate of association or dissociation between a ligand and receptor is dependent on the concentration of both the ligand and receptor.At equilibrium, the rate of formation of the ligand complex should be equal to the rate of depletion of ligand and receptor concentrations, that is, Te ratio in equation ( 19) can be used to calculate the dissociation constant of the reaction, Te unit of the equilibrium dissociation constant is the molar.Te reciprocal of the equilibrium dissociation constant, k D , that is, k A is a parameter called the afnity of the biochemical reaction.k A is measured in unit per molar ([M − 1 ]).One can calculate the afnities of ligand-receptor interactions using a technique called equilibrium dialysis.Tis involves separating two solutions with a semipermeable membrane and waiting until the concentration of difusible substances on both sides of the membrane, becomes equal.Te equilibrium dialysis process is shown in Figure 7.
In Figure 7 the concentrations at time t � 0 of both ligands and receptors are [R 0 ] and [L 0 ], respectively, in the biosensor chamber.Te complex concentration changes over time until equilibrium is reached, and this dynamic evolution is given by the equation, To solve the equation for the ligand-receptor reaction, the approximation of pseudo frst-order reaction is utilized, as the reaction is a second-order reaction.Tis approximation assumes that the ligand concentration is in excess of the receptor concentration, denoted by [L 0 ] ≫ [R 0 ].Tus, the amount of ligand involved in the binding interactions is negligible relative to the ligand's initial concentration.Tis can be expressed as Te pseudo frst-order's solution, equation ( 22), equation is When the reaction has reached equilibrium at time t � τ, the concentration of complexes in the chamber increases to [C τ ].Tis is the point in time where the chamber is typically washed with a bufer, such as water, to remove any unbound ligand or receptor and to prevent any further binding from occurring.Te decreasing rate of the complex is expressed as When the binding reaction begins, the signal change in the association region is observed (as receptor-ligand complexes form) up to the point where the binding surface is fully covered, at which point we observe a constant signal of the steady state; when we begin fushing out the ligand, we observe a sharp decline in the dissociation region, which is a result of rapid unbinding of the receptor-ligand complexes.

International Journal of Optics
As the bufer is introduced to the system, the concentration of the complex [C] starts to decrease exponentially with time from the beginning of the elution process t � τ.Tis is due to the fact that the concentration of the ligand [L] decreases to zero with the introduction of the bufer.Figure 8 gives a picture of ligand-receptor interactions and how this can be mapped onto a sensorgram from which the binding kinetics can be extracted.
From Figure 8 when the bufer is pushed in, a resonance drop is observed, after which we let the analyte fow causing the resonance drop changes in intensity versus the SPR angle plot.Tis change can be tracked over time on a sensorgram, which measures the change in the SPR angle over time.
Using the angle shift model (equation ( 25 Following the calculations by Xiao [63], a mathematical model can be constructed that gives the sensorgram of the angle drift versus time.Te mathematical model is used for theoretical biochemical reaction simulations which helps us understand the binding kinetics that occur (see the example in Figure 4(b), where they are investigating the interaction of bovine serum albumin (BSA) with rabbit anticow albumin antibody IgG1 (anti-BSA)).Te model is as follows.

Δθ
Te change in resonance angle (Δθ) is a function of time (t), observable rate constant (k s ), and A ∞ and A τ , which are related to the sample concentrations.Te rate constant (k s ) can be calculated as the sum of the product of the association constant (k a ) and the initial ligand concentration (L 0 ), and the dissociation constant (k d ), such that k s � k a [L 0 ] + k d .By plotting Δθ against time, a sensorgram can be obtained, and the association and dissociation parameters can be extracted from it.
We can also study the kinetics of intensity-based sensorgrams (see the example in Figure 4(b), where they are investigating the interaction of BSA with rabbit anticow albumin antibody IgG1 (anti-BSA)).For the transmittance sensorgram (when there is a linear relationship between the concentration of the complex [C] and the transmittance T), we can write [63] where T ∞ is a constant (this constant is dependent on the initial receptor and ligand concentrations as well as the receptor and ligand thickness above the metal layer), k A represents the afnity constant, the other constant, T τ , depends on T ∞ , and k s is the association constant [61].A rise in the complex concentration [C] leads to an increase in the value of ϵ a , which consequently increase T, when the angle of incidence of light remains constant.

Uses of SPR
t= ∞ Figure 7: Te schematic representation demonstrates the application of equilibrium dialysis for assessing the afnity between a ligand and receptor interaction, where a semipermeable membrane is portrayed by the dotted line at the center of a water chamber.Te presumption is that the magnitudes of the association and dissociation rates are equal.Te fgure was drawn from a fgure in reference [63].
10 International Journal of Optics several ways: (1) detection of binding between the interacting molecules, (2) assessment of the strength of the interactions (afnity), (3) measurement of association and dissociation rates (ka and kd), and (4) estimation of the concentration of one of the molecules involved in the interaction.Kinetic parameters are useful for immunology studies, that is, in vaccine design and drug efcacy testing, and for diseases such as malaria, cancer, and Neisseria meningitidis [64].

Quantum Plasmonic Biosensing
Tis section will focus on intensity-based plasmonic sensing; this is because the intensity-based mechanism integrates easily with quantum resources.We examine how quantum optical resources can enhance the precision of SPR-based plasmonic biosensors.Te quantum states discussed in this work are the two-mode squeezed displaced (TMSD), twomode Fock state (TMF), two-mode coherent state (TMC), two-mode squeezed vacuum (TMSV), NOON, and twomode product squeezed (TPS) states (see Figure 9).Despite the successful commercialization of SPR-based optical biosensors, they remain limited and restricted by the fundamental limit on the optical intensity noise (often observed in measurements with a photodiode or a CCD image sensor).Tis limit is a result of quantum fuctuations of light (related to the discreteness of photons) and is known as SNL.
SPR-based biosensors such as those produced by Biacore use laser light to probe biological samples.Lasers produce coherent states of light whose intensity noise cannot go below the shot-noise limit (SNL).It is known that by using quantum states of light such as amplitude-squeezed light (generated by transforming coherent states via nonlinear interactions), we can get a noise below the SNL.Te LoD of our biosensors is dependent on the noise of the signal measured in the biosensor; hence, by using quantum states of light, we can lower the LoD and the SNR ratio of our signal.In this section, we will look at the transition by using quantum states of light from recent SPR work.An example of a two-mode biosensing system is shown in Figure 10.A number of researchers have shown the enhancement that comes from using quantum states of light in SPR-based biosensing.Lee et al. in their paper [60] showed that twomode quantum states of light that have symmetric statistical features in their photon number can improve intensitysensitive SPR sensors.Tis work was followed by an analysis that looked at the TMSD state as a probe state in SPR biosensing where the authors showed an enhancement in the LoD when using this state [65].In another paper [61], the authors studied the quantum-enhanced measurement of kinetic parameters in biochemical binding reactions using a range of diferent photonic quantum states and found that quantum states ofer a great enhancement.Te use of quantum states enables the detection of smaller In the Krestchmann confguration setting, three phases of contact kinetics are generated from the angle shift.In the fgure, after a bufer is passed through the biosensor, a resonance dip is observed at a particular angle, and the position of this dip shifts as the analyte is pushed through the biosensor.Te SPR dip shift corresponds to a change in the index of refraction on the surface of the biosensor as binding reactions occur and can be plotted as a function of time.Te change in the angle-dip position can be plotted as a function of time, resulting in the sensorgram below the intensity versus angle plots.If no analyte is passed through the biosensor, then there will be no change in the dip position, and hence the baseline remains constant.Te change in the signal in the angle change versus time curve is due to binding reactions that occur as the analyte is introduced to the biosensor (this can be interpreted as a change in the concentration of the dielectric above the biosensor or a change in the index of refraction of the dielectric above the biosensor, both of which cause a signal change).Te concentration of the complex [C] and the response of the biosensor are linked by the index of refraction of the analyte n a � �� ϵ a √ of the region above the gold surface, whose change is induced by the sequence of analytes passing through the fow cell: (i) bufer and ligands (association and steady state) and (ii) bufer only (dissociation).Figures were drawn from [33].
concentrations of materials of interest in the same integration time or enables the detection of the same concentration of materials in a shorter integration time.
In this review, we look at new results from [39,60,61,66] where the researchers consider a two-mode Krestchmann sensing model for SPR.Te diference from a traditional single-mode model is that the two-mode setup has an extra path that has nothing done to it.Using this model, researchers showed the enhancement brought about by using quantum states of light in [39,60,61,66].Te enhancement is in the LoD measurement of the refractive index, and for sensorgram measurements, they showed that quantum states of light can improve the precision of measured binding parameters.Due to their ability to overcome the limitations of conventional optical sensors in terms of sensitivity, fexibility, and photostability, SPR-based biosensors have become a powerful tool for biosensing applications [33,67,68].Using quantum states of light can greatly improve these biosensors.As research continues to develop in SPR-based biosensors, diferent detection schemes are being developed which are being incorporated into a wide variety of biological and medical applications.Miniaturization and the improvement of precision measurements are some of the key driving forces behind research in plasmonic biosensors.A tool that can be used to address the miniaturization and development of PoC plasmonic-based devices is LSPR [33].We will not discuss this here, as it is beyond the scope of the article.
Although SPR-based plasmonic nanosensors are commercially available, the precision of current SPR instrumentation is limited when it comes to measuring light refectivity from the sensor.Tis limitation can lead to problems in various applications.For example, examining the movement and behavior of medications targeting various HIV-1 virus variants necessitates a level of accuracy beyond what is presently ofered by commercially accessible SPR-based nanosensors [33].In order to tackle this problem, scientists need to explore the underlying principles of the sensor's physics and take into account the foundational elements of optics, including resources rooted in quantum mechanics [34].
In [60], the LoD of the refractive index, Δn, is calculated for diferent quantum states using the following equation: where  M is a measurement of the diference in the photon number between the two modes of interest, that is, the signal mode and the reference mode.Te variance measurement 〈Δ  M〉 corresponds to the measurement noise in the model.|ψ in 〉 is the input state in the sensing model.Te lower the detection noise, the higher the precision (LoD) of our measured parameter; this is the refractive index, n, in this case.|d〈  M〉/dn| measures the sensitivity of the intensity diference to changes in the refractive index. M is defned as follows: Here,  a †  a and  b †  b are the photon number operators that count the number of photons of the states in mode a (signal) and in mode b (reference).According to Lee et al. [60], this reduces to Here, η a and η b are losses in modes a and b, respectively.Te variance in intensity-diference measurement Here, the parameter Q is a parameter called the Mandel Q parameter and is diferent for diferent states of light (it measures the departure of the occupation number distribution from Poissonian statistics and can thus be used to diferentiate between classical and quantum states of light) and σ is the noise reduction factor (NRF), which measures the noise reduction when diferent states of light are used.9: LoD for diferent quantum states with respect to the angle of incidence.Te purple line is two-mode product squeezed state (TPS is a quantum state but does not give a good LoD for this experiment), the orange line is an entangled quantum state called the NOON state (it is not a good quantum state for this type of sensing but is good for phase-sensing experiments), the green line is two-mode squeezed vacuum state (TMSV is a quantum state which performs just as well as the coherent state for this type of sensing), the black line is the twin coherent state (TMC, in this case, is the baseline for the best classical state performance), and the red line is two-mode Fock state (TMF is a quantum state which has the best LoD in this case, and it far outperforms all the other states).It is clear that the Fock state gives the better enhancement since the LoD is the lowest; thus, we can see that quantum states can outperform classical states such as the coherent state, justifying the need for quantum biosensors to enhance precision in SPR-based biosensors.Tis was taken from reference [40].International Journal of Optics Tere are many states of interest which we can compare to the coherent state, which we are taking as giving the best LoD achievable by a classical state (they are the minimum nonzero uncertainty states).As an example, when we consider the coherent and Fock states (calculations for these states and other states are shown in [60]), the 〈Δ  M〉 values are calculated as follows: for the coherent state and for the Fock state, Comparing the intensity-diference variance measurements for the coherent and Fock states, the measurement noise in the Fock state will be signifcantly lower than that of the coherent state whose intensity noise is bound by the shot-noise limit (we look at the coherent state as the classical limit state, that is, the classical state that will have the lowest intensity noise), but here we will focus only on the Fock state because, for intensity-based sensing, it gives the best in LoD for refractive index measurements.Clearly, quantum states such as Fock states can provide a better LoD of the refractive index.A ratio R is calculated, and it compares the noise of the classical case with the quantum and measures the enhancement in precision brought about by using the quantum states.R is defned as R � 〈Δ  M〉 Coherent /〈Δ  M〉 Fock .It follows that |α| 2 � N; hence, the above equation reduces to Setting η a � η b � η, we can reduce the expression even further and have A plot of the ratio which measures the enhancement in noise reduction, R, is shown in Figure 11(a) for changing the resonance angle and in Figure 11(b) for changing the refractive index.In the region of the resonance dip angle and the resonance dip of the refractive index, the enhancement ratio, R, is maximized.Te measured uncertainty of the Fock state becomes minimized compared to that of the coherent state, and from an experimental perspective, this would be the region where we would want to take measurements.Usually, in the Kretschmann setup, a laser serves as the light source, and within the quantum framework, laser light is aptly depicted as a coherent state [69].Tus, we can begin to see the limitations of the currently available SPR-based biosensors.Limitations in the LoD of the refractive index can be overcome by studying the quantum nature of light.
Quantum states exhibiting non-Poissonian photon number distributions or nonclassical correlations have demonstrated the ability to surpass the SNL [70,71].In this section, we consider a result from a two-mode system in which the LoD, Δn, is considered for a particular experiment for diferent quantum states.We see that the LoD is a function of the incident angle and is dependent on the quantum state of the light used.By using Fock states of light, we can greatly improve the precision of our biosensor.Mpofu et al. [61] extend this work by looking at the enhancement in the precision of the measured kinetic parameters when quantum states are used.Tey considered a range of loss conditions when modeling the kinetics simulation but showed that Fock states consistently give better precision compared to the classical coherent states.It has also been shown experimentally that quantum states of light (single photons) give better precision in the measured kinetic parameters [60,61].

SPR Experimental Setup
Te goal of this section is to describe experimentally how an SPR experiment can be setup in the lab by proving an example and also briefy describing how a classical SPR setup can be converted into a quantum SPR source by integrating a single photon source as input to the setup (see Figures 12(a) and 12(b)).Te paper by Pluchery et al.
[72] is a great reference for experimentally setting up an SPR experiment.Te SPR experiment using the Kretschmann confguration is a relatively simple experiment that can be easily conducted in a laboratory setting using optical components.From Figure 12(a) for the classical part, we see how diferent components can be arranged; an example laser source can be a 785 nm wavelength semiconductor diode unpolarized monochromatic 50 mW laser light source.To implement the quantum SPR experiment, this laser source will be replaced with a single photon source.A laser source can be used in conjunction with a microscope objective with a calibrated aperture, typically with a magnifcation of 20x, to function as a beam expander.Te characteristics of the microscope objective cause the laser beam to grow and change into a collimated beam as a result.To increase the intensity of the beam coming from the microscope objective, M coherent 〉 with 〈Δ  M fock 〉.We see enhancement when Fock states are used versus the coherent state.(a) In this scenario, the resonance angle is fxed, and the analyte's refractive index is detected.Te analyte has a refractive index of 1.39.(b) In this scenario, both the refractive index and the resonance angle of the analyte being sensed are set.Refractive index is 1.39, and the resonance angle is 73 °.Te coherent state is represented by the black line, and the Fock state by the red line.In the plot, it is assumed that there is no system loss in the setup, that is, η a � η b � 1. Photon number N � |α| 2 � 10.(c) Enhancement ratio R. (d) Case in which the index of refraction of the analyte that is detected is fxed while the resonance angle is changed.Te refractive index is 1.39 (b) in this scenario, the resonance angle of the analyte being sensed is fxed, and the index of refraction changes.Te resonance angle is 73 °.In the plots, it is assumed that there is no loss in the setup, that is, η a � η b � 1. Photon number, N � |α| 2 � 10.Tis was taken from referecne [40].International Journal of Optics light is focused from the objective towards a half-wave plate.From the microscope objective, the light is directed towards a half-wave plate which is used to maximize the intensity of the beam from the objective.From the half-wave plate, the light is directed towards a polarizing beam splitter (PBS) that lets through the horizontally polarized component of the beam (p) and refects out vertically polarized component of the beam (s).Another half-wave plate can be added after the PBS, which can be used to rotate the polarization between the s or p polarization; it is useful if the researcher is interested in investigating the s polarization.
From the second half-wave plate, the beam is guided towards a high-index equilateral prism (refractive index-� 1.601, for example).SPR glass slides (40 nm thick, goldcoated slides) are perched on the prism.In order to prevent an additional layer (air, for example, which would be a dielectric and introduce a variable refractive index) from being present between the surface of the prism and the gold slide, index-matching liquid is supplied.Tis keeps the refractive index approximately constant.Employing gold slides rather than applying a thin gold flm on the prism surface ofers multiple benefts.First, it enables the utilization of optimized gold thin flms, since the flm's thickness is crucial and governs the extent of coupling that takes place.Second, it facilitates the convenient substitution of the gold surface in case of damage.Rotation stages which are automated, for example, can be used to adjust the incident angle θ ext (Te mirrors and rotation angles are not shown in Figure 12(a)).
Te beam refects of the prism and is aimed towards a CCD detector that is connected to a computer for the classical case.For the quantum case, single-photon detectors are used, and intensity data are stored and processed on a computer system.Figures 12(a) and 12(b) which show the arrangement of components in the setup are known as the Kretschmann confguration, and the setup is shown in Figure 12(a).Over the range of angles where there is 100% internal refection from the prism, surface plasmons are stimulated; hence, the critical angle of the setup needs to be identifed during the calibration of the setup.It is important to note that in experiments, surface plasmons can only be excited in metal flms that are thin enough (with the specifc thickness determined by the type of metal used) and only with a specifc polarization of light, known as the p-polarization.

Using SPDC Source for Single Photon Plasmonic
Biosensing.In this section, we explore how quantum plasmonic sensing experiments can be carried out using quantum states of light in an SPR biosensing setup, achieved by merging the SPR and SPDC confgurations, as shown in Figure 12.In [66], a single-photon-based proof-of-principle experiment is presented that uses quantum sensing techniques to improve the precision of the kinetic parameters (that is, an experiment with the confguration as shown in Figure 12(b)).In their study, the researchers examined the interplay between BSA binding to gold through an electrostatic process.Tey utilized single photons (created by parametric down-conversion) to investigate the BSA-gold interaction within a plasmonic resonance sensor.Researchers reported sub-shot-noise level fuctuations in the sensor signal that improved precision for values in the kinetic parameters, highlighting the potential use of quantum states of light for sensing in biochemical research.While a coherent beam with many photons might ofer better noise reduction, the use of single photons is sufcient to demonstrate the relative enhancement ofered by quantum plasmonic sensing.Additionally, the use of single photons can help avoid unwanted efects that might arise from the collective behavior of large numbers of photons [60].In the quantum SPR experiment, Figure 12 Figure 12: (a) An experimental setup used for SPR biosensing.Before striking the prism with the gold slide, a 785 nm unpolarized laser beam passes through two half-wave plates, a PBS, and then excites the surface plasmons at the metal-air interface.A CCD camera is used to capture the prism's refected light.Tere is a fow cell on top of the gold layer that contains the analyte of interest.(b) A modifed version of the SPR setup; it is modifed in that the light source is no longer a laser but a single-photon source (SPDC source).Tis is an experimental example of single-photon plasmonic sensing from an SPDC source.In this case, the detectors will be single-photon detectors.Te detectors measure coincidences and counts of single photons.Tis was taken from reference [40].
International Journal of Optics measured and used in the analysis.Mpofu et al. [61] showed that an improvement in precision of up to 31.8% is possible in the values of the kinetic parameters.Tis can be attributed to the diminished noise associated with single-photon statistics, corroborating recent theoretical forecasts.Te study demonstrates that employing quantum light sources for sensing kinetic parameters is a viable option, ofering enhanced accuracy compared to traditional methods.Future work in this area could involve looking at sources such as the TMSD state and the TMSV state to enhance biosensing precision.

Conclusion and Outlook
In this paper, we review classical SPR-based biosensing and look at recent progress made in terms of the development of quantum SPR biosensors and the advantages that quantum biosensing brings.We reviewed fundamental details about SPR and looked at how to excite a surface plasmon as well as the conditions in which surface plasmons are excited.We looked at interaction kinetics and how they can be useful for biosensing and drug discovery applications, and we also detailed mathematical models for binding reactions.We compared classical SPR biosensing with quantum SPR-based biosensing and showed that using quantum states of light ofers a great enhancement in the precision of our biosensor, as they are able to go below the SNL of intensity noise detection.We also looked at an experimental implementation of the SPR biosensor and a quantum experimental implementation that involved a single-photon source.Here, we see that quantum technologies are the next stage in enhancing biosensor performance, as they are already showing that they are capable of going beyond the SNL and facilitating a lower LoD for biosensing applications.Quantum biosensing ofers great potential for better biosensing devices, and some of the information in this paper can ofer a baseline from which to begin the application of quantum resources in biosensing.Tere is a lot of future work that can be carried out in both the classical and quantum areas of research.In the quantum space, researchers can begin to look at the two-mode squeeze displaced states and how they can be used for biosensing enhancement.

Figure 1 :
Figure 1: Exciting surface plasmons in prism confguration setups.(a) Krestchmann confguration setup and (b) Otto confguration setup.Figures (a) and (b) show how surface plasmons are generated in a prism-based SPR setup.Tere are two common confgurations that can be used to excite surface plasmons, that is, the Krestchmann confguration shown in (a) and the Otto confguration shown in (b).Both confgurations are practically useful, but the Krestchmann confguration tends to be easier to confgure and is more commonly used.In these setups, light is sent at a varied angle of incidence towards a glass prism (BK7) that has a gold layer on its surface (or has a gold-coated slide sitting above it).Tis gold layer is only a few nanometers thick.At a certain angle, the incident light couples with oscillating electrons in the thin gold layer, resulting in an evanescent wave propagating on the gold surface, which is called the surface plasmon wave.When this surface plasmon wave forms, we observe a dip in the intensity of the refected wave, which is collected by the detector.Te surface plasmon wave is sensitive to changes in the dielectric properties of the analyte (with a refractive index n 2 ) on the surface of the gold.Tis sensitivity changing dielectric conditions on the surface of the gold is used for biosensing applications.

Figure 2 :
Figure2: Te dependence of the measured refectance on the incident angle of light and the analyte's refractive index.As the angle of incidence of light hitting the prism changes, the intensity of the refected beam R changes.Tis is because the incident beam begins to couple with the plasmons in the gold metal exciting them and generating a surface plasmon wave in the process; the energy of the beam is thus converted to this wave, and we start to get a drop in power.As the coupling becomes stronger and the evanescent waves become more intense, the refected beam becomes weaker until it reaches the minimum point, that is, the resonance dip.(a) SPR dip with resonance angle.Te same resonance dip efect can be observed when the incidence angle is kept constant and the refractive index changes instead.Tis is shown in (b) with an SPR dip plotted against the dielectric material's refractive index.(c) A 3D representation of the SPR dip is shown, considering the refractive index of the dielectric material and the angle of incidence.In this example, the coupling occurs at an incident angle of θ � 73 °and a dielectric material of refractive index n 2 � 1.39 and the refractive index of the prism n 1 � 1.52.Tis was taken from reference[40].

kFigure 3 :
Figure 3: How the dispersion relation informs whether or not coupling can occur.(a) Diagram illustrating the propagation relationships in various media.ω is the surface plasmon frequency.Te high-index medium is indicated by the letter "h" in the wave number k's subscript.Tis shows that if the electromagnetic wave does not propagate through some high refractive index medium frst before hitting the gold layer in the setup, then there would be no coupling as the propagation coefcients never meet; however, if the light goes through some high-index medium, then there will be coupling.Tis fgure was drawn from an example in reference[42].(b) Tis graphic displays the propagation relations in a Kretschmann prism confguration as well as the reasons surface plasmons at the metal-glass contact cannot be activated (coupling conditions are not satisfed).If a prism (high refractive index medium, such as a BK7 SiO 2 ) is incorporated into the setup and the light passes through it frst, surface plasmons are excited at the metal-air interface.Tis is demonstrated by the intersection of the blue prism dispersion line and the red surface plasmon dispersion curve at the metal/air contact.Surface plasmons at the metal/glass contact cannot be stimulated, according to the green curve.Tis fgure was drawn from an example fgure in reference[42].

Figure 6 :
Figure6: Tree phases of interaction kinetics.Te response unit may be related to a change in resonance angle, intensity, or refractive index, depending on what you are measuring.When the binding reaction begins, the signal change in the association region is observed (as receptor-ligand complexes form) up to the point where the binding surface is fully covered, at which point we observe a constant signal of the steady state; when we begin fushing out the ligand, we observe a sharp decline in the dissociation region, which is a result of rapid unbinding of the receptor-ligand complexes.
) below we can ft a model on the sensorgram from which we can get the binding parameters.When conducting experiments, the incident angle is typically fxed within a region around the SPR dip.Te left side of the dip is the most sensitive region, and the change in refectance is measured.Te angular drift or shift Δθ or the refectance/intensity drift or shift ΔR can be used interchangeably (i.e., you can go from the SPR angle change sensorgram to the intensity/transmission change sensorgram) in the sensorgram.

Figure
Figure9: LoD for diferent quantum states with respect to the angle of incidence.Te purple line is two-mode product squeezed state (TPS is a quantum state but does not give a good LoD for this experiment), the orange line is an entangled quantum state called the NOON state (it is not a good quantum state for this type of sensing but is good for phase-sensing experiments), the green line is two-mode squeezed vacuum state (TMSV is a quantum state which performs just as well as the coherent state for this type of sensing), the black line is the twin coherent state (TMC, in this case, is the baseline for the best classical state performance), and the red line is two-mode Fock state (TMF is a quantum state which has the best LoD in this case, and it far outperforms all the other states).It is clear that the Fock state gives the better enhancement since the LoD is the lowest; thus, we can see that quantum states can outperform classical states such as the coherent state, justifying the need for quantum biosensors to enhance precision in SPR-based biosensors.Tis was taken from reference[40].

Figure 10 :
Figure10: Two-mode biosensing setup.Model showing how a Krestchmann confguration setup can be integrated into a two-mode intensity-diference system.Te signal mode goes to the prism and probes the system whereas the reference mode goes directly to the detector.Te input states can be any state quantum or classical.Intensity-diference measurements have the added advantage of removing any background noise from the signal we are interested in.Tis was taken from reference[40].

Figure 11 :
Figure 11: (a) A comparison of 〈Δ M coherent 〉 with 〈Δ  M fock 〉.We see enhancement when Fock states are used versus the coherent state.(a) In this scenario, the resonance angle is fxed, and the analyte's refractive index is detected.Te analyte has a refractive index of 1.39.(b) In this scenario, both the refractive index and the resonance angle of the analyte being sensed are set.Refractive index is 1.39, and the resonance angle is 73 °.Te coherent state is represented by the black line, and the Fock state by the red line.In the plot, it is assumed that there is no system loss in the setup, that is, η a � η b � 1. Photon number N � |α| 2 � 10.(c) Enhancement ratio R. (d) Case in which the index of refraction of the analyte that is detected is fxed while the resonance angle is changed.Te refractive index is 1.39 (b) in this scenario, the resonance angle of the analyte being sensed is fxed, and the index of refraction changes.Te resonance angle is 73 °.In the plots, it is assumed that there is no loss in the setup, that is, η a � η b � 1. Photon number, N � |α| 2 � 10.Tis was taken from referecne[40].
(b) coincidences and single counts of idler arms (top) and signal (bottom) were