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Abstract
Recent studies on ultrafine particles (UFP), which are smaller than 100 nm, emphasized their hazardous potential to the human organism. They are comparable in size to typical nano-organisms such as viruses and can penetrate physiological barriers in a similar way. Currently, there are no low-cost and miniaturized detectors for UFP available. In our first experiments with an integrated evanescent field particle detector, we could already successfully detect single 200 nm polystyrene latex (PSL) spheres, although the implemented waveguide geometry was only rudimentary optimized with costly 3D simulations. We developed a fast and structured optimization model for waveguide geometry and operation wavelength of an integrated evanescent field particle detector in order to exploit its full potential for the detection of discrete analytes in the UFP size range. The optimization model is based on a modified formulation of Mie theory and its computational effort is reduced by a factor of 100 compared to 3D simulations. The optimization potential of the sensor response signal is demonstrated for several waveguide geometries that can be produced with established semiconductor fabrication technology at high production volumes and low costs. An optimized silicon nitride waveguide features sensor response signals that are about one order of magnitude higher compared to previous experiments, which pushes the limit of detection even further down to particle sizes below 100 nm. A small integrated evanescent field particle detector based on this optimized waveguide will be used for the first low-cost and miniaturized devices that can monitor the personal exposure to UFP.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical particle counters (OPC) that measure the scattered light of particles traversing a laser beam are the established technology in the field of low-cost particle sensors. Since there is room for interpretation, low-cost, in our understanding, corresponds to devices that are cheaper than $500 \$$ per piece. Commercially available OPCs feature a lower limit of detection in terms of particle size of about $300$ nm [1,2]. This limit is sufficient for the detection of particle classes like PM$_{2.5}$, which comprises airborne particles that are $2.5$ $\mu$m in size and smaller. Since PM$_{2.5}$ is usually quantified as mass density, the mass contribution of the particle fraction smaller than $300$ nm is negligible. However, recent studies investigating the toxicity of small particles indicate that in particular ultrafine particles (UFP; < $100$ nm) affect organs and the brain and cause serious diseases in the human body [3–5]. Due to their small size, they can penetrate the physiological barriers and travel within the circulatory systems of the host, similar to nano-organisms such as viruses, which are in the same size range: eg. influenza virus $\approx 100$ nm; severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) $\approx 60 - 140$ nm [6–8]. At the moment there are no low-cost sensors available, which are capable of measuring the personal exposure to UFP as a wearable device. A further reduction of the lower size limit of OPCs is matter of recent research activities, but it turned out that they are not in the low-cost regime anymore [9]. Laboratory-proven detection concepts for UFP, such as condensation particle counters (CPC), are too bulky and expensive for integration into a miniaturized device [10]. New particle detection methods are investigated in ongoing research activities in order to realize small-sized and low-cost devices for the detection of UFP. First experiments with particle detectors utilizing the evanescent field of a dielectric waveguide show their potential for the detection of small particles, especially in comparison with low-cost OPCs [11,12]. The experiments were carried out on small sized dielectric waveguide chips that can be produced with high-volume semiconductor fabrication technology at very low cost.
Sensors based on dielectric waveguide structures are well-established in the field of chemical and biological sensors. Such devices utilize the exponentially decaying field surrounding the waveguide, where it is referred to as evanescent [13,14]. The same measurement principle was applied for gas sensing in the past [15,16]. Promising results have been achieved with biosensors based on whispering gallery mode resonators (WGMR) with the detection of single nanoparticles and even molecules. The sensor effect is caused by perturbations in the evanescent field of resonating cylindrical or spherical symmetric structures, which changes the optical properties of the resonator [17]. However, high Q-factors of the resonator as well as a sophisticated read out mechanism to detect changes in the resonance are required, which makes fabrication and operation rather complex. In order to decrease the complexity and costs of the sensor, our concept is based on a power measurement through a non-resonant integrated waveguide, which has low requirements in terms propagation losses and can be fabricated with high-volume semiconductor technology. Nevertheless, our findings for the optimum waveguide geometry can also be applied to integrated WGMR, since the interaction volume of the evanescent field influences their performance as well.
Investigations of the general sensing performance of integrated evanescent field sensors by Parriaux & Dierauer showed that there is an analytical correlation between wavelength and optimum waveguide thickness for a slab waveguide, which corresponds to the configuration where the power transported in the superstrate (= top cladding) is a maximum [18]. This power converges to the same value for all wavelengths at optimum waveguide geometry and depends essentially just on the optical properties (eg. refractive index) of the materials used for the setup. These correlations apply for analytes that interact homogeneously with the waveguide surface, such as gases. However, if you deal with discrete analytes like solid particles, these findings have to be reconsidered. We investigated the interaction of small particles with the evanescent field of a dielectric waveguide in recent experiments [12]. The main interaction mechanism is extinction (= scattering $+$ absorption) of the evanescent field, which is incident on the particle. This results in a relative decrease of the transmitted light power through the waveguide. First sensor prototypes provided single counting sensitivity for polystyrene latex (PSL) spheres in the size range below $1$ $\mu$m down to $200$ nm in diameter. In order to quantify the sensor response signal theoretically, 3D simulations of the particle-waveguide interaction were used in the past [11,12,19]. Since simulations in the 3D domain are computationally very time-consuming, the optimization of the waveguide geometry was performed just with limited degrees of freedom (eg. waveguide width optimization with fixed height and wavelength). Nevertheless, a structured optimization with respect to wavelength and waveguide geometry is necessary to exploit the full potential of this kind of sensors in terms of single counting sensitivity and lower limit of detection regarding particle size.
We developed a fast and structured optimization model for an integrated evanescent field particle detector in order to increase its sensitivity for the detection of small discrete analytes such as UFP. The optimization model is based on a 2D simulation of the electromagnetic field. A modified version of the results from Mie theory for evanescent fields by Quinten et al. was applied to quantify the interaction between particle and light [20]. The results of this simplified model are in very good agreement with the full 3D simulation and provide the advantage of a reduction in computation time by a factor of 100. By that means an optimization of a strip waveguide was carried out in order to achieve the highest sensitivity for the detection of particles in the UFP size range. A linear correlation over a large parameter space between waveguide height, width and wavelength for maximum sensing performance could be determined, which is comparable to the findings of Parriaux & Dierauer for a slab waveguide and homogeneous media [18]. A silicon nitride strip waveguide geometry was identified that features sensor response signals, which are one order of magnitude higher compared to previous experiments. This enables the detection of particles smaller than 100 nm with a purely optical concept. Such optimized waveguide geometries will be implemented in first low-cost evanescent field particle detectors that are sensitive to discrete analytes down to the UFP size range.
2. Evanescent field of an integrated waveguide
A dielectric guiding structure consists of a waveguide core with higher refractive index than the cladding that surrounds it. Figure 1(a) shows a cross section through the guiding structure of a typical evanescent field sensor. The top cladding is either air or an aqueous environment, which contains the analytes that are the subjects of investigation. If light propagates through the waveguide core, the electromagnetic field does not vanish abruptly at the interfaces, but decays exponentially as a so-called evanescent field [21]. A considerable amount of the power is transported in this evanescent region, if the waveguide is operated in single mode. The evanescent field ratio (EFR) is the ratio of the power transported in the top cladding to the total power [15].It can be calculated with a surface integral of the time-averaged Poynting vector $\boldsymbol {S}$ over the top cladding normalized by the same integral over the total area. The mathematical formulation of EFR is
For the calculation of EFR, the electromagnetic field distribution needs to be known, which is dependent on the corresponding propagation mode. Propagation modes in dielectric waveguides are described with an effective refractive index
which is dependent on the refractive index $n_g$ of the waveguide and the propagation angle of the mode $\theta _g$ [22]. Common evanescent field sensors consist of a silicon dioxide layer as substrate with a higher refractive index $n_s$ than the top cladding $n_c$, which is air or water. Thus the condition for mode propagation is All modes that fulfill this condition are propagable. A general analytical solution for the electromagnetic field does not exist for strip waveguides [21]. Approximations or numerical FEM methods can be applied to compute the propagation modes, the corresponding $n_{eff}$ and the electromagnetic field distribution. The smaller the waveguide geometry gets, the less propagation modes are supported. By shrinking the waveguide in a size range of about the wavelength of the transmitted light, just the fundamental quasi-TE and quasi-TM mode are propagable (in the following only referred to as TE and TM mode). This condition is necessary to ensure single mode operation where a considerable amount of the total power is transmitted in the evanescent region. Hence, just waveguide geometries that feature single mode operation are considered in the following. For a single-mode waveguide with symmetrical cladding ($n_c = n_s$), the power transmitted in the evanescent region increases as the waveguide size and the effective index decreases, reaching its maximum at the lowest $n_{eff}$ that fulfills the propagation condition (Eq. (3)). However, common evanescent field sensors are asymmetrical ($n_c < n_s$), which results an EFR that reaches a maximum at a certain waveguide size and effective index, which is not the smallest possible according to the propagation condition [18]. Sensitivity optimization of evanescent field sensors for biochemical applications or gas sensing is connected with maximizing the EFR. However, a closer look at the interaction between particles and light reveals, that the optimization model needs to be reconsidered for an evanescent field particle detector, which is described in the following.
Fig. 1. (a) Cross section of a typical setup for an evanescent field sensor comprising a dielectric strip waveguide and (b) the z-component of time averaged Poynting vector $S_z$ of the fundamental TE mode in propagation direction.
3. Particle-light interaction in an evanescent field
A common formalism to describe the interaction of spherical particles with light is Mie theory [23]. Since its formulation in 1908 numerous extensions and modifications have been made to adapt the formalism to specific problems. The fundamental formulas that are based on plane wave excitation have to be modified in order to describe the interaction of spherical particles with an evanescent field [20,24,25].
A possible representation for quantifying the interaction between particles and light is the extinction efficiency
which is the sum of the scattering and the absorption efficiencies $Q_{sca}$ and $Q_{abs}$ [26]. Another possible representation is in terms of the extinction cross section $\sigma _{ext}$, which can be derived by multiplication of the extinction efficiency $Q_{ext}$ with the geometrical cross section of the particle. In the following considerations it was preferred to use efficiencies rather than cross sections. The extinction efficiency $Q_{ext}$ is a factor, which quantifies how much of the light power that is incident on the particle $P_{inc}$ is extincted If the particle is non-absorbing, the absorption efficiency is $0$ which leads to $Q_{ext} = Q_{sca}$ and $P_{ext} = P_{sca}$. Evanescent field particle sensors detect changes in the transmission through a waveguide that are caused by extinction of a single particle. The sensitivity of these sensors is ideal, if the extincted light $P_{ext}$ and therefore both factors of Eq. (5) are at maximum. The incident power $P_{inc}$ mainly depends on waveguide geometry and wavelength. Therefore, it is the crucial parameter for the optimization of the waveguide and is discussed in more detail in Section 3.2. The extinction efficiency $Q_{ext}$ on the contrary, mainly depends on the wavelength and particle itself. For the optimization model this efficiency needs to be considered as well and will be discussed in the following Section 3.1.3.1 Extinction efficiency $Q_{ext}$
The extinction efficiency $Q_{ext}$ is dependent on wavelength, particle size and the refractive indices of particle and medium. For particles that are much smaller than the wavelength, there are analytic expressions for the calculation of $Q_{ext}$ available (Rayleigh regime) [27]. If the wavelength is in the same range as the size of the particles, more sophisticated equations based on Bessel functions are used to calculate $Q_{ext}$. A lot of literature can be found for a plane-wave incident on the particle [26,27]. Quinten et al. derived equations for the extinction cross section $\sigma _{ext}$ that can be applied to an incident evanescent wave and a spherical particle [20]. We reformulated these equations in terms of the extinction efficiency $Q_{ext}$, which leads to the following to expressions for $Q_{ext}$ depending on the polarization of the incident light (TE or TM)
The Mie coefficients $a_n$ and $b_n$ are defined in [27] and the size factor with wave vector $k$ and particle radius $a$. The definition of $\Pi$ and $T$ contains Legendre polynomials and can be found in [20] as well as the definition of the normalization factor $N$. With the definition of the effective refractive index $n_{eff}$ (Eq. (2)), $N$ simplifies toFig. 2. Semi-logarithmic plot of the extinction efficiency $Q_{ext}$ as a function of the wavelength $\lambda$ for an exemplary non-absorbing ($Q_{ext}$ = $Q_{sca}$) spherical particle in air with a radius of $100$ nm and a refractive index of $1.58$. The evaluation was performed for three different types of incident waves ($n_{eff} = 1.54$ for the evanescent waves). Note: $Q_{ext}$ can become $>1$ according to the extinction paradox [26].
3.2 Incident power on the particle $P_{inc}$
The second parameter that influences the extincted power $P_{ext}$ (cf. Eq. (5)) is the power that is incident on the particle $P_{inc}$. Quinten et al. redefined the incident irradiance on the particle, because the time-averaged Poynting vector is not constant over the geometric cross section for evanescent excitation [20]. We transformed this definition to our case and found an expression for the incident power on the particle
as surface integral of the dot product of incident time-averaged Poynting vector $\boldsymbol {S}_{inc}$ of the propagation mode and surface normal $\boldsymbol {n}$ in propagation direction over the particle cross section [20]. The dot product in Eq. (11) simplifies to the z-component of the time-averaged Poynting vector $S_z$. It is defined by the electromagnetic field, which depends on waveguide geometry and wavelength (cf. Fig. 3).Fig. 3. Cross section of the z-component of the time averaged Poynting vector ($S_z$) of the fundamental TE mode in a strip waveguide, which depends on the height, width and wavelength. $S_z$ is used for the calculation of $P_{inc,rel}$ (Eq. (12)).
A closer look at Eqs. (11) and (1) shows that $P_{inc}$ and EFR are closely related. Based on these two equations we defined a new parameter, which quantifies the fraction of the totally transmitted power that is incident on the particle
3.3 Relative extincted power $P_{ext,rel}$
Inserting $P_{inc,rel}$ in Eq. (5) yields the relative extincted power
which corresponds to the extincted power of a single sphere $P_{ext}$ relative to the total transmitted power $P_{total}$ through the waveguide. This relative decrease of the transmitted power is the sensor response signal observed in experimental measurements and will be used for the optimization in terms of single particle detection sensitivity. In principle, every possible particle-waveguide configuration can be optimized with Eqs. (13), (12) and (4), if the material parameters are known and the electromagnetic fields are derived.3.4 Comparison with experimental data
In order to validate our derived model for the relative extincted power $P_{ext,rel}$, the results were compared with experimental data for a rectangular silicon nitride strip waveguide ($550$ x $160$ nm) and a wavelength of $660$ nm (TE polarization) from previous experiments with PSL spheres ($200$ and $500$ nm radius) [12]. The effective refractive index $n_{eff}$ and the time averaged Poynting vector $S_z$ were calculated in COMSOL Multiphysics and the relative incident power $P_{inc,rel}$ was calculated with Eq. (12) in MATLAB. The simulation environment is described in more detail in Section 4.1. The extinction efficiency $Q_{ext}$ was calculated with Eq. (6) using the corresponding $n_{eff}$ and refractive index $n_{PSL}=1.58$ for two PSL sphere radii of $100$ and $250$ nm. Finally, $P_{ext,rel}$ was derived with Eq. (13). The waveguide geometry underlies production tolerances of $\pm 10$ nm. Its height is the dominating parameter in the calculation of the relative incident power $P_{inc,rel}$ and the effective refractive index $n_{eff}$. Hence, a waveguide height of $160 \pm 10$ nm was taken into account for the calculation. Variations of the waveguide width have much less influence in the calculation and were not considered. Table 1 shows that our model is in good agreement with the experiments that we performed previously [12]. In the next step the model was applied for the optimization of the waveguide geometry in order to increase its sensitivity to UFP.
Table 1. Calculated values for the relative extinction efficiency $P_{ext,rel}$ of a silicon nitride waveguide (W $= 550$ nm, H $=160 \pm 10$ nm) 660 nm for PSL spheres of two different radii $a_{PSL}$ in comparison with experimental data from [12]
4. Optimization of a strip waveguide for the detection of UFP
The performance of the optimization model is demonstrated in the following for specific particle-waveguide configurations. Silicon (Si) and silicon nitride (Si$_3$N$_4$) strips deposited on a silicon dioxide (SiO$_2$) substrate are chosen for the waveguide, because the geometry can be fabricated with conventional semiconductor fabrication technology and the materials are established in the field. Chips containing this kind of waveguides can be produced at high production volumes and very low cost. Polystyrene latex (PSL) spheres in the size range of UFP were chosen as target particle. PSL spheres are commonly used for the characterization of particle sensors including the first proof of concept experiments with strip waveguides as evanescent particle detectors [11,12]. The electromagnetic fields that are necessary for the optimization model are derived numerically.
4.1 Determination of the optimum waveguide geometry in 2D
The numerical simulation was performed on a workstation PC with a 16 core processor and $128$ GB memory. The commercial FEM software COMSOL Multiphysics Version 5.4 including the wave optics module was applied to calculate the electromagnetic field of single-mode operated strip waveguides for all possible combinations of waveguide geometry and wavelength according to Table 2. Absorption of Silicon below 1200 nm and of silicon nitride below 400 nm determines the lower wavelength limit. The wavelength-dependent refractive indices for Si, Si$_3$N$_4$ (waveguide core) and SiO$_2$ (substrate) were taken from literature [28–30]. The refractive index of the top cladding was set to $1$, because in common evanescent field particle detectors it consists of air. The domain size was set to $4x4$ $\mu$m with the waveguide core located in the center. Propagation modes and corresponding electromagnetic fields were derived with a mode analysis and the time-averaged Poynting vector was computed for the fundamental TE and TM mode. Subsequently, the incident power on the particle $P_{inc,rel}$ was calculated with Eq. (12) for different particle sizes in the UFP range located at the center of the waveguide where the interaction is a maximum. The relative extincted power $P_{ext,rel}$ was finally calculated using Eq. (13) with $Q_{ext}$ and the maximum values of $P_{inc,rel}$ at optimum waveguide geometry for every wavelength. The corresponding $Q_{ext}$ was calculated for a PSL sphere with Eq. (2) using values for the refractive index from literature [31].
Table 2. Parameter space for the numerical calculation of the effective refractive index $n_{eff}$ and the time-averaged Poynting vector in propagation direction $S_z$ for a silicon (Si) and silicon nitride (Si$_3$N$_4$) strip waveguide in COMSOL Multiphysics.
4.2 Comparison with a 3D simulation of the full particle-waveguide interaction
In previous experimental work it was shown that the interaction between a particle and the dielectric waveguide of an evanescent field detector can be sufficiently described with a full numeric 3D model of particle and waveguide [12]. 3D FEM simulations are computationally very time-consuming and not suitable for a systematic optimization of waveguide geometry and operation wavelength over a large parameter space. The number of nodes in the mesh of the full 3D model with the same element size and, accordingly, also the computation time is by a factor of 100 larger than in the 2D model of the waveguide cross section. However, these simulations are a helpful tool, if the waveguide geometry and wavelength is fixed or just a few configurations are investigated. COMSOL Multiphysics was applied to perform 3D simulations for selected optimized configurations and the result for the extincted power $P_{ext,3D}$ was compared with $P_{ext,rel}$ from the optimization.
The geometry of the model is sketched in Fig. 4 with the in and out port of the waveguide marked as blue faces. For the simulation of the particle-waveguide interaction, a PSL sphere located at the center of the top surface of the waveguide was used in the model (refractive index from [31]). The material parameters of the other components were the same as for the simulation of the electromagnetic field in 2D and the domain size was set to $4x4x3 \mu$m with scattering boundary conditions of second order.
Fig. 4. Sketch of a strip waveguide model that was used for the numeric 3D simulation of the interaction with a PSL sphere, which results in a decrease of the light power at the out port (P$_{out})$ with respect to the in port (P$_{in})$.
A boundary mode analysis at the in and out port was performed in order to derive the propagation modes and the transmission through the waveguide was calculated without the particle $T_0 = 1$ ($P_{in} = P_{out} = P_0$) and with the particle present $T_{part} < 1$. Quantitatively this loss is defined as relative extincted power
which is equivalent to the relative extincted power $P_{ext,rel}$ from the optimization model (Eq. (13)).4.3 Results of the optimization
The optimization was performed for silicon and silicon nitride strip waveguides according to the definition in Section 4.1 $P_{inc,rel}$ was calculated with Eq. (12) and the time-averaged Poynting vector was derived by numerical calculation of the electromagnetic field for all possible combinations of wavelength, waveguide height and width according to Table 2. In order to evaluate the influence of the particle size, simulations were performed for particle radii $a$ of $50$ and $100$ nm. An exemplary result for the calculation of $P_{inc,rel}$ for a particle radius of $100$ nm on a silicon nitride strip waveguide is shown in Fig. 5 as heat maps over waveguide height and width for all simulated wavelengths and both mode polarizations (TE and TM) separately. The results for silicon strip waveguides and for other radii look qualitatively the same. Thus, the plots are not shown explicitly, but they were used for the quantitative evaluation below. The color scaling is set globally equal for all wavelengths. Configurations (H,W, $\lambda$) that did not fulfill the propagation condition (cf. Eq. (3)) or where a clear distinction of the mode (especially in the analysis of the TM mode) was not possible, lead to physically meaningless results and were neglected. As a consequence, certain areas of the plots in Fig. 5 are blank.
Fig. 5. Result for the calculation of $P_{inc,rel}$ for a silicon nitride strip waveguide and a particle radius of $100$ nm, for (a) TE polarization and (b) TM polarization. $P_{inc,rel}$ is the fraction of the transmitted power that interacts with the particle in %. The plots were derived with a numerical calculation of $S_z$ and Eq. (12) for different heights and widths of the waveguide and wavelength according to Table 2. There is a separate result for each wavelength starting from $400$ nm from the top-left to $800$ nm at the bottom-right. Blank areas are configurations where mode propagation or a clear distinction of the mode polarization is not possible. The global maximum for TE polarization $P_{inc,rel} = 5.9 \%$ at $W = 300$ nm, $H = 60$ nm and $\lambda = 400$ nm and for TM polarization $P_{inc,rel} = 1.9 \%$ for $W = 225$ nm, $H = 110$ nm and $\lambda = 400$ nm.
The evaluation of $P_{inc,rel}$ shows a local maximum for each simulated wavelength and propagation mode (cf. Fig. 5). The wavelength dependency of all maxima of $P_{inc,rel}$ and the corresponding optimized waveguide geometry (height and width) for particle radii $50$ and $100$ nm and both polarization modes is discussed in the following. Due to the larger integration area of the particle, it is evident that the values for radius $100$ nm are higher than for $50$ nm. Figure 6 shows the result for a silicon nitride and Fig. 7 for a silicon strip waveguide.
Fig. 6. Simulation results for a silicon nitride strip waveguide, particle radii of $50$ and $100$ nm and the fundamental TE and TM mode: (a) Maximum of $P_{inc,rel}$ at each simulated wavelength from Fig. 5, (b) the corresponding optimum waveguide height $H_{opt}$ and (c) width $W_{opt}$. A linear fit was performed to describe the correlation between wavelength and optimum waveguide geometry.
Fig. 7. Simulation results for a silicon strip waveguide, particle radii of $50$ and $100$ nm and the fundamental TE and TM mode: (a) Maximum of $P_{inc,rel}$ at each simulated wavelength (2D plots not shown explicitly - qualitatively similar to Fig. 5), (b) the corresponding optimum waveguide height $H_{opt}$ and (c) width $W_{opt}$. A linear fit was performed to describe the correlation between wavelength and optimum waveguide geometry.
An evaluation of the simulation results for the silicon nitride strip waveguide in Fig. 6(a) shows that $P_{inc,rel}$ at optimum waveguide geometry increases towards smaller wavelengths and is higher for TE mode compared to TM mode. Figures 6(b) and 6(c) show the wavelength dependency of the optimum waveguide height $H_{opt}$ and width $W_{opt}$ (at maximum $P_{inc,rel}$) for each mode and particle radius. Since $Q_{ext}$ is not explicitly dependent on the waveguide geometry, these values correspond to the optimum for a silicon nitride strip waveguide in terms of particle detection. Similar to the findings of Parriaux & Dierauer for a slab waveguide and homogeneous media it is possible to fit a linear correlation between the optimum waveguide geometry (height and width) and wavelength in all cases [18]. The optimum values for $50$ nm and $100$ nm deviate slightly from each other. A thinner waveguide provides an evanescent field with larger penetration depth, which is beneficial for larger particles.
The results for a silicon strip waveguide in Fig. 7 are similar to the results for silicon nitride with a less pronounced wavelength dependency. The optimum waveguide height of TE and TM mode differs significantly, whereas the target particle size has minor influence on the optimum geometry. For the TE mode, silicon features a higher incident field on the particle ($P_{inc,rel}$) in comparison with silicon nitride with a maximum at $1200$ nm wavelength. Below this limit the absorption of silicon increases considerably and wave propagation is not possible anymore.
The sensor signal is determined by a decrease of the transmission through the waveguide caused by extinction of single particles. In order to evaluate the sensing performance of the optimum waveguide geometries derived for each wavelength in Figs. 6 and 7, the corresponding maximum relative extincted power $P_{ext,rel}$ (Eq. (13)) was calculated. The result is plotted in Fig. 8 as function of the wavelength for PSL sphere radii of $50$ and $100$ nm, TE and TM polarization.
Fig. 8. Semi-logarithmic plot of the maximum $P_{ext,rel}$ at each simulated wavelength for (a) a silicon nitride strip waveguide and (b) a silicon strip waveguide. $P_{ext,rel}$ is calculated for each optimized geometry (cf. Figs. 6 and 7) with $P_{inc}$ and corresponding $Q_{ext}$ (Eq. (13)) and describes the power extincted by a single PSL sphere relative to the transmitted power. PSL spheres with radius $50$ and $100$ nm and the fundamental TE and TM modes were taken into account. The results are compared with costly 3D simulations ($P_{ext,3D}$) of the particle-waveguide interaction (green and cyan crosses).
A 3D simulation of the full particle-waveguide interaction was performed for selected optimized geometries and the result for $P_{ext,3D}$ was compared with the outcome from the optimization model for $P_{ext,rel}$. Figure 8 shows that both results are in good agreement. The slightly lower values of the 3D simulations can be explained by re-coupling of scattered light into the waveguide, which is not taken into account by the optimization model for $P_{ext}$. However, this does not affect the result for the optimum geometry of the waveguide in Figs. 6 and 7, making the presented model an excellent tool for the fast optimization of the geometry of an integrated evanescent field particle sensor. The evaluation of $P_{ext,rel}$ for a silicon nitride strip waveguide shows that the values for TM mode are slightly enhanced compared to the TE mode (with an exception), although the incident power $P_{inc}$ (Fig. 6(a)) is much lower for TM mode. The difference is compensated by a higher extinction efficiency $Q_{ext}$ for the TM mode. Just if the ratio between wavelength and particle radius becomes smaller than $5$ the TE mode is superior. This is also induced by the extinction efficiency $Q_{ext}$ (cf. Figure 2). In case of a silicon strip waveguide, the TE mode is superior for all investigated cases, because the much lower incident field $P_{inc,rel}$ (Fig. 7(a)) for the TM mode cannot be compensated by its higher extinction efficiency $Q_{ext}$. The evaluation of the overall maximum of the relative extincted power $P_{ext,rel}$ shows that the wavelength has much larger influence than the modes. Especially the extinction efficiency $Q_{ext}$ is strongly increased towards smaller wavelengths, which results in a preference of silicon nitride waveguides, since they support shorter wavelengths in comparison to silicon.
Compared to experiments with a silicon nitride waveguide ($W = 550$ nm, $H = 160$ nm) and $\lambda = 660$ nm wavelength, where PSL with $100$ nm radius were successfully detected ($P_{ext,rel} = 0.29 \pm 0.05 \%$ [12]), the maximum relative extincted power in optimum configuration (silicon nitride, $W = 300$ nm, $H = 60$ nm, $\lambda = 400$ nm) would be increased by a factor of about $10$ ($P_{ext,rel} = 2.7 \%$). The optimum configuration for radius $a = 50$ nm provides a relative extincted power of $P_{ext,rel} = 0.2$ - $0.3 \%$ depending on the polarization. Previous experiments showed that sensor response signals for $P_{ext,rel} > 0.1 \%$ can be measured reproducibly. Experiments that aim to detect single particles down to the UFP size range with an evanescent field particle detector based on optimized silicon nitride strip waveguides are planned for the future. A corresponding chip layout is already in the design phase.
5. Conclusion
Evanescent field particle detectors based on dielectric waveguides were successfully used for the detection of small particles down to $200$ nm. The presented model allows an optimization of waveguide geometry and wavelength over a large parameter space in terms of sensor response (relative extincted power $P_{ext,rel}$) in order to fully exploit the potential of this measurement method for the detection of single particles down to the size range of UFP (< $100$ nm). An extension of the Mie theory for evanescent fields is used to calculate the extinction efficiency $Q_{ext}$, which shows different quantitative results for TE and TM mode. The incident power on the particle $P_{inc,rel}$ was determined with a numerical calculation of the electromagnetic field and the time-averaged Poynting vector over the waveguide cross section for the corresponding propagation mode. This calculation was performed in the 2D domain, making it 100 times faster than the previously applied 3D simulations of the full particle-waveguide interaction. The sensor signal is described by the relative extincted power $P_{ext,rel}$, which depends on $Q_{ext}$ and $P_{inc,rel}$. The derived model for $P_{ext,rel}$ describes the particle-waveguide interaction adequately and shows good agreement with previously performed experiments. Thus it was applied for the optimization of silicon and silicon nitride strip waveguides in order to evaluate their potential for the application as an integrated evanescent field particle detector for UFP. The optimization model shows that there is an optimum waveguide height and width for every waveguide material, wavelength and polarization. The wavelength dependency of the optimum waveguide height and width is linear, which is comparable with the findings from Parriaux & Dierauer for slab waveguides and homogeneous media [18]. A closer look on the maximum relative incident power $P_{inc,rel}$ shows a preference for the TE mode and shorter wavelengths. The extinction efficiency $Q_{ext}$ is enhanced towards shorter wavelengths as well, but it is higher for the TM mode compared to the TE mode. In the final result for the maximum relative extincted power $P_{ext,rel}$, the dependencies on the polarization mode of $P_{inc,rel}$ and $Q_{ext}$ will approximately compensate, whereas their wavelength dependencies amplify each other, so that silicon nitride is superior to silicon strip waveguides with the largest value at $400$ nm wavelength. Below $400$ nm wavelength light propagation is prevented because silicon nitride absorbs too much. The optimized silicon nitride waveguide provides sensor response signals $P_{ext,rel}$ for single particles with $50$ nm radius between $0.2$ - $0.3 \%$ (depending on the polarization), enabling their detection with this purely optical concept. In the optimized configuration, the sensor signal is expected to be at least one order of magnitude higher compared to the waveguide used for first proof of concept experiments. An integrated evanescent field detector based on this optimized silicon nitride waveguide can be produced with established semiconductor fabrication technology at low costs and will be applied in first miniaturized devices, which are capable of detecting single particles down to the UFP range.
Funding
Österreichische Forschungsförderungsgesellschaft (864341).
Acknowledgments
This publication was supported by the TU Graz Open Acess Publishing Fund. The authors gratefully acknowledge our collaborators from ams AG (V. Sidorov, G. Meinhardt, A. Singulani, J. Pulko, M. Sagmeister) and Silicon Austria Labs GmbH (J. Pribosek, A. Tortschanoff) for the fruitful discussions in the research consortium.
Disclosures
The authors declare no conflicts of interest.
References
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