Effect of Scattering Loss on Optimization of Waveguide Enhanced Raman Spectroscopy

Waveguide enhanced Raman spectroscopy (WERS) is a promising sensing technology requiring accurate waveguide optimization to increase the pump/signal surface intensity. Traditionally, WERS has been focused on single-mode operation, which results in stringent waveguide parameter control and increased propagation losses. In this work, we have studied theoretically and experimentally the impact of planar waveguide thickness on surface scattering losses and the waveguide propagation losses. In our study, we consider radiation from a Raman-emitting dipole on a waveguide can be captured back into the waveguide in polarization and spatial modes different from the pump mode, provided the waveguide can support them. In the case of randomly-distributed dipoles, we consider the Raman gain coefficients corresponding to all possible combinations of pump and signal polarization and spatial modes. We have introduced a new generalized FOM to optimize planar waveguide-based WERS sensors, which is a promising candidate for flexible disposable biomedical sensing, under multimode excitation/collection operation and waveguide-thickness- and mode-dependent propagation losses. The FOM is shown to increase with the mode order as a result of the combination of substantial reduction in propagation loss, increased number of collecting modes, longer optimum sensing lengths, and this occurs despite the concomitant surface intensity reduction. This implies that in the case of randomly distributed dipoles, the single-mode pump excitation is not strictly required, and multimode pump excitation will give superior conversion efficiencies.

of maturity, but commercializing SERS faces reproducibility and robustness challenges due to its reliance on complex nanoscale noble metal structures.With the development of integrated photonic circuits and demand for rapid sensing, dielectric waveguide based WERS, which in contrast to SERS builds up with length and shows a "distributed" enhancement, is attracting renewed attention due to its reproducibility, robustness, manufacturability, and ease of integration into a photonic chip.In planar WERS, one whole side of a waveguide to the target analyte, which is impossible in a hollow-core or liquid-core fiber.This allows analytes to quickly get in and out of the evanescent field, enables reusable sensors, and enables careful, precise design of this evanescent field interaction.
Considerable work has been done to optimize waveguides for WERS, including optimizing the waveguide core material and waveguide structure.Experimental comparison of core materials [6] included four candidate waveguide platforms (Al 2 O 3 , Si 3 N 4 , Ta 2 O 5 and TiO 2 ), and demonstrated that Ta 2 O 5 shows a good compromise in optical properties for WERS.The Ta 2 O 5 core typically has a refractive index of 2.1, which is smaller than materials such as Si or TiO 2 , resulting in a lower Raman signal conversion efficiency, but the loss is significantly lower (measured in narrow strip waveguides) [6].Compared to lower index materials, such as Al 2 O 3 , the signal conversion efficiency is much higher, although the loss of Al 2 O 3 waveguides was half that of the Ta 2 O 5 waveguides [6].Si 3 N 4 has slightly lower index, so the loss and background noise is slightly lower, however, the Raman signal conversion efficiency is half that of the Ta 2 O 5 waveguides [6].Planar (slab) waveguides, as an elementary form, have been investigated for power budget analysis [7] and comparison of collection configurations [8].Various potential waveguide structures compared using various figures-of-merit (FOMs) for WERS have been investigated.Strip waveguides [9] have been studied due to their stronger confinement of field from two side-walls compared to the open slab waveguide, which benefits from more confined pump and better collection efficiency in the sense of guiding the collected light to the end of the chip and improved potential for device integration.Further, strip waveguides that are etched to form slot waveguides [10], [11] and subwavelength grating waveguides [12] have been studied to access more of the evanescent field to excite the analyte.The numerical optimization of the corresponding FOMs of these state-of-the-art waveguides mainly focus on factors including the efficiency of exciting or capturing the Raman signal, or the pump/signal conversion efficiency per unit length [13], [14].
However, with a few exceptions (see [14], [15]) scattering loss, which is a fundamental factor that limits WERS overall conversion efficiency has received limited attention.Usually, it has been either neglected or oversimplified as a constant, independent of the waveguide parameters, in current conventional FOMs.As a strong evanescent field exposed at the surface of the waveguide core is expected in WERS, scattering from the roughness of the surface is unavoidable.Scattering loss is not independent of the waveguide geometry, as surface modal field varies with geometry (i.e., thickness for planar (slab) waveguides), and a full optimization should take the variation of scattering loss as a function of waveguide core thickness into consideration.Kita et al. [14], [15] have considered the impact of slot waveguide parameters on side-wall induced scattering losses and used it to optimize the proposed FOM.This study, however, is limited to single-mode slot waveguide operation.Even optical fibers, which exhibit minimal losses reaching the limits of Rayleigh scattering, exhibit increased losses when single mode operation and strong evanescent field is needed for sensing or other similar purposes [16].A ∼dB/cm scattering loss could result from a nanometer surface fluctuation in this case [17].
The mechanism of scattering loss from a waveguide has attracted attention since the late 1960s, resulting in four main approaches to estimate it.The simplest one is based on ray optics, and a simple analytical expression was deduced by Tien [18], suitable mainly for multimode waveguides.Marcuse et al. in the 1960s studied the loss caused by imperfections of the surface by using the mode conversion theory for cylindrical waveguides [19] and slab waveguides [20].Marcuse's model was based on the weakly-guiding approximation, which makes it unsuitable for WERS devices using high index contrast waveguides.The volume-current method [19], [20] was introduced in the 1970s as a practical way to calculate scattering losses in waveguides [21], [22].Kita et al. [14], [15] adopted this method in their numerical model for calculating FOMs for different waveguides for sensing, and it is the only work so far that takes into account scattering loss into the waveguide optimization and WERS FOM.Payne-Lacey's (PL) theoretical analysis [23], [24], based on solving the wave equation via Green's function at a random interface, shows that the scattering loss is a function of the waveguide thickness and spatial frequency (characterized by correlation length) of the rough surface.Moreover, Schmid's et al recent theoretical analysis and experimental work, which is developed from PL's theoretical work, has shown that surface-roughness scattering also interferes with reflections from other interfaces of the waveguide and significantly modifies the resultant scattering loss [25].In other words, conventionally optimized waveguide parameters for maximum evanescent field strength at the interfaces might not be optimal for WERS due to the concomitant increased scattering loss.
Another point that has been rarely addressed is the full impact of the multimode signal collection and cross-polarization excitation (i.e., signal and pump in different polarizations, which is discussed by Dhakal et al [26]).Conventionally, the waveguide is designed for the fundamental mode because of its strong surface modal field.However, due to additional requirement for minimized scattering loss, the waveguide parameters may become optimum for WERS when the waveguide is multimode and Raman is collected in multiple modes supported by the waveguide.Thus, the signal could be collected by the waveguide in a higher-order mode and different polarizations as the Raman signal is emitted by randomly oriented dipoles.These additional considerations would further impact the optimum waveguide length (or "saturation" length, in the case of backscattered collection), which is simply the reciprocal of the loss coefficient for single mode excitation and collection.
In this work, we expand previous approaches and extend the WERS waveguide design into the multimode regime by considering the combined effects of reduced surface intensities, reduced scattering losses (increased interaction lengths) and increased number of pump excitation/signal collection spatial/polarization modes.In Section II-A, we expand the random dipole-to-waveguide coupling into multimode polarizationresolved operation.In Section II-B, we consider multimode pump-excitation/signal-collection and generalize previously derived FOMs, applicable to single-mode operation, for application in the multimode regime.In Section III-A, we extend previously derived modal field, coupling efficiency and scattering loss calculations into the multimode regime.The model on scattering loss is extended with the transfer matrix method so that the multilayer interference effect on scattering loss is considered.In Section III-B we show detailed comparison between planar waveguide scattering loss theoretical calculations and experimental measurements.In Sections III-C and III-D, the traditional singlemode-pumping/singlemode-collection FOM calculations are used to compare with the new singlemode-pumping/ multimode-collection FOM.In addition, we have also calculated and compared the signal-background-ratio.In Section IV, we summarize and present the final conclusions.

II. THEORETICAL MODEL
In Fig. 1, a typical three-layer WERS waveguide system is shown.The pump is coupled into the waveguide from a coupler and excites the Raman signal from the analyte, which is modeled as an oscillating dipole, on the top of the core of the waveguide excited by the evanescent field, while the guided power is scattered by the rough core-cladding interface.
The randomly oriented excited molecule radiates the Raman signal incoherently and isotropically.Therefore, the signal can be collected in TE and TM polarization regardless of the pumping polarization from the top of the waveguide (free-space collection) and the ends of the waveguide.However, it has been shown that much more power can be collected from the ends of the waveguide [8], and hence, the top-collection is not considered further here.The forward-/back-collected signal is the integration of the signal excited and captured by the waveguide with the supported spatial mode along the waveguide from z = 0 to z 0 .Following Ref. [14], the Raman gain coefficient η j for the jth mode, which indicates the efficiency of converting pump to guided jth mode Raman signal per unit length, is the product of the efficiency of dipole excitation by the pump and the coupling efficiency of the power radiated from the dipole into the guided mode.

A. Dipole-to-Waveguide Coupling Efficiency; Pump/Signal Polarization Dependence
The coupling efficiency into the jth waveguide mode of the emission from a dipole on a layered structure is calculated in terms of the waveguide mode power (Pj) can be calculated with the theorem on dipole radiation from a layered structure.The total emission power (P ) normalized to the free space emission power (P 0 ), in general, is given as [26], [27], [28]: where μ and k 3 = n 3 k 0 are the dipole moment and wavenumber in the superstrate, while the subscripts y and ρ denote the vertical and horizontal (x-z plane) components, respectively.r T M and r T E are the plane-wave reflectivities for the p (TM) and s (TE) polarizations of the whole structure, including the waveguide and any other interface(s) in the substrate (see Fig. 1).r 0 is the distance between the dipole and waveguide core.For monolayer detection, the analyte is deposited on the waveguide core, so r 0 = 0. TM/TE waveguide modes are given by the poles of the r T M /r T E reflectivities and the integration is calculated along the infinitesimal semicircle in the complex plane surrounding the corresponding pole [26], [27] to calculate the waveguide mode power (P j ).Calculation of P j is detailed in Appendix C after introducing the free space dipole radiation in Appendix A and multilayer reflectivity/transmissivity in Appendix B. It is noteworthy that waveguide modes and therefore the corresponding poles of the r T M / r T E plane-wave reflectivities are considerably different in the few-mode high index-contrast slab waveguide, so there will be negligible crosstalk and so no significant additional power contributions between modes.The horizontal dipole moment (μ ρ ) can be further analyzed into two orthogonal components (μ x and μ z ).Randomly oriented dipoles can also be divided into three equally-populated x-, y-, z-oriented subgroups and the horizontal and vertical dipole moment contributions in (1) are usually taken as μ ρ /μ = 2/3 and μ y /μ = 1/3 [26], [27].Strictly speaking, this applies when the dipole excitation is isotropic with respect to the pump field vector (i.e., it depends only on the pump intensity Ý|E p | 2 ), as is the case when considering gain effects in doped planar waveguides [29].However, in WERS the x-, y-, z-oriented subgroups will be predominantly excited by the x-, y-, z-components of the pump field, respectively, and, therefore, we will explicitly use μ x /μ = μ y /μ = μ z /μ = 1/3, along with the appropriate pump field components, in the subsequent analysis.Equation (1) gives the total collected signal power including TE and TM polarized modes.The collected signal partition into TE and TM waveguide modes is given by the terms containing explicitly the r T E and r T M coefficients.The free-space radiation power normalized to pump power P p from a dipole excited by the evanescent field of a planar waveguide is [30] where σ R is the scattering cross-section and is given a representative value of 10 −30 cm 2 in the following calculations, while 2 is a constant specified by Long [31], n g is the group index and Ãeff is the effective mode area defined in [30].The Raman gain coefficient, η j for the jth signal mode is calculated by multiplying the signal coupling efficiency (1) after substituting n g and Ãeff (detailed in appendix) and excitation efficiency (2), namely: From ( 1)-(3), we can derive the following polarization resolved dipole excitation and signal collection cases.|E p | 2 is the power normalized squared electric field.The TE polarization x /μ 2 = 1/3 and (1) and (3) reduce to: where l denotes the infinitesimal contour of integration in the complex k ρ plane around the r TE and r TM poles of jth mode (see Appendix C).It is apparent that TE polarized pump excites the x-oriented dipole subgroup and contributes to both TE and TM signal coupling.
2) TM Pump: z /μ 2 = 1/3, μ 2 y /μ 2 = 1/3 and from (1) and (3) we obtain: It is evident that TM polarized pump excites both the z-and y-oriented dipole subgroups (with strengths proportional to the |E z | 2 and |E y | 2 pump components, respectively) and contributes to both TE and TM signal coupling.It is also shown that under TM polarized pump excitation both TE and TM signal modes are collected.It should be mentioned that in previous studies [26], [27] the full pump field |E p | 2 was considered with and 1) and (3).

B. Raman Conversion Efficiency -Effect of Scattering Propagation Loss
Due to the scattering loss, the collected signal at the waveguide ends is a function of both the loss coefficient and waveguide length, and for the jth− mode collected Raman signal, P sj , can be described by the following differential equations, for the forward collection configuration (6(a)) and the back collection configuration (6(b)), dP s,j dz = η j P p (z)e −α s,j (z 0 −z) (a) where z 0 is the total sensing length, and α s,j is the loss coefficient for a signal collected in the jth mode.The pump decays exponentially along the waveguide with loss coefficient α p , and is expressed as P p (z) = P p (0)e −α p z .The jth mode collection efficiency is obtained from (6) as Equation ( 7) determines the ability of the waveguide to convert the pump power into Raman signal in the jth mode, where f f (b), j is the integrated function regarding the loss of signal and pump along the waveguide for forward (indicated by subscript f) and backward collection (indicated by subscript b), and is expressed as As mentioned in the introduction, the geometry of the waveguide (thickness of the planar waveguide core) changes the scattering loss as well, through two mechanisms.Firstly, the core thickness determines the strength of the modal field at the core-cladding interface, and a stronger surface modal field excites stronger scattering.Waveguide scattering loss is usually calculated by Payne-Lacey's model [24].This model has been modified by Schmid et al. [25] to include the contributions of the interference effects inside the two core interfaces that change the scattering significantly.The interference effect has been proven to have significant impact on the overall field distribution and, therefore, the scattering loss [25].This is a second mechanism affecting scattering loss, also dependent on the core thickness.The modified scattering loss coefficients are given by [25], where E x 0 (E y 0 , E z 0 ) is the normalized TE (TM) mode field at the n 1 /n 3 interface, n jeff is the effective refractive index of the corresponding waveguide jth mode.r T E/T M and t T E/T M are the total reflectivity and transmittance for TE/TM waves reflected and transmitted through the whole structure of the waveguide.b = ∫ ∞ −∞ n 2 (y)|E y (y)| 2 dy is the normalization constant.R is the correlation function of the roughness which takes the widely-accepted auto-correlation form [24] R (k) = 2σ 2 L/(1 + L 2 k 2 ), characterized by the root-meansquare height variation of the fluctuating surface σ, and L is the rough-surface correlation length.It can be inferred that the core thickness is the decisive factor in the scattering loss for waveguide with a certain surface roughness.The surface modal field, (E x 0 for TE and E y 0 , E z 0 for TM), is a function of core thickness, which can be obtained by solving the dispersion equation (c.f.Appendix D for our implementation).The total transmittance and reflectivity are also functions of the core thickness, which is detailed in Appendix B. The expressions (10) and ( 11) are originally for a waveguide core with two rough interfaces, while, for the planar waveguide the substrate interface is normally much smoother and therefore contributes negligible loss.Therefore, the loss coefficients used in the following study are half the ones given by ( 10) and (11).For given roughness parameters, changes in the waveguide parameters (core thickness and/or refractive indices) will impact the scattering loss through the variation of the n 1 \n 3 interface field strength, as well as the changes in r T E/T M and t T E/T M .

C. Raman Conversion Efficiency -Figure of Merit (FOM)
Normally, optimization focuses on strengthening the surface field for a higher Raman conversion coefficient, while the impact on the scattering loss is either ignored or simply treated as a constant, independent of the waveguide parameters.In this case, the fundamental mode is always preferred due to the strongest modal field, and the loss for pump and signal are assumed to be the same ( α s = α p ).By taking the conventional optimal length, z 0 = 1/α p [14], into (8), for forward collection, the function f f takes its max value, an appropriate FOM for forward collection is given by [14] The constant e is omitted in reference [14] for simplicity, but here it is kept because it allows direct quantitative comparison of forward and backward configurations.In order to calculate the FOM for bulk analyte, we assume the dipole is evenly distributed with a density ρ R along y direction.For backward collection, the signal increases with waveguide length monotonically to its extreme value P sj (z 0 → ∞) = η j P p (0)/(a p + α s ), which is indicated by (9).The corresponding backward collection FOM b is [6], [15] This implies that all else being equal, backward collection is always ∼36% better than forward collection.
As can be inferred from (10) and (11), for fixed surface roughness parameters and refractive indices, the loss coefficient is a function of the thickness of the waveguide h.It is therefore expected that the thickness that maximizes the surface mode field does not necessarily maximize the overall WERS conversion efficiency.For example, due to reduced scattering loss, the optimum thickness could be greater than the conventional FOM indicates.Furthermore, a thicker waveguide may guide more modes and higher order modes may be used to increase the overall Raman collection.The captured signal and the pump in the core can be guided with different modes, in which case the loss coefficient will no longer be the same.Therefore, here the new FOMs, which is denoted as nFOM in contrast with conventional FOM, for signal collected in each individual (jth) mode are expressed using a more generalized formula including the ratio between the collected signal power and pump power, following (7), The signal collected at the ends of the waveguide is incoherently excited and captured by all the modes and polarizations that the waveguide supports.In this case, generalizing the singlemode excitation/collection nFOM mentioned above, we introduce a FOM applicable to single-mode excitation/multimode collection WERS arrangements, namely: In (15), nFOM m f is the total FOM for forward signal collection (subscript f), obtained under mth (TE or TM) mode pumping, by summing up the individual figures of merit FOM m fjk corresponding to individual jth signal mode with k (TE or TM) polarization pumped by the mth mode.η m jk is the simplified Raman coefficient given by ( 4) and ( 5) for the mth pump mode and the jth signal mode with k (TE or TM) polarization.f m fjk is given in ( 8) and corresponds to the propagation factor with the propagation loss coefficient of the mth pump mode and the jth signal mode with k (TE or TM) polarization given by ( 10) and (11).A similar formula applies for the case of backward collection (with f replaced with b).The new generalized nFOM incorporates geometry dependent loss and multi-interface interference effects and includes collection in multiple modes and both polarizations for polarized waveguide excitation.
This study covers both monolayer and bulk analyte detection, where the analyte fills the entire cladding.For monolayer, the value of density ρ R along y direction is apparently 1, while for bulk analyte with a typical molecule size of ∼nm, the value 10 9 is taken into our calculation.To calculate the nFOM for bulk detection, we integrate the excitation efficiency η over the cladding area.This allows for a comprehensive evaluation of WERS performance for different analyte detection scenarios.Typically, the surface of the slab waveguide in biomedical sensing is chemically modified for the target molecules so that more target molecules that conducted by lateral flow or liquid drop can be attached onto the core-cladding interface as a monolayer where the modal field is strongest in the free space.Therefore, monolayer nFOM is our major focus.nFOM calculations are usually for monolayer detection, unless otherwise specified.
To demonstrate the improvement of the waveguide design with our nFOM, the relative FOM (rFOM) is defined as follows, the ratio between single-mode/total nFOM and conventional FOM of TE 0 mode.Following the conventional design, the conventional FOM is calculated using ( 12) and ( 13) with the waveguide thickness that maximizes the surface modal field of the pump.

III. RESULTS AND DISCUSSION
In this section, modal field, coupling efficiency and scattering loss calculations and compare with experimental waveguide loss measurements is carried out.The nFOM calculations for polarization resolved single-mode pumping and single-mode/multimode signal collection is also performed.The calculations presented here correspond to a planar waveguide with Ta 2 O 5 core deposited on a SiO 2 substrate.The cladding is assumed to be water.The refractive index for cladding, core, and substrate used throughout this section are 1.333, 2.12, and 1.4535 at a wavelength of 785 nm, respectively.The pumping wavelength is set to be 785 nm, which is a balanced choice in terms of sensitivity and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.fluorescence background noise [4].First, the dispersion equation is solved to obtain the electric field of the waveguide modes, normalized to carry 1W/m power.Implementation of our model and relevant parameters are shown in Appendix D.

A. Modal Field, Coupling Efficiency and Scattering Loss Calculations
The surface electric field for various waveguide modes as a function of core thickness is shown in Fig. 2(a).TM modes show the highest core/cladding surface field, making them preferable to the TE modes for WERS.In Fig. 2(b), the efficiency for coupling the radiation from a horizontal dipole to guided modes is shown, assuming the dipole is radiating at the same wavelength as the pump (785 nm).Fig. 2(c) shows the case for a vertical dipole.It is shown that, in this case, there are no TE signal modes excited or collected.We also observe that for TM 0 pumping the collection efficiency can be >1 due to the strong Purcell effect.Fig. 2 shows that the TM mode is not only preferable for pumping but also for signal collection.
Both the thickness of the waveguide and the roughness parameters impact the scattering loss.The loss coefficient is proportional to the square of the RMS roughness amplitude.In our model, the RMS amplitude is set to 1nm in line with recently developed state-of-art roughness measurements [32], [33], [34], which is also a value close to the lattice constant of Ta 2 O 5 (a = 0.620 nm, b = 0.366 nm and c = 0.389 nm) [35].
The correlation length is related to the spatial frequency components of the roughness and, combined with the core thickness dependent propagation constant and surface field, impacts the scattering.In contrast to the RMS amplitude, the correlation length has been less systematically studied.Ladouceur et al. considered that the correlation length, L, for a rough waveguide surface is in the range between 0.1 to 1 μm [36].Kita considered L to be ∼100 nm [14] and Schmid's measured value is 180 nm, with which a good experimental and analytical agreement is achieved [25].For deposited Ta 2 O 5 thin film, the measured correlation length are ranging from ∼100 nm [37], 400 nm [38] and 49 to 282 nm [39].Due to the rather wide range of the correlation length reported, the relationship between scattering loss and correlation length is first investigated by calculating the loss of fundamental modes as a function of correlation length and core thickness using (10) and (11).Fig. 3(a) and (b), for the TE 0 and TM 0 modes respectively, show that the scattering loss does not change dramatically when the correlation length is in the range from 50 nm to 400 nm.The correlation length of 180 nm measured by Schmid et al is inside range for the measured value for Ta 2 O 5 thin film, and both roughness for the sidewall and deposited thin film are of exponential autocorrelation with similar RMS value, so 180 nm is adopted in this work as a representative value for correlation length.Another significant point is that the thickness that maximizes the surface field, marked by the red lines in Fig. 3(a) and (b), generates strong scattering, although it does not necessarily maximize it.This shows that the impact of the core thickness on scattering loss should be quantified and taken into account into WERS waveguide optimization.
Fig. 3(c) shows the propagation loss coefficient for both TM and TE polarizations versus core thickness for the first three modes.Obviously, the loss coefficient for different modes is not the same, so that the assumption that the loss coefficient for pump and signal are the same is not valid for signal collected in different modes.To illustrate the effect of the interference effect on scattering, the scattering loss calculated by the widely-used Payne-Lacey's model, which does not consider interference, is also included in the same figure.The interference effect within the waveguide layers results in pronounced undulations and strong suppression of the scattering from ∼10 dB/cm down to ∼3.5 dB/cm.Fig. 3(d) shows the scattering loss from a Ta 2 O 5 film deposited on the Si substrate coated with a 2 μm oxide layer.The reason for simulating this multilayer substrate is that oxidized silicon wafers have excellent surface quality, the silica layer enables low-loss waveguiding on a silicon substrate and they are convenient substrates to experimentally study the scattering loss.Due to the extra layer, the reflectivity and transmissivity for calculating the scattering loss is determined by the transfer matrix method [40], [41].The calculation shows that the extra layer impacts the interference, and consequently the scattering loss, significantly.Although Payne-Lacey's model is currently widely accepted, our measurements show that inclusion of interference effects and further expansion of the multilayer interference are required to properly account for the scattering loss dependence The above study shows that the waveguide core thickness impacts the surface modal field, efficiency of coupling the signal and the scattering loss significantly.Thus, the optimization for WERS should take these parameters into account, especially the scattering loss that strongly depends on the core thickness as it impacts the surface field and interference.

B. Experimental Loss Measurements
The scattering loss is shown to have a more complicated dependency on the core-thickness but has been always oversighted or oversimplified.Moreover, experimental study has rarely carried out especially with the multilayer interference considered.To verify the theoretical prediction for the scattering loss, a systematic measurement of the loss coefficient vs core thickness was carried out.Ta 2 O 5 thin films with thicknesses between 90 nm and 360 nm were deposited on silicon wafers with a 2 μm oxidized layer.To achieve minimal surface roughness, the sputtering parameters, including A r : O 2 ratio, chamber pressure, and RF power, were optimized.After deposition, all samples were annealed at 550 °C for three hours to fill any oxygen vacancies lost during the sputtering process and thereby reduce absorption.The coated Ta 2 O 5 surfaces were characterized by a stylus profiler.The stylus profiler has a similar resolution as atomic force microscopy (AFM) in the vertical direction for the rough surface, while its field of view is much larger, thus making the RMS amplitude measured from a wider range more reliable.The horizontal resolution for the stylus profiler is much lower than an AFM, so the correlation length obtained from the stylus profiler was not applied in this study.Instead, as discussed above, 180 nm was adopted in our model as a representative value for the correlation length.The auto-correlation function was calculated from the raw data from the stylus profiler so that the RMS amplitude is obtained for theoretical calculations.
The experimental measurements and theoretical calculations are shown in Fig. 4. The experimental loss coefficients are normalized to the mean square of the fluctuation of the rough surface, σ 2 , since the scattering loss coefficient is proportional to it (see (10) and ( 11)), and the ideal roughness created by an optimized deposition recipe should be around 1nm in practice as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I MEAN SQUARE OF FLUCTUATION AND LOSS FOR DIFFERENT FABRICATED
CORE THICKNESSES mentioned above.The samples are deposited by RF plasma sputtering with different recipes, which were aiming at exploring the optimal recipe for the surface roughness, and their corresponding mean square of the fluctuation and original loss are given in Table I.The loss coefficient measurements were conducted with the prism coupling using a Metricon instrument operating at 632.8 nm in air, so the theoretical curves are re-calculated for this wavelength and are different to the ones in Fig. 3(d).Only when the phase-matching angle is achieved, the incident laser power can be coupled into the waveguide in a certain mode.By recording and analyzing the decaying light in the thin film with the built-in photodetector, an exponential curve can be fitted, thus obtaining the modal loss coefficient.It is worth noting that the 120 nm thick film was exposed to a longer plasma strike than the rest of the samples during the deposition process which may have damaged its surface and led to the increased loss values.Moreover, the 359 nm core exhibits higher loss coefficients than the theoretical prediction for fundamental modes most likely because the core is too thick to allow the annealing to oxidize all the tantalum.Therefore, with the fundamental modes mostly confined in the core higher absorption contributes to higher propagation loss.Both theoretical and experimental values show that a thicker waveguide suffers less scattering loss, and, for a waveguide with a specific thickness, TM modes typically will experience less loss when the Ta 2 O 5 core is fabricated with the optimized recipe to reach an ideal roughness.
It worth highlighting that the Payne-Lacey's model, widely accepted at present, predicts 3-times higher scattering loss than Schmid et al.'s model.The fundamental difference is that the latter considers the impact of interference effect of the scattered light within the waveguide interfaces.The good agreement between experimental and theoretical results validates the use of the Schmid's et al augmented scattering loss coefficient model and our extension of considering multilayer interference with the transfer matrix method.

C. (n)FOM Calculations -Single-Mode Pumping / Single-Mode Collection
In a practical device, the Raman signal is collected as the sum of each of the multiple spatial and polarization modes supported by the waveguide.We first calculate the individual nFOMs obtained with a single pump mode and a single collected signal mode, from a conventional three-layer waveguide.Results of nFOM for TM 0 and TE 0 pumping are shown in Figs. 5 and 6.
As shown, the signal collected by the lowest order fundamental mode has the highest efficiency, but the collection efficiency into higher order modes is not negligible, with weaker dependence on waveguide thickness and length.Comparing TM and TE excitation, the former is shown to give larger efficiencies.The conventional FOM is calculated using ( 12) and ( 13) with the waveguide thickness that maximizes the surface modal field of the pump, as discussed in Section III-A.The conventional FOM for TE 0 and TM 0 pumping and collection and the corresponding waveguide parameters are listed Table II.It is shown that for randomly distributed dipoles, compared to TE 0 pumping/collection, TM 0 pumping/collection results in 2.3 times larger Raman conversion efficiency, due to higher optimum surface modal field and lower thickness-dependent scattering loss (see Figs. 2(a) and 3(c)).In addition, the optimum waveguide thickness and length are ∼2 times larger.FOM for bulk analyte has also been presented in Table II, providing supplementary information for planar WERS.Considering the analyte molecule with ∼nm scale, the bulk FOM will be greater by approximately two orders, which means the signal from monolayer is only around a percent of that from bulk analyte.In this scenario, signal power is prior and is our nFOM addressed.
The relative FOM (rFOM, c.f. ( 16)) and the related optimal parameters are listed in Tables III and IV for forward and backward collection, respectively.It is evident that the new optimum waveguide parameters are significantly different from the conventional ones shown in Table II.The new optimized waveguides are of longer length and larger thickness, due to the corresponding lower propagation loss in the presence of roughness.The TE 0 Raman conversion efficiency is 1.18 times larger when compared to the optimum conventional TE 0 case.The TM 0 pumping/collection Raman conversion efficiency, on the other hand, is 2.5 times larger when compared to the optimum conventional TE 0 case.This figure decreases to 1.1 when compared to the optimum conventional TM 0 case, due to the smaller difference in scattering losses in the corresponding optimum thicknesses (c.f.Fig. 3(c)).
As can be clearly seen, the waveguide parameters are significantly from the conventional FOM values due to the impacts of scattering loss, and the cross-polarization collection generates nonnegligible signal.

D. (n)FOM Calculations -Single-Mode Pumping / Multi-Mode Collection
The single-mode pumping/collection cases studied in Section III-C have clearly demonstrated the impact of the waveguide thickness on the pump/signal surface intensities and the corresponding scattering loss, captured by the nFOM definition.However, in practice, the signal is always collected by all guided modes and the optimum waveguide parameters should maximize the overall Raman conversion efficiency.The nFOM is calculated and shown in detailed contour map for the conventional three-layer waveguide at first.The overall rFOM for the four-layer structure, the Ta 2 O 5 layer deposited on the silicon  wafer with an oxidized layer which creates extra reflections and interference to reduce the scattering loss, is also calculated in this section.Using (15) and summing up Raman collection in all the spatial and polarization modes, the nFOM that characterizes the total collected signal is shown in Fig. 7 for forward collection.In general, the TM modes show better performance compared to TE modes excitation in terms of the nFOM as well as core thickness and the length fabrication tolerances.From the contour maps in Fig. 7(b) and (c), it is shown that the total nFOM of higherorder mode pumping is comparable to the fundamental mode performance, despite the fact that higher-order modes have a much smaller surface modal field (c.f.Fig. 2(a)).This is due to larger number of collected modes, the much lower scattering losses (c.f.Fig. 3(c)) and the corresponding longer optimum sensing lengths.
The backward collection is also calculated and shown in Fig. 8.As expected from (9), the nFOM increases with the sensing length monotonically and saturates to a maximum value.Compared to forward collection, backward collection results in stronger Raman signal conversion efficiencies and wider sensing length fabrication tolerances.Therefore, backward collection

TABLE V OPTIMAL NEW RELATIVE TOTAL FOM FOR EACH SINGLE-MODE PUMPING AND THE CORRESPONDING OPTIMUM WAVEGUIDE THICKNESS AND EFFECTIVE LENGTH FOR THE FORWARD (F) AND BACKWARD (B) SIGNAL COLLECTED BY
ALL SUPPORTED MODES would be a better choice provided that the presence of the pump in-coupling method does not hinder the Raman signal collection.The contour maps in Figs.7 and 8 show multiple peaks due to the multimode collection.However, in addition to the nFOM value, the waveguide thickness and sensing length fabrication tolerances need to be considered in real WERS systems and applications.
The optimum rFOM and the corresponding waveguide parameters are listed in Table V.It should be mentioned that in this case the optimum waveguide thickness and sensing length do not correspond to a particular mode but are average values that maximize the overall collection efficiency.With the growing thickness, the propagation loss drops and the corresponding waveguide length becomes longer, and deviates from the conventional reciprocal of the loss coefficient.When compared with the results in Tables III and IV, it is clear that multimode collection is more efficient than the single-mode collection.It is also shown that the different TE/TM pumping mode rFOM are comparable.This implies that in the case of randomly distributed the single-mode pump excitation is not strictly required, and multimode pump excitation will give comparable conversion efficiencies.This is an important result as it relaxes considerably the fabrication and other experimental tolerances.
In Table V, to present a full picture of the optimization, the rFOM and relevant waveguide parameters for bulk analyte are also provided, along with the relative signal-background-ratio r(SBR), which indicates the signal quality.The signal noise ratio (SBR) is calculated following Ref.[6], [15] (see appendix).The SBR normalized to the value calculated from the core thickness from conventional FOM, and is listed in Table V as rSBR.Concerns may arise from the possibility that a thicker core could lead to an increased generation of background to the signal due to a greater overlap of the waveguide material with the modal field.However, the calculated rSBR shows that signal quality is not significantly affected by our optimized waveguide parameters, and configurations for high-order TM modes can even enhance the SBR.The indicators, rSBR and nFOM, for bulk analyte are following the trend for monolayer nFOM.
The same analysis is also carried out for the waveguide with oxidized silicon wafer substrate, whose scattering loss behavior is shown in Fig. 3(d).The results are summarized in Tables VI and VII.Compared to the previous simpler three-media planar waveguide, this composite waveguide provides much superior conversion efficiency performance.This is due to the improved scattering losses, as a result of the additional interference effects from the oxidized SiO 2 /Si interface.This is important as it clearly demonstrates the significance of interference effects on  Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

IV. CONCLUSION
Traditional optimization of waveguides for advanced WERS sensor applications is based on defining the waveguide dimensions that maximize the pump and signal surface intensity under pump/signal single-mode operation.However, this approach overlooks that fact that maximization of the surface intensity results inevitably in increased surface-induced scattering losses, increase the waveguide propagation losses and reduce the overall Raman collection efficiency.
In this work, we have first studied theoretically and experimentally the impact of planar waveguide thickness on surface scattering losses and the waveguide propagation losses.It is shown that surface scattering losses are strongly influenced by interference effects due to reflections at the core/cladding and core/substrate interfaces, as well as additional reflections from other interfaces at adjacent substrate layers.It should be mentioned that this is similar to the reduction of propagation losses in anti-resonant waveguides [42] and hollow-core antiresonant fibers [43], [44].In the case of Ta 2 O 5 waveguides in silica-on-silicon substrates with typical surface roughness parameters, it is shown that the peak propagation losses reduce from ∼10 dB/cm to below ∼3 dB/cm for TE 0 modes and below ∼2 dB/cm for TM 0 modes, when the interference with additional reflections from the substrate interfaces is taken into account.The propagation losses are even lower for higher-order modes.These results are in good agreement with experimental data.
In our study, we consider radiation from a Raman-emitting dipole on a waveguide can be captured back into the waveguide in polarization and spatial modes different from the mode carrying the pump, provided the waveguide can support them.In the case of randomly distributed dipoles, we consider the Raman gain coefficients corresponding to all possible combinations of pump and signal polarization and spatial modes.It should be noted that, in this case, single-polarization pump will produce both co-polarized and cross-polarized Raman signal.Our initial WERS experimental results have confirmed both co-polarized (TE) and cross-polarized (TM) signal collection from a TE pumped waveguide, and the ratio of the observed power in different polarizations agrees well with our theory.
We have introduced a new generalized nFOM to optimize planar waveguide-based WERS sensors under multimode excitation/collection operation and waveguide-thickness-and modedependent propagation losses.Our study covers core thicknesses up to 800 nm supporting up to 6 modes.However, as our FOM calculations have shown, thicker cores need longer waveguides for maximum signal power collection, since the scattering loss is reduced.However, long waveguides are less favorable in practice, especially in the case of disposable point care applications.In addition, thicker cores could be dominated by core material absorption, which is highly dependent on the fabrication and adds cost.A waveguide length of ∼1 cm, optimum for core thicknesses of ∼100 to 600 nm, which support few modes, is considered to be a sweet spot for practical biomedical applications.Our nFOM suggests, for example, that once the core thickness increases from ∼80 nm, which is the optimum value given by conventional FOM for a Ta 2 O 5 on SiO 2 TE 0 waveguide at 785 nm, the signal power will increase significantly due to the reduced scattering loss and higher number of modes supported by waveguide, as shown in Table V.It is shown that in comparison with the traditional optimization, our optimized larger-core waveguides result in a WERS signal collection efficiency increase by a factor of ∼1.5 when using TE 0 pumping and a factor of ∼5 when using TM 0 pumping.It is also shown that the improvement increases to ∼4 when TM 1 and TM 2 pump is used.When the waveguide is deposited on a multilayer oxidized silicon wafer substrate, improvements by a factor of ∼6 to ∼7 are observed when the pumping mode order increases from TM 0 to TM 2 (see Table VII).The nFOM increase with the mode order is a result of the combination of substantial reduction in propagation loss, increased number of collected modes, longer optimum sensing lengths and occurs despite the concomitant surface intensity decrease.This implies that in the case of randomly distributed dipoles, the single-mode pump excitation is not strictly required, and multimode pump excitation will give superior conversion efficiencies.This is an important result as it relaxes considerably the fabrication and other experimental tolerances, and it is expected to result in more robust, reproducible, and cheaper WERS sensors.
Compared to a singlemode waveguide counterpart, use of multimode waveguides and multimode signal collection is likely to affect the spectroscopic measurements, especially by reducing the resolution of the spectra, but not normally below that required for measurement of non-gaseous species.This will be the case, if the multimode waveguide signal output is collimated and directly fed into the spectrometer.However, in most WERS system implementations, a multimode fiber is used at the output to collect the signal and feed it into the spectrometer [7], [8], [45], [46], [47], In this case, the input to the spectrometer is highly multimoded even in the case of a singlemode waveguide and it will not further affect the spectroscopic measurements.Furthermore, under the condition of sufficient signal power, the deconvolution method can be applied to enhance the resolution of WERS spectra, as demonstrated in Ref. [47].
Lastly, our theoretical model and experimental findings, along with our new FOM, have the potential to be extended to more complex 3D waveguide structures, such as ridge/channel waveguides.The scattering losses arising from core-cladding interface roughness in slab waveguides and sidewall roughness in 3D waveguides share theoretical similarities.Consequently, it becomes possible to mitigate scattering losses from the sidewalls by manipulating the multilayer interference effect.Similarly, in 3D waveguide-based WERS, the impact of sidewall scattering can be alleviated by widening the waveguide (reducing the lateral modal field at the sidewalls).Furthermore, the increased width of the 3D waveguide could facilitate Raman signal collection with multiple cross-polarization lateral modes, leading to enhanced efficiency in WERS.

A. Free Space Dipole Radiation Due to Evanescent Field Pump
Based on Fermi golden rule, the Excitation efficiency for a molecule emitting Raman power P 0 with the frequency of ω s excited by the evanescent field of pumping power P pump with the frequency ω p close to the surface of the waveguide core is given by [30] where Ãeff ( r 0 , ω p ) is defined as the effective mode area: σ R (ω p , ω s ) is the cross-section of the scatter.n g (ω p ) is the group refractive index for the pump.k v = 1.26 × 10 23 C −2 V 2 m 2 is a constant given by Long [31].ε( r, ω p ) is the relative dielectric function of the pump.e( r, ω p ) is the electric field of in its mode area.
The group refractivity is defined as the velocity of light in vacuum, c, divided by the group velocity of the mode, while Snyder proves the group velocity of a slab waveguide to be the modal power P j divided by, W j , the total stored energy per unit length of the waveguide [48] For non-dispersive core [48], while the power can be calculated by integrating the Poynting vector.So, the item of n g / Ãeff now is taking a more straightforward form, where the power-normalized modal field can be defined to simplify the formula, Therefore, the free-space radiation due to the evanescent field from a slab waveguide can be written in a simple form Equation ( 23) is also applicable to the background generated by the pump in the core and substrate area.The excitation efficiency for background is written as [6], [15], where σ bg is the cross-section of background.The group index n g is calculated with (20).The integration is calculated along the region where background is generated (core and substrate).As discussed in Ref. [15], the signal background ratio and signal noise ratio are given as and It is evident that the background power P bg ∼ ρ bg η bg , where ρ bg is the density of background dipole.To calculate the SNR/SBR, the dipole densities (ρ bg and ρ R ) and cross-sections (σ bg and σ R ) must be accurately known.However, when calculating the relative SBR (rSBR, SBR normalized to value calculated from the core thickness from conventional FOM for TE 0 mode), the parameters, ρ bg , ρ R ,σ bg and σ R , will cancel out.

B. Transfer Matrix for Reflectivity and Transmissivity
Standard transfer matrix method for calculating the multilayer structure illustrated in Fig. 9 is given by Ref. [40], [41]: Light transfer in same layer n from upper interface y n−1 to lower interface y n where the transfer matrix for layer n is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
in which φ n = k yn h n is the phase shift in layer n, where k yn and h n are the y component of the wave number in layer n and thickness of layer n.For calculating the reflectivity and transmissivity for the dipole coupling efficiency ((1) in main text): And for calculating the loss coefficient (( 10) and ( 11)), assuming the scattering is from layer 0-1 interface, with a scattering angle θ (0∼π), The matrix for the interface can be written as where r and t are the Fresnel reflectivity and transmissivity from a single layer.The transfer matrix describes the light transfer from layer n to n+1 is where M nn+1 = m n m nn+1 The transfer matrix for the whole structure is the product of all matrixes for each layer where, i.e., Hence

C. Coupling Efficiency of Dipole Radiation to Guided Mode
It is well known that the guided modes correspond to the poles of the total reflectivities, r T E and r T M of the layered structure, while the coupled power can be calculated by integrating (1) along the infinitesimal contour around the pole in the complex plane [26], [27], [28], as shown in Fig. 10.
The coupled jth guided mode power P j is calculated as follows,

TABLE VIII PARAMETERS USED IN THIS PAPER
where l is semicircle with infinitesimal radius , i.e., on the path l, k ρ = β j + exp (iθ) , θ ∈ (0, π) , (38) It worth noting that the reflectivity/transmissivity is also a function of k ρ , as indicted by (36).

D. Implementation of the Model
The calculations for this paper are conducted with free scientific Python packages, NumPy [49] and SciPy [50].
First, the propagation constant for each mode is obtained by solving the dispersion equation, which is a transcendental equation with the univariable of core thickness for the waveguide with certain material.The transcendental equation is solved with the optimize module in SciPy [50].The distribution of the electric field is expressed in terms of the propagation constant and normalized to carry 1 W/m modal power.The mode parameters are calculated using the transfer matrix method, which is implemented with ndarray module in NumPy [49], while the other scientific calculations, including integration etc., are realized with NumPy package.
Parameters used in this paper are given in Table VIII.

Fig. 1 .
Fig.1.Illustration of a WERS system with a three-layer waveguide.The refractive index for cladding, waveguide core, and substrate are n 3 , n 1 and n 2 respectively.The geometry of the waveguide for further calculation is depicted.The root-mean-square (RMS) fluctuation is σ, as marked.

Fig. 2 .
Fig. 2. Excitation and waveguide coupling for a dipole on the cladding-core interface; (a) normalized squared electric field at the core-cladding surface.(b) Signal capture efficiency in different modes for horizontal dipole.(c) Signal capture efficiency in different modes for vertical dipole.The curves corresponding to TE modes are not shown because their efficiency is 0.

Fig. 3 .
Fig. 3. Propagation loss coefficient is plotted as a function of core thickness and correlation length for (a) TE 0 mode and (b) TM 0 mode.The surface roughness RMS amplitude is 1 nm.The red lines mark the thickness that maximizes the surface modal field.(c) Loss coefficient for the first three modes as a function of thickness with correlation length fixed at 180 nm.The waveguide comprises a water-Ta 2 O 5 -SiO 2 three-layer structure.The dot-dash line corresponding to the right y-axis is calculated by the widely used Payne-Lacey's model.(d) The scattering loss from the film on Si substrate with a 2 μm oxidized (SiO 2 ) layer in between.The waveguide comprises a water-Ta 2 O 5 -SiO 2 -Si four-layer structure.The waveguide structure is used for experimentally validate the model on scattering loss in the following section.

Fig. 4 .
Fig. 4. Measured loss coefficient and comparison with the theoretical value for different modes.The Theoretical value is calculated by assuming the RMS, σ, is 1nm.The measured experimental loss coefficient is normalized to the mean square of the RMS amplitude, σ 2 .

Fig. 9 .
Fig.9.Illustration of the light propagation, reflection (r) and transmission (t) in a multilayer structure with an input (marked as 1) from layer 0.

Fig. 10 .
Fig. 10.Integration path to calculate in the presence of the jth guided mode with propagating constant β j .Integration around the semicircle of infinitesimal radius, , at the pole corresponding to the power coupled to jth the mode.

TABLE II OPTIMAL
CONVENTIONAL FOM FOR TE0 AND TM 0 MODES PUMPING AND THE CORRESPONDING OPTIMUM CORE THICKNESS AND OPTIMUM (FORWARD COLLECTION) / EFFECTIVE (BACKWARD COLLECTION) WAVEGUIDE LENGTH TABLE III OPTIMAL RELATIVE FOM FOR TE 0 AND TM 0 MODES PUMPING AND THE CORRESPONDING OPTIMUM WAVEGUIDE PARAMETERS FOR FORWARD COLLECTED SIGNAL IN EACH INDIVIDUAL MODE

TABLE IV OPTIMAL
NEW RELATIVE FOM FOR TE 0 AND TM 0 MODES PUMPING AND THE CORRESPONDING OPTIMUM WAVEGUIDE THICKNESS AND EFFECTIVE LENGTH FOR BACKWARD COLLECTED SIGNAL IN EACH INDIVIDUAL MODE

TABLE VI OPTIMAL
CONVENTIONAL FOM FOR TE 0 AND TM 0 MODES PUMPING AND THE CORRESPONDING OPTIMUM CORE THICKNESS AND OPTIMUM (FORWARD) / EFFECTIVE (BACKWARD) WAVEGUIDE LENGTH FOR THE MULTILAYER OXIDIZED SILICON WAFER SUBSTRATE

TABLE VII OPTIMAL
NEW RELATIVE TOTAL FOM FOR EACH SINGLE-MODE PUMPING AND THE CORRESPONDING OPTIMUM WAVEGUIDE THICKNESS AND EFFECTIVE LENGTH FOR THE FRONT AND BACKWARD SIGNAL COLLECTED BY MULTIMODE FOR THE MULTILAYER OXIDIZED SILICON WAFER SUBSTRATE the waveguide loss performance and the WERS sensor overall efficiency.