Surface sensing with integrated optical waveguides : a design guideline

Waveguide-based biochemical sensors exploit detection of target molecules that bind specifically to a functionalized waveguide surface. For optimum sensitivity, the waveguide should be designed to mediate maximum influence of the surface layer on the effective refractive index of the guided mode. In this paper, we define a surface sensitivity metric which quantifies this impact and which allows to broadly compare different waveguide types and integration platforms. Focusing on silicon nitride and silicon-on-insulator (SOI) as the most common material systems, we systematically analyze and optimize a variety of waveguide types, comprising simple strips, slot and double slot structures, as well as subwavelength gratings (SWG). Comparing the highest achievable surface sensitivities, we provide universal design guidelines and physically interpret the observed trends and limitations. Our findings allow to select the appropriate WG platform and to optimize sensitivity for a given measurement task. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (130.6010) Sensors; (130.0130) Integrated optics; (280.1415) Biological sensing and sensors; (230.7370) Waveguides. References and links 1. S. Schmidt, J. Flueckiger, W. Wu, S. M. Grist, S. T. Fard, V. Donzella, P. Khumwan, E. R. Thompson, Q. Wang, P. Kulik, J. Kirk, K. C. Cheung, L. Chrostowski, and D. Ratner, “Improving the performance of silicon photonic rings, disks, and bragg gratings for use in label-free biosensing,” Proc. SPIE 9166, 91660M (2014). 2. V. M. N. Passaro, B. Troia, M. La Notte, and F. De Leonardis, “Chemical sensors based on photonic structures” in Advances Chemical Sensors, W. Wang, ed. (InTech, 2012). 3. S. Janz, A. Densmore, D.-X. Xu, P. Waldron, J. Lapointe, J. H. Schmid, T. Mischki, G. Lopinski, A. Delâge, R. McKinnon, P. Cheben, and B. Lamontagne, “Silicon Photonic Wire Waveguide Sensors” in Advanced photonic structrues for biological and chemical detection, X. Fan, ed. (Springer, 2009). 4. P. Kozma, F. Kehl, E. Ehrentreich-Förster, C. Stamm, and F. F. Bier, “Integrated planar optical waveguide interferometer biosensors: A comparative review,” Biosens. Bioelectron. 58, 287–307 (2014). 5. A. Densmore, D.-X. Xu, S. Janz, P. Waldron, J. Lapointe, T. Mischki, G. Lopinski, A. Delåge, J. Schmid, and P. Cheben, “Sensitive label free Sensitive label-free biomolecular detection using thin silicon waveguides,” Adv. Opt. Technol. 2008, 725967 (2008). 6. L. Chrostowski, S. Grist, J. Flueckiger, W. Shi, X. Wang, E. Ouellet, H. Yun, M. Webb, B. Nie, Z. Liang, K. Cheung, S. Schmidt, D. Ratner, and N. Jaeger, “Silicon photonic resonator sensors and devices,” Proc. SPIE 8236, 823620 (2012). 7. X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013). 8. J. Flueckiger, S. Schmidt, V. Donzella, A. Sherwali, D. M. Ratner, L. Chrostowski, and K. C. Cheung, “Subwavelength grating for enhanced ring resonator biosensor,” Opt. Express 24(14), 15672–15686 (2016). 9. H. Yan, L. Huang, X. Xu, S. Chakravarty, N. Tang, H. Tian, and R. T. Chen, “Unique surface sensing property and enhanced sensitivity in microring resonator biosensors based on subwavelength grating waveguides,” Opt. Express 24(26), 29724–29733 (2016). Vol. 26, No. 16 | 6 Aug 2018 | OPTICS EXPRESS 19885 #326159 https://doi.org/10.1364/OE.26.019885 Journal © 2018 Received 14 Mar 2018; revised 10 May 2018; accepted 13 May 2018; published 24 Jul 2018 10. B. Sepúlveda, J. S. Del Rio, M. Moreno, F. J. Blanco, K. Mayora, C. Domínguez, and L. M. Lechuga, “Optical biosensor microsystems based on the integration of highly sensitive Mach-Zehnder interferometer devices,” J. Opt. A, Pure Appl. Opt. 8(7), 561–566 (2006). 11. A. Densmore, D.-X. Xu, P. Waldron, S. Janz, P. Cheben, J. Lapointe, A. Delâge, B. Lamontagne, J. Schmid, and E. Post, “A silicon-on-insulator photonic wire based evanescent field sensor,” IEEE Photonics Technol. Lett. 18 (23), 2520–2522 (2006). 12. S. T. Fard, V. Donzella, S. A. Schmidt, J. Flueckiger, S. M. Grist, P. Talebi Fard, Y. Wu, R. J. Bojko, E. Kwok, N. A. Jaeger, D. M. Ratner, and L. Chrostowski, “Performance of ultra-thin SOI-based resonators for sensing applications,” Opt. Express 22(12), 14166–14179 (2014). 13. A. Kargar and C.-Y. Chao, “Design and optimization of waveguide sensitivity in slot microring sensors,” J. Opt. Soc. Am. A 28(4), 596–603 (2011). 14. L. Huang, H. Yan, X. Xu, S. Chakravarty, N. Tang, H. Tian, and R. T. Chen, “Improving the detection limit for on-chip photonic sensors based on subwavelength grating racetrack resonators,” Opt. Express 25(9), 10527– 10535 (2017). 15. H. Sun, A. Chen, and L. Dalton, “Enhanced evanescent confinement in multiple-slot waveguides and its application in biochemical sensing,” IEEE Photon. J., A1(1), 48–57 (2009). 16. I. Khodadad, N. Clarke, M. Khorasaninejad, D. Henneke, and S. S. Saini, “Optimization of multiple-slot waveguides for biochemical sensing,” Appl. Opt. 53(23), 5169–5178 (2014). 17. C. A. Barrios, “Analysis and modeling of a silicon nitride slot-waveguide microring resonator biochemical sensor,” Proc. SPIE 7356, 735605 (2009). 18. T. Claes, W. Bogaerts, and P. Bienstman, “Experimental characterization of a silicon photonic biosensor consisting of two cascaded ring resonators based on the Vernier-effect and introduction of a curve fitting method for an improved detection limit,” Opt. Express 18(22), 22747–22761 (2010). 19. F. Dell’Olio and V. M. N. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15(8), 4977–4993 (2007). 20. J. Gonzalo Wangüemert-Pérez, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, D. Pérez-Galacho, R. Halir, I. Molina-Fernández, D.-X. Xu, and J. H. Schmid, “Evanescent field waveguide sensing with subwavelength grating structures in silicon-on-insulator,” Opt. Lett. 39(15), 4442–4445 (2014). 21. A. Dhakal, A. Raza, F. Peyskens, A. Z. Subramanian, S. Clemmen, N. Le Thomas, and R. Baets, “Efficiency of evanescent excitation and collection of spontaneous Raman scattering near high index contrast channel waveguides,” Opt. Express 23(21), 27391–27404 (2015). 22. D.-X. Xu, J. Schmid, G. Reed, G. Mashanovich, D. Thomson, M. Nedeljkovic, X. Chen, D. Van Thourhout, S. Keyvaninia, and S. Selvaraja, “Silicon photonic integration platform-Have we found the sweet spot?” IEEE J. Sel. Topics Quantum Electron., A20(4), 8100217 (2014). 23. M. G. Scullion, T. F. Krauss, and A. Di Falco, “Slotted photonic crystal sensors,” Sensors (Basel) 13(3), 3675– 3710 (2013). 24. J. Hu and D. Dai, “Cascaded-ring optical sensor with enhanced sensitivity by using suspended Si-nanowires,” IEEE Photonics Technol. Lett. 23(13), 842–844 (2011). 25. U. Manual, Version 2016, CST-Computer Simulation Technology GmbH, Darmstadt, Germany, 2017, http://www.cst.com. 26. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015). 27. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94(3), 033903 (2005). 28. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012). 29. H. Kogelnik, “Theory of optical waveguides” in Guided-wave Optoelectronics, T. Tamir, ed. (Springer, 1988). 30. C. Koos, Nanophotonic Devices for Linear and Nonlinear Optical Signal Processing (Univ.-Verlag Karlsruhe, 2007). 31. J.-M. Brosi, Slow Light Photonic Crystal Devices for High Speed Optical Signal Processing (Univ.-Verlag Karlsruhe, 2009). 32. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

The sensor principle relies on an optical WG, guiding a mode which significantly extends into the cladding medium that surrounds the WG core.The interaction between the optical mode field and the varying surface layer properties alters the effective refractive index by e n  and thus the optical phase shift accumulated during propagation.The strength of this effect is expressed by the so-called surface sensitivity, which, in combination with the effective-index sensitivity from the phase measurement of the underlying resonator or interferometer, determines the overall sensitivity.Proper optimization of the WG towards high surface sensitivities is hence key for realizing high-performance sensors.
In this paper, we define a universal surface sensitivity which is broadly applicable to layers of different thicknesses and refractive indices.Focusing on Si3N4 and Si as the most common integration platforms, we systematically analyze a wide variety of WG types, comprising simple strip WG, slot and double slot WG, as well as sub-wavelength grating (SWG) WG.For each of these WG types, we identify the optimum geometry for both TE and TM polarization, and we compare the highest achievable surface sensitivities, taking into account implementation limitations that are associated with state-of-the art fabrication technologies.The focus of our analysis is on WG types that can be reliably mass-produced by optical lithography and single-etch structuring of WG on solid substrate layers.Note that even higher sensitivities can be achieved by more sophisticated WG concepts comprising ultrasmall features [23] or suspended WG sections [24].These schemes, however, require dedicated fabrication processes which are not yet accessible through scalable foundry processes.Exploiting the scalability of Maxwell's equations with respect to refractive index and geometry, we derive and physically explain general trends and design rules to corroborate the numerical results.Our findings can be used as design guidelines to select the appropriate WG platform and to optimize sensitivity for a given measurement task.

Basic sensing principle: Wave propagation and effective refractive index
For illustrating the basic sensing principle, we regard homogeneous sensing with a waveguide core embedded in an infinitely extended cladding medium.First, we define a few quantities: The propagation of monochromatic plane waves with vacuum wavelength  in a homogenous medium is determined by the propagation constant 0 nk   (refractive index n , vacuum propagation constant 0 2 k

 
). Dielectric WG consist of a high-refractive index core core () n and a low-refractive index cladding medium ( M n ).For integrated optical WG made from silicon or silicon nitride, the core is usually supported by a buried oxide layer (BOX, bottom cladding) with refractive index BOX n .The evanescent parts of the WG mode, which are essential for the sensing process, extend into the cladding region.The actual field distribution in the various materials determines the WG propagation constant  , which can be expressed by an effective refractive index e0 Figure 1 summarizes the essential design elements for a WG: The material platform, the WG type, the polarization, and the WG geometry.
Platform Common integration platforms for optical sensors rely on a layer stack of a silicon (Si) or silicon nitride (Si3N4) device layer on a several micrometer thick BOX (SiO2) as a bottom cladding, mechanically supported by a Si substrate.WG are structured in the device layer, and the BOX thickness is chosen to avoid leakage into the high-refractive index silicon substrate as well as to optimize grating coupler efficiency.
While Si WG are operated in the near infrared (NIR),

34
Si N WG are suitable for operation across the whole visible (VIS) and NIR spectrum.As the target media for biosensors are usually provided in the form of aqueous solutions, sensor operation at VIS wavelengths is much less impaired by water absorption than in the NIR.However, a large wavelength allows relaxed WG fabrication accuracies, and reduces scattering loss due to WG roughness.In this paper, we thus consider Si WG operated at NIR telecom wavelengths around 1550 nm, where low-cost laser sources are readily available, as well as 34 Si N WG operated at 600 nm as an example of low-wavelength sensors, Fig. 1(a).
Waveguide type For both integration platforms, we study four typical WG types, denoted as strip, slot, double slot and subwavelength grating (SWG) WG, Fig. 1(b).The last three types require significantly smaller feature sizes than the strip WG and thus lead to more challenging fabrication processes.We concentrate on single-mode WG, combining stable operation with high surface sensitivity.
Polarization We study the fundamental mode of both (quasi-)TE and (quasi-)TM polarization, Fig. 1(c).The term (quasi-)TE refers to the case where the dominant transverse electric field component is oriented parallel to the substrate.Similarly, (quasi-)TM denotes a configuration in which the dominant transverse magnetic field component is parallel to the substrate.For the TE polarization, an enhancement of the electric field x E exists at the WG of the displacement must be continuous, while for TM polarization this field enhancement is to be seen at the top and at the bottom WG surfaces.
Geometry For all combinations of platform, WG type and polarization, we study the impact of the WG geometry in terms of height, width, and length (SWG only) of WG features, Fig. 1(d).Three standard device layer heights h of 220 nm, 250 nm and 340 nm are considered for Si WG structured on silicon-on-insulator (SOI) wafers.Wafers for structuring Si3N4 WG are available with a maximum Si3N4 thickness of several hundred nanometers, if stoichiometric growth is important, and we therefore consider typical WG heights h of 200 nm, 300 nm and 400 nm.For all WG types, we vary the overall width w in steps of 10 nm.For slot and double-slot WG, we additionally vary the slot width slot w in four steps.For SWG WG, we vary the period a and the spacing gap d between the WG elements, which can also be quantified by the fill factor gap FF ( ) a d a  . We chose a minimum feature size of 80nm to meet commercial technological conditions.
In the following, we define the surface sensitivity as a quantitative metric and maximize it by varying the various WG parameters.Given a certain sensor application, this data set allows to find the optimum design for a WG-based sensor.

Sensitivity of waveguide surfaces with respect to attached molecules
In WG-based sensing, two basic approaches are commonly used, which are compared in Fig. 2. For so-called homogeneous sensing, a bare WG core is exposed to a typically aqueous homogeneous target medium with refractive index M n , Fig. 2(a).For surface sensing, a WG core is functionalized such that target molecules from an aqueous solution can bind to the core forming a surface layer with effective thickness SL t and refractive index SL n , Fig. Δn cannot be traced back to a specific substance in the target medium if it is unknown which constituent is actually changing.
For surface sensing, the refractive index M n of the aqueous solution remains essentially fixed, while the molecules, bound to the surface layer, influence the optical mode.By functionalizing the WG surface with dedicated capture agents, surface sensing can be used for Δn is the stronger, the more the optical field is concentrated within the surface layer.The definition of an effective layer thickness SL t accounts for a possibly inhomogeneous distribution of target molecules within the surface layer.With biological samples this effective thickness is in the nanometer range, and SL n is typically around 1.5.
For the remainder of this paper, we concentrate on surface sensing, offering a wide variety of applications.We are hence interested in the detailed influence of the surface layer on the electric and magnetic field distribution.
Figures 3 and 4 show the simulated electric field magnitudes of the fundamental quasi-TE mode of strip, slot and double slot WG, and for subwavelength grating WG, respectively.Details on the simulation parameters can be found in Appendix A. White contours mark the surface layer where molecules bind, leading to a change of the local refractive index.
A higher field concentration in these regions increases the surface sensitivity.While in a typical strip WG, Fig. 3(a), a large portion of the power is located inside the WG core, slot and double slot WG concentrate the power between the rails, Fig. 3(b) and Fig. 3(c).This means that the surface layer experiences higher field strengths in slotted WG, leading to a larger e n  compared to a strip WG.SWG WG are composed of a multitude of individual WG elements, thus increasing the surface layer area per unit length of the WG. Figure 4(a) shows a section of a SWG WG with three periods along the propagation axis (pink arrow).For determining the surface sensitivity, a 3D elementary cell has to be simulated using periodic boundary conditions in the axial direction [25].The three coloured planes mark the cross sections where the field distributions of Fig. 4(b)-4(d) are plotted.In Fig. 4(c), high field strengths (red regions) are located at the vertical edges of the blocks.

Surface sensitivity and field perturbation approach
In the following, we define the surface sensitivity (surf)  S as a quality metric for quantifying which WG design leads to the potentially highest overall sensor sensitivity.In order to obtain (surf )

S
for a specific WG with a specific surface layer, a full simulation of the WG with and without the surface layer would be required.This would include various combinations of surface layer properties like layer refractive index SL n and layer thickness SL t and thus requires a multitude of high-resolution simulations to resolve the surface layer with a thickness in the range of a few nanometers.Exploring the whole parameter space as discussed in Section 2 and Fig. 2 would hence be a time-consuming and probably unrealistic endeavour.
To overcome this problem we use a perturbation approach.To this end, we perform a single finite-element method (FEM) simulation of a bare WG for each WG geometry, store the resulting fields and compute the influence of an additional surface layer using a field interaction factor.This technique is only valid for small perturbations, i.e., the modal field does not change significantly with the surface layer, and hence the effective refractive index e n changes only slightly, too.In our case, binding events of molecules change the refractive index of a surface layer of thickness SL t from M n to SL n , where SL n is the refractive index of the attached molecules and M n denotes the background refractive index of the solution.The conditions for the perturbation approach hold, if either the thickness is smallthen the refractive index change SL M nn can be larger -or if SL M nn is small, in which case SL t can be larger.
We will first consider the general case of a z-variant SWG WG.According to Eq. ( 14) and Eq. ( 15) in Appendix C, the local perturbation in the surface layer can be translated into a change Note that the field interaction factor , because it describes also slow-light propagation, see Eq. (10).For computing L SL S Γ ( ) t , we simulate the fields of a unit cell for a bare SWG WG without surface layers, and integrate 2 E over the fictitious surface layer volume SL V within that unit cell of length a .The result is proportional to the electric energy SL W in the surface layer per unit cell, Eq. ( 7) and Eq. ( 10) for SL i  , normalized to the cross-sectional power P of Eq. ( 5).The integral in the denominator extends over the entire (x,y)-plane.For z-invariant WG such as strip, slot, or double slot structures, Eq. ( 1) can be simplified by exploiting the fact that the integral over 2 E is invariant along z.Simplifying the volume integral in the numerator, we obtain for the special case of z-invariant WG.We use Eq. ( 2) to analyze L SL S Γ ( ) t for a set of widely different silicon strip WG geometries, operated in quasi-TE polarization, see in Fig. 5(a) "Perturbation".As a comparison, we extract the numerically exact values obtained by FEM simulations of quasi-TE fields that include the actual surface layers, and plot them in Fig. 5(a) with a dashed line "Simulation".The agreement is very good except for large surface layer thicknesses tSL on narrow and high WG.In this case the majority of the field is contained within the surface layers, violating the assumption of a small field perturbation.The electric field outside the core decays approximately exponentially.For a penetration depth much larger than SL t , the decay of the field magnitude and of the power within the surface layer can be approximated by a linear function.As a consequence, SL  in Fig. 5(a) exhibits a region where it depends linearly on SL t .We now want to define a sensing sensitivity which isin the framework of the perturbation approachindependent of the surface layer thickness.This can be achieved by looking at the derivative   Because SL n influences the result, we define the surface sensitivity as the derivative of the field interaction factor with respect to the surface layer thickness, Within the framework of the perturbation approach, the surface sensitivity (surf) S is independent of both the thickness and the refractive index of the surface layer, thus providing a universal guideline for a sensor design.To confirm this, we calculate the derivatives for the same WG types as used in Fig. 5(a), and plot them semi-logarithmically in Fig. 5(b) in a range SL (1 350) nm t  .We see that   rr , owing to the approximately linear dependence of SL  on SL t in this region.We extrapolate the curve to SL 0 t  , where the computation fails due to the finite spatial discretization.In Fig. 5(b) these extrapolated values are marked with filled circles and denoted by (surf)  S , see Eq. ( 2) and Eq. ( 3).Note that a WG design for best homogeneous sensitivity (hom)   M SL SL lim t S      is not necessarily optimum when it comes to surface  in surface layer of thickness SL t .For each geometry, we compare SL  computed with a perturbation approach (based on a single FEM simulation without a surface layer) with (sim) SL  obtained from numerically exact calculations (FEM simulations with different surface layers).For small SL t the agreement is very good.For large SL t (marked by ) and SL M nn  , the field interaction factor SL  approaches M  and thus the homogeneous sensitivity sensing, see Eq. ( 16) in Appendix C for rigorous definition of (hom)  S .This can be inferred by comparing (hom)  S of the two top strip WG in Fig. 5(a) (blue and red arrows) to the corresponding (surf)  S of the same WG in Fig. 5(b) (blue and red dots at SL 0 t  ): A larger homogeneous sensitivity does not lead to a difference in surface sensitivity.In sensing applications, the desired measurement quantity is the effective surface layer thickness SL t .From a measurement of e Δn and for known bulk refractive indices SL n and ( ( ) nn   ) rr , SL t can be inferred from Eq. (3).A proper choice of the WG then maximizes (surf) S and therefore the measurement sensitivity for SL t .

Results
We extract the surface sensitivity (surf)  S according to Section 3.2 for all kinds of WG outlined in Section 2.2.For each TE-or TM-operated WG core in Si3N4 or alternatively in Si, a few typical values of heights h , slot widths slot w , periods a and fill factors FF are considered.For each combination of these parameters, the total WG width w (see Fig. 3) is then optimized to obtain optimum surface sensitivity (surf)   opt S . To this end, the WG width is swept with a step size of 10 nm.The range of w was chosen to maintain single-mode operation and to avoid substrate leakage for a BOX thickness of 2μm .The following WG surface sensitivity analysis relates to Si3N4 cores operated in the VIS (Section 4.1) as well as to Si cores operated in the NIR (Section 4.2).

Silicon nitride -VIS
For Si3N4 strip WG operated at a vacuum wavelength of 600 nm, Fig. 6(a) displays the surface sensitivity (surf)  S as a function of the WG width w for three different WG heights h and for both polarizations.Surface sensitivity generally benefits from large electric field strengths in the region of the surface layer as indicated in Fig. 3. Within their single-mode range, each WG shows a maximum (surf)   opt S at a distinct width opt w , marked by  for TE and by  for TM.Any deviation from this optimum width opt w reduces the interaction factor of the mode field with the surface layers, e.g., by concentrating the modal field to the WG core in the case of large w , or by spreading the mode field over a wider cladding region for small w .In the second case, a large portion of the modal field propagates in the BOX, where it cannot contribute to sensing.In Fig. 6(b), the extracted opt w (green, left vertical axis) and the corresponding (surf)   opt

S
(magenta, right vertical axis) are plotted as a function of the WG height h .The largest (surf)   opt S are found for large h and small w for both polarizations, with TE operation being more sensitive than TM operation.In the TE case, the two sides of the WG exploit the field enhancement, and only a small portion of the mode propagates in the BOX, see Section 5.2 for a more detailed discussion.
We apply the same procedure to TE-operated 34 Si N slot and double slot WG and plot the results in Fig. 7.As for the strip WG, the sensitivity increases for higher WG.Decreasing WG width first leads to increasing sensitivity, which drops again as the width becomes too small and the mode extends far into the cladding, Fig. 7(a).Smaller slot widths increase the field strength in the slot, Fig. 7(b).This, together with the growth of the total core surface as compared to a strip WG, increases (surf)  S .The slot WG behaves like two strip WG if a slot r il w w .In this case, each of the strips carries half the power of the slot WG and both the optimum rail width and the optimum sensitivity converge to those of a single optimum strip WG, see top light green line () and light magenta line () in Fig. 7(b).The double slot WG behaves like three separate strip WG if a slot r il w w .In this case, each strip carries one third of the power of the slot WG.As before, both the optimum rail width and the optimum sensitivity converge to those of a single optimum strip WG, see top light green line () and light magenta line () in Fig. 7(c) and in Fig. 7(d).w for different slot widths.For decreasing rail width, the sensitivity first increases to its maximum value and then drops.The drop is caused by the fact that the mode extends far into the cladding or is even lost to the substrate for very small rail widths.The optimum sensitivities (surf)   opt

S
at the corresponding optimum widths opt w are marked by dots.(b,d) Optimized geometries for slot and double slot Si3N4 WG.Sensitivities are larger for small slot widths and higher WG, the fabrication of which is limited by technological constraints.The light solid lines indicate the limits for slot w  (light green and light magenta).

Silicon -NIR
For silicon WG operated in the NIR at 1550nm   , we apply the same technique as described in Section 4.1 for extracting the optimized widths j ( , ) ( ) t te   E r E r and sensitivities (surf) opt S .In Fig. 8(a) we display the surface sensitivity (surf)

S
of a strip WG with different heights h and polarizations as a function of the strip width w .As for Si3N4 strip WG, Fig. 6(a), we see a sensitivity maximum within the range of single-mode operation.The TE sensitivity exhibits a pronounced maximum when varying the WG width, whereas the maximum of the TM sensitivity is much less pronounced, Fig. 8(a).
We extract the optimum sensitivity and find that, similar to Si3N4, the TE sensitivity can be optimized by choosing a high WG with a narrow width, while the optimum TM sensitivity depends only weakly on the WG height, Fig. 8(b).This is in contrast to the findings for the Si3N4 WG, Fig. 6(a) and Fig. 6(b).We attribute this to the fact that the refractive index of Si is significantly larger than that of Si3N4 and hence the asymmetry introduced by the underlying BOX is less significant for the case of Si WG.Consequently, for the WG heights h under consideration and for small w , the TM modal field does not extend into the BOX as strongly as for the Si3N4 WG.Regarding Si slot and double slot WG, Fig. 8(c) and Fig. 8(d),

S
for Si strip WG with three standard heights h as a function of WG width w in TE and TM operation.For decreasing WG width, the sensitivity first increases to its maximum value and then drops.The drop is caused by the fact that the mode extends far into the cladding or is even lost to the substrate for very small WG widths.The optimum sensitivities (surf)    the trends for optimum WG parameters are similar to those of Si3N4 WG, Fig. 7, and thus the same conclusions as in Section 4.1 can be drawn.For achieving better sensitivities, completely different WG structures in form of subwavelength gratings (SWG) were proposed [8,14,20], see Fig. 4. For a systematic comparison to strip and slot WG, we calculate and optimize the surface sensitivities of SWG WG, see Fig. 9. Figure 9(a), left y-axis, shows the results obtained from two exemplary families of TEoperated SWG, which share the same period a , but differ in height h , and feature three different fill factors (50%, 60%, 70%, differing in opacity).For the smaller WG height ( 220 nm h  , black lines) the sensitivity does not depend strongly on the WG width.Strip and slotted WG show a fast decrease of sensitivity for widths larger than wopt, since the fields become smaller at the sensitive side walls and upper surfaces and are more confined to the core.This effect is less pronounced in the SWG WG, since the sensitive surface in between the individual blocks are still experiencing high field strengths.The maximum sensitivity shifts to smaller w, when the FF increases.This is to be expected: If the FF becomes larger, the effective refractive index increases and the modal field is stronger confined to the SWG "core".For a sufficient influence of the analyte, the field must then extend into the SWG "cladding", which is achieved by decreasing the WG width.
Interestingly, for higher SWG WG ( 340 nm h  , blue lines), no optimum sensitivity within the observed range of w is found.Instead, the sensitivity increases strongly for larger FF and wider WG.This increase is caused by a decrease of the group velocity: Larger FF and increased w lead to an increase of the Bragg wavelength associated with the SWG structure.Once the Bragg wavelength comes close to the operation wavelength of the sensor, the device enters the so-called slow-light regime [26], which is characterized by a greatly reduced group velocity and a greatly increased effective group index eg n . .In this case, the sensitivity does not change significantly with w.For higher SWG WG ( 300 nm h  , blue lines), the sensitivity traces do not exhibit an optimum before entering into the so-called slow-light regime, where the sensitivity is dominated by a largely increased effective group index eg n (dotted lines for two different heights for FF 50%  ).For these traces, we choose the inflection points (blue crosses) to define reasonable sensor designs that are not subject to the impairments associated with slow-light operation.Note that this does not represent an optimum in a strict mathematical sense.(b) Overview of optimized SWG WG sensitivities and geometries outside the slow-light regime for different heights, periods, and fill factors.As a trend, the sensitivity increases for high WG and small gap size gap (1 FF) da  .Sensitivities obtained from inflection points at the transition to slow-light operation are again marked by blue crosses.Note that the traces for FF = 70% and a = 200 nm or a = 250 nm do not appear in the plot since the associated gap sizes are below the minimum feature size of 80 nm.
This leads to strong interaction of the guided mode with the surface layer and hence to a sharp increase of the sensitivity even though the field confinement (conf)   i  of the surface layer does not significantly increase, see Appendix C, Eq. ( 7).Note, however, that operation in the slow-light regime strongly increases the detrimental impact of environmental influences and fabrication tolerances [27] on mode propagation.Amongst others, the sensitivity and the optical propagation loss can vary significantly, such that reproducible device properties and hence a reliable sensor read-out are difficult to obtain in practice.This susceptibility to tiny changes can be seen exemplarily for the sensitivity of WG operating in the slow-light regime, Fig. 9(a), which is why we exclude them from the following discussion.For sensitivity traces entering into the slow-light regime without showing a pronounced maximum in Fig. 9(a), we choose the inflection points (blue crosses) to define reasonable sensor designs that are not subject to the impairments associated with slow-light operation without being optimum in a strict mathematical sense.Note that there is no distinct transition from "normal" SWG operation to the slow-light regime and that a strict classification is therefore difficult.choice of the inflection point is motivated by the fact that it marks the width at which the influence of the increased effective group index eg n starts dominating over the impact of the field confinement factor (conf)  , Eq. ( 10) in Appendix C, and dotted lines for two different heights for FF 50%  , see Fig. 9(a).The optimal sensitivities of the SWG WG and those obtained for the transition to the slow-light regime are summarized Fig. 9(b), indicating again a general sensitivity increase with WG height.Sensitivities obtained from inflection points at the transition to slow-light operation are marked by blue crosses.As a trend, () Er increases for small gap (1 FF) da  .

Comparison of optimized sensitivities
As a summary of the last two subsections, we compare the sensitivities for the Si3N4 platform in the VIS (λ = 600 nm) and for the Si platform in the NIR (λ = 1550 nm).In Fig. 10, we display the optimized surface layer sensitivities (surf)   opt S of four WG types (strip, slot, double slot, SWG) with optimized widths opt w and three different heights h for each platform.The left (right) panel shows the results for TE (TM) polarization.Three SWG positions are empty because either the gap size is below the minimum feature size of 80 nm, or because the structures do not support a well-guided WG mode ("poor waveguiding").
Generally, WG operated in TE polarization tend to be more sensitive than their TMoperated counterparts.Taking into account technological constraints such as a minimum feature size, the sensitivity of sophisticated WG designs can be increased by approximately a factor of 2 compared to a simple optimized strip WG.Moreover, slotted WG exhibit higher surface sensitivities than strip and SWG WG.Generally, the silicon platform exhibits higher sensitivities than the silicon nitride platform, which can be attributed to the fact that a higher index contrast leads to a smaller penetration depth into the WG cladding and hence to a stronger interaction with a thin surface layer.Larger WG heights tend to be generally better.For the silicon photonic platform, a sensitivity improvement of a factor of 4 can be achieved by using optimum double slot WG operating in TE polarization instead of non-optimum standard strip WG featuring, e.g., WG dimensions of 2 450 220 nm wh    for TM operation.A detailed physical explanation of these trends is given in Section 5.2.

Discussion
For a better understanding of the trends summarized in Fig. 10, we first discuss in Sect.5.1 the influence of the cladding asymmetry and we introduce universal scaling laws of Maxwell's equations that allow to generalize the findings to other wavelengths and material platforms.Subsequently, Section 5.2 explains in detail the trends when varying platform, operating wavelength, polarization, and WG geometry.Section 5.3 discusses our results in view of an overall sensor performance, including the impact of mode loss and analyte delivery as well as a practical instruction on finding global sensitivity optima.

Physical interpretation -Cladding asymmetry and scaling laws
Cladding asymmetry We consider WG cores that are supported by a SiO2 substrate (BOX) and are surrounded by an aqueous medium.Since the refractive index of SiO2 is larger than that of water, the cladding of the WG is asymmetric.It is hence impossible to increase the interaction of the guided light with the functionalized surface of the WG core by simply indefinitely reducing the core dimensions.Any attempt to do so will predominantly increase the fraction of the mode fields in the BOX, which do not contribute to sensing, and therefore decrease the fraction in the surface layer, leading to a decrease of surface sensitivity.
Scaling laws According to Appendix D, Table 1, line (1), scaling the geometry of a WG by g  while simultaneously scaling the operating frequency by leads to a simple geometrical scaling of the eigenfunctions of the electric and magnetic fields by a factor of g  .This effect has no impact on the homogeneous sensitivity, since the relative portion of the mode fields in the cladding medium remains the same.However, the surface sensitivity Eq. ( 3) as defined by a derivative with respect to the layer thickness SL t scales by 1 does not change the distributions of the electric and the magnetic field.Note, however, that the ratio of the electric-field eigenfunction () Er and the magnetic field eigenfunction () Hr must be scaled by a factor of 1 n  , see Eq. (19)   and the corresponding discussion.In total, this leaves the sensitivity unchanged, see Appendix D, Table 1, line (2).The optimum surface sensitivity of a WG with fixed geometry does not change for a scaling of the RI profile along with an inverse scaling of the frequency.

Trends for increased surface sensitivity
High index-contrast platform The surface sensitivity is maximized by an increased field concentration at the core surface which is in contact with the analyte-delivering aqueous medium.For a given wavelength, the surface sensitivity tends to increase if the refractive index ratio core M nn between core and cladding is increased.This is caused by the fact that the enhancement of surface-normal electric-field components is proportional to the square   2 core M nn of the index ratio.Similarly, the surface sensitivity increases when the refractive Fig. 10.Comparison of the optimized sensitivity for the Si3N4 platform in the VIS (λ = 600 nm) and for the Si platform in the NIR (λ = 1550 nm).We consider four WG types with three standard heights, operated in TE and TM polarization.Sensitivity can be increased by higher WG cores, by using Si rather than index profile is scaled up by a factor of n  > 1 while the geometry is scaled down by gn 1   to maintain the same operating frequency, see Appendix D, Table 1, line (4).Short operating wavelength For a given platform, higher surface sensitivities can be achieved with shorter wavelengths.The geometry of the WG has to be chosen guided by the scaling law in Appendix D, Table 1, line (1).A Si3N4 WG operated in the VIS will achieve a higher optimized surface sensitivity than an up-scaled Si3N4 WG operated in the NIR.When comparing optimized Si3N4 WG operated in the VIS with Si WG in the NIR, we see similar sensitivities, see Fig. 10.This is due to a combination of scaling the operating frequency along with WG geometry and an increase of the index contrast, which have contrary effects on the surface sensitivity that partially cancel each other.However, optimized WG in the high index-contrast Si platform will be as good as or better than those in the Si3N4 platform, as the limitation caused by the cladding asymmetry is more severe for lower core refractive indices.
TE polarization The TE-mode sensitivity generally outperforms that of the TM-mode.For high index-contrast WG as considered in this work, the surface sensitivity is dominated by regions exhibiting a large field enhancement due to field discontinuities at the corecladding interface.In the case of strip WG and of vertical-slot WG, TE mode operation exploits more of these surfaces showing a field enhancement compared to TM mode operation, where the interface to the bottom oxide cladding (BOX) does not contribute to the sensitivity.
It has to be noted that in typical telecom applications, WG with large widths and small heights are used (e.g. ).These established WG geometries are often directly transferred to sensing applications.For these WG and for improper choices of WG widths exceeding 400 nm, the TM-polarized mode may exhibit higher sensitivity than the TE, see Fig. 8(a).However, under the assumption that the width w is chosen properly, TEpolarized modes exhibit optimized sensitivities.If an application demands Si WG operated with TM polarized light, SWG WG are best, see Fig. 10.
Enlarged surface Larger core surfaces can generally lead to higher sensitivities.This is exploited by slot, double slot and SWG WG, which introduce additional sensor surfaces compared to strip WG.For slot and double slot WG operated in TE polarization, these additional surfaces lead to an enhancement of the dominant transvers electric-field component and can hence further increase surface sensitivity, see Fig. 3.For SWG WG, the enhancement at the additional surfaces between the blocks affects the weaker longitudinal electric-field component, Fig. 4. The sensitivity gain of SWG WG is hence smaller than that of slot and double slot WG and depends only weakly on the polarization.
Narrow waveguides Optimized sensing WG have a narrower core than typical routing WG designed for low-loss light transport.Narrow WG increase the sensitivity for both TE and TM polarization.When narrowing the WG, the sensitive area of the surface layer remains essentially constant for TE polarization, but the surface field strength initially increases.For very small WG widths, the modal field expands into the cladding and the field interaction with the surface layer decreases, hence reducing surface sensitivity.For TM polarization, an additional effect comes into play: When reducing the WG width, the top surface, containing high field strengths due to the field discontinuity, becomes smaller, while the overall field strengths at the core surface become more dominant as the mode expands into the cladding.This interplay leads to a less pronounced optimum with respect to the WG width than for TE polarization, see Fig. 6(a) and Fig. 8(a).
High waveguides Higher WG cores outperform thinner WG cores.Higher WG cores reduce the relative portion of the fields located in the BOX (cladding asymmetry).Since this area does not contribute to sensing, less field strength in the BOX leads to larger sensitivities.In addition, and especially for the TE mode, higher WG cores enlarge the sensitive sidewall regions that are subject to field enhancement.
Small slot widths and gap sizes Smaller slot widths and gap sizes generally increase the sensitivity.For slot WG, the sensitivity gain for TE is more pronounced than for TM polarization.

Overall sensor system performance and the impact of mode loss
It is important to note that, when implementing these WG into functional sensors, the overall system performance will also depend on additional aspects that are outside the scope of our current analysis.One of the most important aspects is the propagation loss of the optical mode: Adsorption of target molecules to the WG surface is usually measured by recording the phase shift accumulated over a certain propagation length L .In technical implementations of sensors, the phase measurement accuracy depends on the precision with which amplitudes can be measured in an interference setup.This precision and ultimately the detection limit of the sensor system decreases with increasing propagation losses, which, like the sensitivity, depends on WG platform, type, polarization and geometry.A rigorous system optimization would ideally have to take into account all these effects to account for mutual trade-offs and to find a global optimum of the WG design.This, however, would require a quantitative relationship between the WG geometry and the associated power propagation loss   exp L  characterized by the loss constant  , which is impossible to state in a general and reliable manner across different integration platforms.
For finding optima for the overall system sensitivity sys S for a specific sensor implementation, we need to know three dependencies: The surface sensitivity (surf )  S as a function of the WG geometry, provided in this paper, the WG loss constant  as a function of the WG geometry, which must be experimentally determined for the specific WG type and technology, and the influence of this loss on the effective-index sensitivity ( ' ) , ( ' ) into a measurable output signal, e.g., a current ( [1,4,28]).If the linewidth of the light source is neglected, the overall system sensitivity can be expressed by the product of the surface sensitivity and the effective-index sensitivity The results presented in this paper can hence serve both as guidelines for identifying and selecting promising WG designs and as quantitative measures to determine the overall system sensitivity sys S once the technology-dependent loss constant  and the architecture-specific relationship   e S  are known.
Another aspect that may influence the overall performance of practical sensors is the analyte delivery to the sensor surface.In this context, narrow gaps or other high aspect-ratio voids tend to have less exchange with the surrounding liquid or might even not be accessible to large target objects such as cells or cellular vesicles.In these cases, simple strip WG geometries might exhibit advantages in comparison to more complex concepts featuring narrow strips and slots.

Summary
We have performed a comparative study of different WG types for application in label-free detection of chemical or biological analytes that specifically bind to functionalized WG surfaces.To this end, we have introduced the so-called surface sensitivity (surf )  S as a general quality metric that is broadly applicable to surface layers of different thicknesses and refractive indices.This metric allows to optimize the WG design and to compare the fundamentally achievable sensor performances across different WG types and integration platforms.We specify optimized WG designs along with the corresponding surface sensitivities for the most common WG parameter combinations.We also introduce and explain a computationally efficient recipe for applying the methodology to additional WG types that are not covered by our current study.
The following key findings result from our study: First, sensitivity trends can be explained by the interplay of a cladding asymmetry, the scaling laws of Maxwell's equations and the effect of field enhancement.Second, WG with optimum surface sensitivities are typically high and narrow, are fabricated on high-index contrast platforms and operated in TE polarization.Third, the surface sensitivity of highly sophisticated WG such as slot, double slot or sub-wavelength grating exceeds that of optimized strip WG by a factor of 2, and that of standard telecom strip WG by a factor of 4 if realistic feature size constraints are observed.

A. Simulation parameters and mesh considerations
In the following, we describe the relevant parameters used for the simulation, which are performed with CST (Computer Simulation Technology GmbH) Microwave Studio.Modal fields of the WG are calculated in the frequency domain.Simulation boundaries in the cross section are perfectly absorbing.The computational mesh is tetrahedral, and an ultra-fine mesh is required around the WG core to obtain accurate field data within the surface layers.The field interaction factor SL  is well-described by a linear approximation for SL t smaller than the penetration depth, see Fig. 5(a).The surface sensitivity (surf )  S can be reliably extracted from the derivative SL SL Γ t  at SL 0 t  by a linear extrapolation of data within the first 10 nm, see Fig. 5(b).We have found that a mesh size of < 3 nm around the WG core region is sufficient.The choice of the mesh size in the cladding region was left to the program.The simulation area had a total size of 4 µm × 4 µm, where the bottom half of the simulation region was occupied by the 2 µm thick BOX.For WG significantly narrower than opt w , the WG mode is not well-confined to the core anymore, and much larger simulation windows are required.These results were omitted from the discussion, but lie at WG geometries far away from any sensitivity optimum (Fig. 11).The refractive indices were assumed to be Si

C. Propagation and sensitivity in waveguides
In the following discussion we assume a positive time dependence i .Cross-sectional power, stored energy, and field confinement factor The crosssectional power P associated with a guided mode is constant along the propagation direction z (unit vector z e ) if losses are neglected.This is true even if the WG geometry varies periodically in the direction of propagation, as in the case of SWG WG.The cross-sectional power is expressed by the real part of the time-averaged complex Poynting vector in the direction of propagation, integrated over the WG cross-section The time-averaged stored energy per unit length is obtained by an integration of the modal electric and magnetic energy densities over a unit cell volume V [29].The expression can be simplified with the help of the space-dependent material group index g n n n      and with the identity EH [29], where, for non-magnetic materials, 1 Note that in this relation, W is a length-related energy density with unit J/m and denotes the ratio of stored energy within a WG section, e.g., a unit cell, and the length of the section, e.g., the unit cell length a.For z-invariant WG, the volume element is simply dd V a A  .The field confinement factor (conf )   i  represents the ratio of the mode energy in a partial volume i V related to the energy in the total volume V of a unit cell, Here, 00 1 ( ) Zc   is the free-space wave impedance.Combining Eq. ( 7) and Eq. ( 9) we define the field interaction factor n , see next section.The effective group index can be expressed by a sum of the partial field interaction factors i  multiplied with the respective material group indices.The field interaction factor is determined by the relative mode energy (conf )   i  in region i and by the time the mode dwells in that region, expressed by the effective modal group index eg n , which can be larger than , gi n .This is exploited in slow-light applications.For this case the field interaction factor can become larger than one.

Dielectric profile perturbation and definition of sensitivities
Typical index differences are in the order of 0.1 0.2 n  , which justifies the approximation in Eq. (11).For homogeneous sensing, only the refractive index of the aqueous cladding medium (partial volume M i  ) in the volume M V changes by M n  , see Fig.For surface sensing, we consider a thin surface layer (partial volume SL i  ) of refractive index ( ' )  Hr around the WG core, which locally replaces the aqueous medium with refractive index M n .The refractive index change is confined to and constant within the surface layer volume SL V of thickness SL t , see Fig.
We introduce the perturbations defined by Eq. ( 11)-( 13) at a fixed frequency into the general variation theorem of Eq. ( 8).Since the permittivity perturbation is limited to SL V in the case of surface sensing, the change of the propagation constant Similarly, for homogeneous sensing, the permittivity perturbation is limited to M V , the change of the propagation constant (hom)   is obtained from Eq. ( 14) by replacing the superscript "(surf)" by "(hom)" and by integrating over M V instead of SL V .A direct link between Δβ and the field interaction factor  Eq. ( 10) is observed.For a sensor it is important how the effective modal index e n changes with respect to the local refractive index perturbation n  .For surface (homogeneous) sensing, we have For our sensitivity analyses, we calculate the field interaction factors for a WG by numerically calculated mode fields.We directly define the homogeneous sensitivity (hom) S to be identical with the corresponding field interaction factor M  , measuring the impact of the refractive index perturbation  (19) refractive index of the WG cladding changes, β and hence e n are modified, which impacts the phase shift e0 propagation length L .Due to the large optical frequencies, a change of e n is measured with high accuracy.For a given WG length L , the measured phase shift can be referred to a change e n  of the effective index, which finally allows to sense a change of the cladding index.The larger e n  becomes for a certain cladding index change, the more sensitive the device becomes.2.2 Parameters for WG-based sensing Numerous parameters determine how sensitive e n  reacts on a cladding index change M n  .

Fig. 1 .
Fig. 1.Essential design elements for maximizing surface sensitivity.For each combination of (a) material platform, (b) WG type and (c) polarization, the optimum (d) geometry can be determined, observing reasonable technological constraints.
2(b).For homogeneous sensing, a change in M n causes a change e Δn of the effective index.This change is the stronger the more the mode optical field extends into the target medium.Homogeneous sensing is usually unspecific, i.e., e

Fig. 2 .
Fig. 2. Plain and functionalized strip WG on a buried silicon oxide (BOX) layer.(a) Homogeneous sensing.The aqueous target medium with refractive index M n forms the

Fig. 3 .
Fig. 3. Simulated electric field magnitudes of the fundamental quasi-TE mode in different WG geometries with height h and total width w .White contours mark the surface layer on the WG core.The larger the field strength in this region is, the larger the surface sensitivity becomes.The surface layer is disregarded for the field calculation.(a) Strip WG.The surface layer experiences only moderate field strengths.(b) Slot WG.A large field strength is located in the surface layers of the slot.(c) Double slot WG.The field strength in each of the two slots is smaller than for a single slot, but the relevant surface layer area has doubled.

Fig. 4 .
Fig. 4. Section of a subwavelength grating (SWG) WG and simulated electric field magnitudes of the fundamental quasi-TE mode.Compared to a strip WG, the surface layer area per unit length of the WG is increased.(a) Schematic of a SWG WG with width w , height h , period a and gap size gap d .The pink arrow marks the direction of propagation.Specific crosssections 0 z  , 0 y  , 0 x  are indicated with colored planes.Front, Top and Side mark the associated views.(b)-(d) Electric field magnitudes.White contours mark the surface layers on the WG core, which are disregarded for the field calculation.The larger the field strength in this region and the larger the surface, the bigger the surface sensitivity becomes.(b) Front view at 0 z  .The surface layer experiences only moderate field strengths. (c) Top view at 0 y  .Large field strengths (red areas) occur at the vertical edges of the blocks.(d) Side view at 0 x  .Moderate field strengths are found at surfaces const.z  refractive index via the field interaction factor

Fig. 5 .
Fig. 5. Validation of perturbation model and definition of surface sensitivity (surf) S for a set of widely different silicon strip WG geometries propagating quasi-TE fields, see legend.(a) Field interaction factor SL in surface layer of thickness SL t .For each geometry, we compare (hom) S .(b) The surface sensitivity SL (surf) SL SL t t S     allows an easy comparison of different WG, irrespective of the actual surface layer properties.

Fig. 6 .
Fig. 6.Optimization of Si3N4 strip WG.(a) Surface layer sensitivity (surf) S of Si3N4 strip WG for three standard heights h as a function of WG width w in TE and TM operation.For decreasing WG width, the sensitivity first increases to its maximum value and then drops.The drop is caused by the fact that the mode extends far into the cladding or is even lost to the substrate for very small WG widths.The optimum sensitivities (surf) opt S at the corresponding optimum widths opt w are marked by dots and triangles.(b) Comparison of optimum WG width (green, left axis) and optimized surface sensitivity (magenta, right axis) as a function of WG height.High and narrow WG (blue markers) are most sensitive.

Fig. 7 .
Fig. 7. Optimization of Si3N4 TE-operated slot and double slot WG. (a,c) Surface layer sensitivity j ( , ) ( ) t te   H r H r of Si3N4 slot and double slot WG for a height of h = 400 nm as a function of rail widthw for different slot widths.For decreasing rail width, the sensitivity first increases to its maximum value and then drops.The drop is caused by the fact that the mode extends far into the cladding or is even lost to the substrate for very small rail widths.The optimum sensitivities(surf)

Fig. 8 .
Fig. 8. Optimization of Si strip, slot and double slot WG.(a) Surface layer sensitivity(surf) opt w are marked by dots and triangles in (a) and compared in (b).For TE polarization, high and narrow WG yield the best results.For TM, no pronounced sensitivity gain is observed for higher WG.(c),(d) Optimized geometries for slot and double slot Si WG.The sensitivity increases for smaller slot widths and higher WG, the fabrication of which is limited by technological constraints.The light solid lines indicate the limits for slot w  (light green and light magenta), which correspond to the case of individual strip WG as shown in (b).Compared to Si3N4, operated in the VIS, the Si rails, operated in the NIR, are typically 100 nm wider while the sensitivity is slightly larger.

Fig. 9 .
Fig. 9. Optimization of Si sub-wavelength grating (SWG) WG.(a) Sensitivity of two SWG WG families with different heights, but common period 300 nm a  as a function of WG width w for varying fill factors FF.Black lines with various shadings stand for a WG height of 220 nm h .In this case, the sensitivity does not change significantly with w.For higher SWG WG ( 300 nm h  , blue lines), the sensitivity traces do not exhibit an optimum before entering into the so-called slow-light regime, where the sensitivity is dominated by a largely increased effective group index eg n (dotted lines for two different heights for FF 50%  ).For these traces, we choose the inflection points (blue crosses) to define reasonable sensor designs that are not subject to the impairments associated with slow-light operation.Note that this does not represent an optimum in a strict mathematical sense.(b) Overview of optimized SWG WG sensitivities and geometries outside the slow-light regime for different heights, periods, and fill factors.As a trend, the sensitivity increases for high WG and small gap size gap (1 FF) da  .Sensitivities obtained from inflection points at the transition to slow-light operation are again marked by blue crosses.Note that the traces for FF = 70% and a = 200 nm or a = 250 nm do not appear in the plot since the associated gap sizes are below the minimum feature size of 80 nm.

g  .
The optimum surface sensitivity of a WG with fixed RI profile hence scales proportionally to frequency and requires an inverse scaling of the geometry.Furthermore, scaling the refractive index profile of a WG by n

Fig. 11 .
Fig. 11.Simulated electric field magnitudes of the fundamental quasi-TM mode in different WG types with heighth and total width w .White contours mark the surface layer on the WG core, which is disregarded for the field calculation.Surface sensitivity generally benefits from large electric field strengths in the region of the surface layer.(a) Strip WG.Large portion of the fields is lost to the non-sensitive interface between core and BOX.(b),(c) Slot WG and double slot WG.In contrast to the TE-Mode (Fig.3) there is no enhancement of the electrical field in the slot.The surface layer at the top of the core and in the slot experiences only moderate field strengths.


complex vectorial electric mode field   , Er and magnetic mode field   , Hr depend on angular frequency  and position vector r .We assume dispersive dielectric and non-magnetic materials which could be periodic along z with a period a .The propagation constant is  and the effective refractive index is e0 region, all refractive indices and all other parameters of the material are assumed to be constant.The various regions are denoted by a subscript

V
the refractive indices i n and the associated group refractive index g,i n are constant.Summing the partial field confinement factors over all partial volumes results in 1 because i i VV   Variation theorem, effective group index, and field interaction factor The influence of small perturbations on the propagation constant   can be examined by extending a variation theorem [29, Eq. (2.2.73)] for dielectric WG, . If we introduce only a frequency perturbation   at a fixed dielectric profile, we find that the effective group index the ratio of the total energy W per unit cell length, and the power P, as the ratio of a local refractive index change i n  in region i and the associated change e n  of the effective modal index e If the 3D refractive index profile () n r with ( , , ) x y z  r of a WG is changed by a small amount ( ( ) nn   ) rr , the associated change of the propagation constant  can be calculated with the perturbation approach, Eq. (8).A small refractive index change (

2
on the effective modal index e n .We further define the surface sensitivity (surf) S as the derivative of the field interaction factor SL  (surface layer volume SL V ) with respect to the surface layer thicknessWe consider dielectric media, which are assumed to be lossless, isotropic, linear and nonmagnetic at the (angular) frequencies 2π f   of interest.The vacuum speed of light is denoted by c .The (real) relative permittivity r ()  r is linked to the refractive index n by Equation (17), together with boundary conditions, defines an eigenvalue problem, where ω is the angular eigenfrequency and () Hr and () Er are the corresponding eigenfunctions of the wave equations.Scaling the geometry.If the geometry of the WG is scaled by a factor 0 g   , i.e.,