Quantitative Diffractometric Biosensing

Diffractometric biosensing is a promising technology to overcome critical limitations of refractometric biosensors, the dominant class of label-free optical transducers. These limitations manifest themselves by higher noise and drifts due to insufficient rejection of refractive index fluctuations caused by variation in temperature, solvent concentration, and, most prominently, nonspecific binding. Diffractometric biosensors overcome these limitations with inherent self-referencing on the submicron scale with no compromise on resolution. Despite this highly promising attribute, the field of diffractometric biosensors has only received limited recognition. A major reason is the lack of a general quantitative analysis. This hinders comparison to other techniques and amongst different diffractometric biosensors. For refractometric biosensors, on the other hand, such a comparison is possible by means of the refractive index unit (RIU). In this paper, we suggest the coherent surface mass density, ${\mathrm{\Gamma }}_{\mathrm{coh}}$, as a quantity for label-free diffractometric biosensors with the same purpose as the RIU in refractometric sensors. It is easy to translate ${\mathrm{\Gamma }}_{\mathrm{coh}}$ to the total surface mass density ${\mathrm{\Gamma }}_{\mathrm{tot}}$, which is an important parameter for many assays. We provide a generalized framework to determine ${\mathrm{\Gamma }}_{\mathrm{coh}}$ for various diffractometric biosensing arrangements that enables quantitative comparison. Additionally, the formalism can be used to estimate background scattering in order to further optimize sensor configurations. Finally, a practical guide with important experimental considerations is given to enable readers of any background to apply the theory. Therefore, this paper provides a powerful tool for the development of diffractometric biosensors and will help the field to mature and unveil its full potential.

Figure 1

Signal and background in diffractometric biosensors. (a) Light is impinging on a molecular grating, which causes diffraction into several orders. The power that is diffracted can be measured by detecting the diffracted light with a detector (camera). A possible image of the diffracted beam is illustrated on the right. It is useful to also monitor the zeroth order to have a power reference, which is required to compute the absolute diffraction efficiency. (b) Noncoherent refractive index distortions such as surface roughness can lead to background light on the camera. Because of the coherent light, this leads to a speckled background in the image and can limit the performance of the sensor.

Figure 2

(a) Illustration of the same total mass density ${\mathrm{\Gamma }}_{\mathrm{tot}}$ in three different distributions (harmonic, canonical, and coherent). The different distributions vary in their proportion of ${\mathrm{\Gamma }}_{\mathrm{coh}}$ and ${\mathrm{\Gamma }}_{\mathrm{inv}}$. Therefore, all three distributions cause a different signal. (b) Illustration of how to use the theory described in this paper. The two sensors (i) and (ii) have a different total amount of immobilized mass and the mass is distributed differently across the sensor area. Nevertheless they both have the same diffraction efficiency $P_{\mathrm{out}}/P_{\mathrm{in}}$. In order to address this properly, the coherent surface mass density ${\mathrm{\Gamma }}_{\mathrm{coh}}$ is used. We define ${\mathrm{\Gamma }}_{\mathrm{coh}}$ as the amount of mass that causes the measured $P_{\mathrm{out}}/P_{\mathrm{in}}$ if the mass is arranged as illustrated in (iii). The parameter ${\mathrm{\Gamma }}_{\mathrm{coh}}$ can be calculated from Eq. (15). It is equal for sensors (i) and (ii) as both have the same optical configuration and the same $P_{\mathrm{out}}/P_{\mathrm{in}}$. When the mass distribution is known (for example, by investigation of the manufacturing process), the analyte efficiency ${\eta }_{[A]}$ can be determined. It can be used to calculate the total mass density from ${\mathrm{\Gamma }}_{\mathrm{coh}}$. In (c) we illustrate that ${\mathrm{\Gamma }}_{\mathrm{coh}}$ can be determined independently of the optical setup. The two sensors (iv) and (v) have the same amount and distribution of adsorbed mass, but use a different optical configuration. This results in a different $P_{\mathrm{out}}/P_{\mathrm{in}}$, but ${\mathrm{\Gamma }}_{\mathrm{coh}}$ determined from Eq. (15) remains the same. This allows us to compare the sensor output from different optical configurations.

Figure 3

(a) Illustration of the ideal mode expansion. (i) The electromagnetic field of a real system is often difficult to solve for by applying Maxwell’s equations directly. Therefore, ideal mode expansion is applied. (ii) A simpler ideal system is taken as reference, for which Maxwell’s equations can be solved easily to yield a complete set of modes [(iii) ${\overrightarrow{E}}_1$, ${\overrightarrow{E}}_2$, …]. (iv) The difference between the ideal and real systems can be described by a perturbation function in a spatially narrow region close to the interface at $z=0$. This perturbation leads to energy transfer between the ideal modes of the system—i.e., coupling. (v) By solving the coupled mode equations the expansion coefficient for each ideal mode $c_{\nu }$ can be calculated. (vi) From the ideal modes and the corresponding expansion coefficients $c_{\nu }$, the real electromagnetic fields can be determined using a linear combination. (b) General representation of the notation and coordinate system used throughout this paper. The vectors ${\overrightarrow{\beta }}_{\mathrm{in}}$ and ${\overrightarrow{\beta }}_{\mathrm{out}}$ are projections of the $k$ vectors of the incident ${\overrightarrow{k}}_{\mathrm{in}}$ and the outgoing mode ${\overrightarrow{k}}_{\mathrm{out}}$ onto the $x$-$y$ plane defined by the boundary of two dielectrics. The $x$ axis is chosen normal to the grating lines (parallel to the grating vector ${\overrightarrow{\beta }}_g$). The molecular grating has an area $A$. The incident and outgoing $k$ vector together with the surface normal define the plane of incidence and the outgoing plane. (c) Plane of incidence: the incident $k$ vector is inclined with respect to the surface normal by an angle ${\theta }_{\mathrm{in}}$. The angle $\theta$ is always measured from the $k$ vector to the surface normal and is positive in the counterclockwise direction when looking in the positive $e_{\mathrm{\perp }}$ direction. (d) Outgoing plane: the outgoing $k$ vector forms an angle ${\theta }_{\mathrm{out}}$ with the surface normal. (e) Top view onto the coupling plane ($xy$) illustrating the coupling condition: the plane of incidence and the outgoing plane are rotated by azimuth angles ${\phi }_{\mathrm{in}}$, ${\phi }_{\mathrm{out}}$ with respect to the positive grating normal direction. Here $\phi$ is again positive in the counterclockwise direction when looking in the negative $z$ direction.

Figure 4

Modes of the ideal interface structure used in the description of two-dimensional diffractometric biosensors. Freely propagating [$F$ in (a),(b)] and guided modes [$G$ in (c)] are distinguished. The $F$ modes are further distinguished according to whether they originate or end up in the cover [$F_c$ (a)] or in the substrate [$F_s$ (b)], respectively. Every mode is either incident or outgoing, and has a TE (s) or a TM (p) polarization. The directions of the electric (blue) and magnetic (green) fields are illustrated for the chosen convention, which result in the signs appearing in Table 2. For a particular mode, the field intensities in the different media are related by the Fresnel coefficients following the definition in Ref. [53]. The fields of the outgoing modes can be found by time reversal of the in arrangement (see Appendix 9b). The angles of the field vector for the guided mode are connected to the effective refractive index of the mode $N=n_f\sin ({\theta }_g)$. The fields involved in the coupling at the molecular grating are evaluated in the cover and their orientation needs to be determined from Snell’s law and its generalization in the case of total internal reflection for beams incident from below the cover ($F_s$ and $G$) [53]. (d) Three exemplary optical configurations for diffractometric sensors on a planar substrate. They couple different incident and outgoing modes (indicated by red arrows): Top: coupling between two $F_s$ modes. Middle: coupling between a $G$ and an $F_s$ mode. Bottom: coupling between two $G$ modes with different propagation directions in the waveguide plane.

Figure 5

Illustration of surfaces with different rms roughness $\sigma$ and correlation lengths $L_c$.

Figure 6

Illustration of the geometry of incident and outgoing modes with TE polarization for (a) $F_{c,\mathrm{in}}$, (b) $F_{c,\mathrm{out}}$, (c) $G_{\mathrm{in}}$, and (d) $G_{\mathrm{out}}$. The free-space modes (a),(b) consist of three beams, an incident, a reflected, and a transmitted beam, or an outgoing and two incoming beams (with electric fields $E_1$ and $E_2$). The power flux in the far field is normalized to the power $P$. The field just above the interface is a superposition of the two beams propagating in the cover. Guided beams (c),(d) are normalized following Ref. [43], which is extended to two dimensions by assuming a homogeneous intensity over the beam width. For the incident mode (c), the beam width is $w_b$. For the outgoing mode (d), the width is given by the orthogonal projection of the molecular grating onto the plane perpendicular to the propagation direction of the mode. We refer to it as the characteristic grating width $w_{\mathrm{ch}}$. While in general the outgoing beam is nonuniform, we show a setup that yields a uniform outgoing beam. The direction of the grating normal is along the $x$ axis of the displayed coordinate system.