1. Introduction

Over the past few years the ever increasing demand of the healthcare and pharmaceutical sectors for more powerful and cost-efficient diagnostic tools has been the main driving force behind the development of optical biosensing devices. Various types of label-free biochemical sensors have been developed which exhibit high-sensitivity and have the potential to be transformed to miniaturized devices for portable applications, such as carbon nanotube field effect transistors (FETs) [1] or silicon nanowire FETs [2]. However, optical label-free sensors [3] seem to be superior mostly because they are non-invasive and immune to external electromagnetic fields. In addition, they do not require reference electrodes, while they offer galvanic isolation of the optical transducer from the electronics. Most importantly, optical detection in many cases allows for real-time multi-analyte monitoring (multiplexing).

Optical label-free biochemical sensors make use of the changes in the refractive index (i.e. the optical path) occurring close to the reaction surface, or more accurately, changes that take place within the penetration depth of the evanescent field, due to a biochemical reaction. In affinity sensors the changes are induced by the binding between a ligand and a receptor immobilized on the sensor surface (e.g. an immuno-complex between antigen and antibody, or the hybridization of complementary DNA strands). The changes in the refractive index are manifested as a change in the photon flux monitored at the output of the sensor. These label-free sensors, widely known as evanescent-field sensors are mainly represented by the reflectometric interference spectroscopy (RIfS) [4], optical porous silicon biosensors [5], surface plasmon resonance (SPR) [6], nanowire waveguides [7], and planar photonic biosensors such as resonators [8,9] gratings [10,11], photonic crystals [12,13] and waveguide-based Integrated Optical Chips (IOCs) [e.g 14–20.]. Compared to the other planar photonic biosensors, the IOC-based label-free biosensors are simpler and more versatile in design and have been proven to be capable of the ultimate degree of integration with the source and detector devices on a single-chip (as in [17] and [18]). More specifically, in the case of dielectric-waveguide IOCs, the Mach–Zehnder Interferometer (MZI) –apart from the incessant interest it attracts for telecommunication and all-optical gate applications (as demonstrated in [21] and [22]) - is one of the most promising biosensing devices due to its high sensitivity and the ability of optoelectronic integration in lab-on-a-chip microsystems [23–27].

In this paper, Frequency-Resolved Mach-Zehnder Interferometry (FR-MZI) is presented as a novel operation principle for affinity sensors that can form the basis of versatile and ultra-sensitive label-free detection schemes. First, the basic principle and the main characteristics of the suggested method are presented and compared with the conventional Single-Wavelength MZI (SW-MZI). Then, a semi-analytical approximation combined with numerical results is presented in order to assess the limits of detection followed by a discussion about the usefulness of integrated FR-MZI affinity sensors as a generic cost-efficient technological platform.

2. FR-MZI principle of operation and advantages over SW-MZI

A SW-MZI is an interferometric device that can detect changes of the effective refractive index of a medium. The light from a coherent light source propagates though an optical waveguide via total internal reflection and then is divided into two branches, the sensing arm and the reference arm, by means of a Y-junction. The two beams are recombined again at a second Y-junction, Fig. 1a . Changing the refractive index over the sensing arm produces a change in the phase velocity between the propagating beams, which is translated to an intensity modulation at the output of the MZI mathematically expressed as:

 

Fig. 1 Schematic representation of (a) a standard planar waveguide MZI configuration, and (b) the cross-section of the simulated ridge-waveguide MZI

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In this work, Frequency-Resolved Mach-Zehnder Interferometry (FR-MZI) is introduced as an alternative, versatile and more cost-efficient detection method compared to conventional SW-MZI. In FR-MZI the “traditional” monochromatic laser input source is substituted by a white-light source. Nowadays, standard silicon microfabrication technology can easily produce planar ridge-waveguides that allow single-mode waveguiding of all wavelengths within the UV-NIR range with low transmission losses. In addition, careful selection of the waveguide thickness in conjunction with the nature of the cladding layers ensures rejection of the TM modes simply by exploiting the mode’s power leakage into the silicon substrate [28]. Therefore, on-chip polarization selection can be achieved and monomodal behavior of the waveguides can be ensured. On the detector’s side, the FR-MZI requires the use of a spectrophotometer. In a more advanced scheme, one can also envision a monolithically integrated approach with the MZI fully-integrated with the LED and photodiode on a single chip [17, 18].

The key point of the substitution of the input laser light source with a white-light source though is not simply to reduce the cost and size of the experimental setup, but to address the above mentioned drawbacks of the conventional SW-MZI device while rendering FR-MZI to an even more powerful detection method:

The first notable feature of the FR-MZI arising from Eq. (6) is that the changes in the output signal do not depend on the length of the sensing arm -as is the case for SW-MZI. Long sensing arms in a FR-MZI only ensure that there can be several local minima within the spectral region of interest which can be analyzed during the signal analysis rendering the statistical analysis even more accurate (from Eq. (3) the longer the arm the larger the number m of λ_m that can satisfy the destructive interference condition).

In addition, it is to be noted that the spectral shifts can be determined with accuracies far better than the wavelength resolution of the spectrometer employed. This is enabled by fitting entire regions of the experimental curve and especially those regions where the wavenength mathematical dependence can be expressed in closed form. For example, one can fit the data near local extrema (minima or maxima) where the signal dependence on the wavelength is of the form b(λ-λ_m)². Here λ_m is the observable to be tracked in time. As it will be shown in the next section, the spectral dependence of the output light has mostly a sinusoidal form and the parabolic fit is quite accurate around the minima/maxima and for a signal change of I_m/4, where I_m is the signal maximum. Then, the coefficient b in the parabolic law becomes b=I_m/(M∆λ)², where M are the number of points inside the parabolic region and ∆λ is the spectrometer resolution or wavelength step. Since any pair of values symmetrically located in the vicinity of the extremum can be used to obtain an independent estimate of λ_m, the set of M data points around λ_m, results in M/2 estimates. The arithmetic average is closer to the ideal λm by a factor of (M/2)^1/2 compared to ∆λ/2. In case of a typical spectrometer with a resolution of ∆λ=0.5nm, the accuracy of a calculated spectral shift δλ can be as low as $\frac{\raisebox{1ex}{$\Delta \lambda $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sqrt{\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ or else ≤0.1Å for M=100. This is the case, of course, if the noise sources do not significantly change the symmetric pairing of the data points on either side of the extremum. For this to be the case, the noise sources in total should be less than b(∆λ)²=I_m/M², which is the smallest signal change next to the extremum. The noise sources include the shot noise of the signal, and the quantization error of the analog to digital converter (ADC) of the spectrometer. The latter is M_s/2ⁿ where M_s is the maximum signal that can be read by the spectrometer and n is the bit resolution of the ADC. If 2ⁿ> M², then the quantization error does not interfere. As far the shot noise is concerned, in the case we keep track of a maximum, the shot noise is (I_m)^1/2, provided the signal is expressed as the number of photons collected at the spectrometer channels per meaurement. Then, the condition I_m/M²>(I_m)^1/2 becomes I_m>M⁴ and applies for the performance to be limited by the spectrometer resolution only. In the case of a minimum, the signal at the minimum and, consequently, the shot noise is zero since ideally the sensing and reference arms are equal in magnitude and 180° out of phase. Near, but not at, the minimum the signal is finite and by applying the same upper limit to the shot noise at the minimum nearest points, where the signal is b(∆λ)², one arrives at I_m>M², which is a much more relaxed condition compared to the quadratic one at the maximum. It is advantageous, therefore, to employ a local minimum since the conditions on the shot noise are far less stringent. At the same time, and due to destructive interference, the position of the minimum is more or less independent of the spectral content of the light that is coupled into the waveguide. Hence, independently of the length of the FR-MZI sensing arm and the input source employed, ultra-high resolution measurements can be achieved.

A second very important result of Eq. (6) is that the sensitivity of the FR-MZI for specific biochemical detection can be increased by decreasing the denominator. The terms of the denominator depend on the differences of the effective refractive indices of the two arms and their respective derivatives. Minimizing the denominator can be achieved by proper selection of the core and cladding layer materials and their thicknesses upon which the values of the effective indices depend.

These two important points will be analyzed in more detail and explicitly demonstrated in the following sections.

3. Semi-analytical approximation of the FR-MZI spectral output

The general detection principle behind all integrated optical (IO) affinity sensors is the interaction of the evanescent field of the waveguided modes with the sample medium. Local changes Δn(z) of the refractive index of the sample n(z) –where z is the distance from the waveguide- result in effective refractive index change δN. This quantity δN is translated into readily measurable output signals in different types of IO sensors, such as changes in the output intensity as in the case of SW-MZI or spectral shifts δλ as in the case of FR-MZI (see previous section). N for planar waveguides depends on the polarization of the guided modes the wavelength λ, the thickness of the waveguiding film t, as well as the refractive indices of the substrate n_sub, the waveguiding film n_wg, and the covering medium n_cov.

Equation (6) includes the effective refractive indices of the sensing and reference arms and their derivatives with respect to the wavelength as well as the so-called in literature “optical sensitivity” $\raisebox{1ex}{$\partial N$}\!\left/ \!\raisebox{-1ex}{$\partial \alpha $}\right.$. The magnitude of the produced shift δλ due to a change of the refractive index δΝ for each wavelength and whether it can be measured constitutes the figure of merit for the FR-MZI. In the following semi-analytical approximation, it will be shown that for an IO FR-MZI no stringent selection rules for the geometrical characteristics of the interferometer are necessary, and that it is possible to create a generic parametric design for a low-cost highly-sensitive miniaturized affinity sensor that can be applied interchangeably to various applications.

The detailed analysis of each term of Eq. (6) is as follows:

In general, the effective refractive index changes δΝ in an affinity sensor can be induced mainly by two different effects:

If both effects are simultaneously present the resulting δΝ can be expressed as:

Two methods have been used in literature to obtain analytical expressions for the “optical sensitivities”. The first method [29] employs the exact mode-guiding condition for a planar two-layer waveguide consisting of the waveguiding film of thickness t and refractive index n _{w g}, and an adsorbed or bound adlayer of thickness d and refractive index n_film sandwiched between a substrate and a cover medium of refractive indices n_sub and n_cov, respectively, and then considers the limit of a very thin layer d<<t. The second method [30] applies the perturbation theory to a planar waveguide with any transverse refractive index distribution. Both methods yield the same results, albeit the second one provides more physical insight. The obtained results for $\frac{\partial N}{\partial n_{\mathrm{cov}}}$and $\frac{\partial N}{\partial d}$are summarized in the formulas below as shown in [30]:

In order to substantiate the high sensitivity and versatility of an IO FR-MZI, the following immunoassay experiment was emulated based on typical literature data. The IO FR-MZI consisted of a silicon nitride (Si₃N₄) ridge planar waveguide with a 2.5nm-thick and 2μm-wide ridge sandwiched between silicon dioxide (SiO₂) cladding layers (Fig. 1b), while the length of the sensing arm was intentionally set to 300μm, a value well below than the mm-long sensing arms of typical SW-MZIs. Such a design can be easily produced with standard microfabrication techniques [23, 25, 28] and allows for single-mode TE₀₀ operation within the VIS-NIR part of the spectrum (450-1200nm). For the simulation purposes, the sensing arm was assumed to be exposed to water and the Cauchy parameters for the calculation of the water’s refractive index were extracted by the experimental data of [31]. The simulations were based on the three-dimensional beam-propagation method (3D-BPM) and have been carried out with commercial software (BeamPROP^®, RSoft Design Group, USA). The thickness of the waveguide was used as a free parameter ranging from 70 to 300nm, and the simulations were performed for a wavelength range between 450 and 1200nm.

A standard immunoassay experiment has in general 4 distinct steps: (1) introduction of a buffer solution (usually Phosphate Buffer Saline or PBS), (2) coating of the sensor surface with a monolayer of probe molecules (an antigen/antibody), (3) introduction of a blocking solution to prevent unspecific binding, and (4) introduction of the target molecules (the corresponding antibody/antigen) within the blocking solution that will attach themselves to the coating adlayer augmenting its thickness. Each step alters the refractive index causing a spectral shift for every wavelength and changing the output spectrum. Considering step (1) as the baseline, the introduction of the coating solution alters the refractive index over the sensing arm of the FR-MZI through the formation of the binding adlayer. Usually, the concentration of the coating solution is in the order of a few μg/ml and does not affect the refractive index of the cover medium in a measurable way compared to the change due to the adlayer formation that dominates as a term in Eq. (7) (in [32] it is argued that for aqueous protein solutions the linear dependence of the refractive index on concentration c for a large range of concentrations is dn/dc=0.18ml/gr, therefore in the case of the c~1μg/ml δn is less than 10⁻⁷ and then δΝ is less than 10⁻⁸). On the contrary, during the blocking step, the introduction of the dense blocking solution (~1-10mg/ml) causes changes in n_cov comparable to the changes induced by the increase of the effective thickness of the coating adlayer because of the filling of empty spots with molecules from the blocking medium. Finally, during the last step of the immunoassay a new change in n_cov and consequently of the final output spectrum is caused by the antigen-antibody binding and the augmentation of the adlayer. For this gedanken affinity experiment, N_s was calculated for the 450 - 1200nm spectrum for all four steps through Eqs. (8) and (9) using as the initial value of N_s the results of the 3D-BPM simulations for water. In order to calculate N_s in the presence of PBS, a difference of δn_cov=0.00167 compared to water was assumed based on [32]. For the second step, a uniform coating adlayer of 4nm effective thickness with a refractive index of n_film=1.45 was further assumed [32, 33]. The introduction of the blocking solution was assumed to alter n_c by δn_cov=0.00019 with respect to the buffer solution, and to increase the coating adlayer effective thickness by δd=1nm. Finally, the binding event was assumed to increase the effective thickness of the adlayer by δd=0.5nm. The effective refractive indices N_s calculated through Eqs. (7)-(9) were used in Eqs. (1) and (2) to calculate the anticipated spectra of two IO FR-MZI with Si₃N₄ thicknesses of 100 and 150nm (Fig. 2 ).

 

Fig. 2 Output spectra of simulated IO FR-MZIs with a (a) 100nm-thick and (b) 150nm-thick ridge waveguide for a hypothetical immunoassay experiment. Binding of 0.5nm of target molecules on a uniform 4nm coating adlayer with n_film=1.45 was assumed. Four distinct steps of the immunoassay are presented: introduction of the PBS buffer solution (circles), the coating of the sensing arm with a 4nm-thick adlayer of n_{f ilm}=1.45 (trangles), introduction of the blocking solution and augmentation of the coating adlayer by 1nm (squares), and binding of the target molecules that increases the effective thickness of the adlayer by 0.5nm (diamonds). Insets: magnifications of the outlined regions.

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In Fig. 2a spectra, a sinusoidal behavior is observed for all immunoassay steps except for the 600-700nm wavelength region approximately where this behavior collapses. In this region, the interference pattern disappears and large variations of the output intensity take place. For wavelengths shorter than 600nm the monitored interference patterns are red shifted with the augmentation of the adlayer effective thickness, while for wavelengths longer than 700nm blue shift is observed. For this reason this spectral region was named transition region. This transition region is a characteristic of FR-MZI detection principle, and its position depends on the waveguide thickness, see for example Fig. 2b. The usefulness of the transition region and the exploitation of the various shifts (blue or red) will be analyzed in the next section.

4. Sensitivity and limit of detection of FR-MZI

From Fig. 2, it can be easily seen that regardless of the waveguide thickness, an effective thickness augmentation by 0.5nm is readily detectable with a FR-MZI since it results in shifts of the order of 10-20nm between the blocking and binding spectra (insets Fig. 2). From the parabolic fit at local minima, it can be derived that the FR-MZI can detect changes of the effective thickness of the binding layer down to 0.025Å (which is in essence translated to being able to detect single-molecule layers or solutions of very small molecules in ultra low concentrations). The limitations of the method are mostly imposed by the resolution Δλ of the spectrometer.

From Eqs. (8) and (9) and the discussion on the perturbational theory of [30], it is established that an important factor determining the sensitivity of a conventional SW evanescent-field sensor is the waveguide thickness. Both “sensitivities” are inversely proportional to the effective thickness of the waveguide, indicating that thinner waveguides can provide higher sensitivity. The physical reason is that thinner waveguides strongly confine the transverse field distribution of the guided waves resulting in high field strength of the “probing” evanescent field. Equations (8) and (9) can be used to calculate directly the effective refractive indices of the sensing arm due to any change δα of the analyte as well as to calculate analytically the numerator of Eq. (6).

The ridge waveguide used in the gedanken immunoassay of the previous section can be approximated to first order by a slab waveguide; that way N can be analytically expressed through the following transcendental equations for TE₀₀ and TM₀₀ modes respectively [34]:

By taking the derivatives of Eqs. (12) and (13) with respect to λ_ο and employing Eq. (14), one ends up with analytical expressions for the derivatives dN_i/dλ for TE₀₀ and TM₀₀ (where the subscript “o” has been omitted for simplicity):

 

Fig. 3 Spectral shifts (circles) as obtained through the analytical expressions of Eqs. (6) and (16) in conjunction with the anticipated spectral difference (squares) of the semi-analytical approximation for the binding of 0.1nm of adlayer on a (a) 100nm-thick and (b) 150nm-thick (open symbols) IO FR-MZI with 300μm-long sensing arms.

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Fig. 4 Anticipated spectral shifts after the binding of 0.1nm of protein with n_film=1.38-1.56 for a 100nm-thick FR-MZI for the emulated immunoassay experiment

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The existence of the transition region can be now explained in physical terms with the aid of Eq. (6) and (16)-(17). For a conventional SW-MZI the sensitivity can be increased by decreasing the thickness of the waveguide; for a FR-MZI the sensitivity (i.e. the spectral shift) can be increased by decreasing the denominator in Eq. (6). The denominator is the difference of the effective refractive indices of the sensing and reference arms and their first derivatives with respect to the wavelength and the physical quantity α. For relatively short wavelengths the effective refractive index of both arms is dominated by the index of refraction of the core material, hence ΔΝ → 0. On the contrary, the absolute values of the derivatives are large and the term Δ(∂N/∂λ) dominates over ΔΝ. Since ∂N_s/∂λ < ∂N_r/∂λ ∀λ, δλ is positive, therefore the spectrum for short wavelengths is red shifted. For longer wavelengths (i.e. by moving closer to the cut-off wavelength) the absolute value of the derivatives becomes increasingly smaller, hence the term Δ(∂N/∂λ) → 0 and ΔΝ<0 dominates and consequently the spectrum is blue shifted. For intermediate wavelengths the two terms of the denominator become comparable and the denominator is close to zero. Mathematically, this leads to infinite shifts; practically, this is the transition region where the red shift turns to blue shift and intense variations in the transmission efficiency occur as illustrated in Figs. 3 and 4.

As a result, when designing an IO FR-MZI the increase of the sensitivity is quite straightforward and is achieved through operation close to the transition region. It was established that irrespectively of the waveguide thickness and for most affinity sensing schemes where aqueous solutions are used (n_film~1.33-1.35) and the refractive index of the adlayers is between 1.38 and 1.56, the spectral shifts for any FR-MZI are within the same order of magnitude and that limit of detection is an effective adlayer thickness of 0.025Å. Therefore for operation close to the transition region one needs to select the core and cladding layer materials and their thicknesses in such a way that the transition region falls close to the maximum of the emission spectrum and of the response curve of the available input source and detector, respectively. This is illustrated in Fig. 5 where the anticipated shifts (per nm of bound target molecule) for waveguides of t ranging from 60 to 300nm are shown for four different wavelengths (500nm, 700nm, 850nm and 1000nm).

 

Fig. 5 Spectral shifts per nm of bound molecule as a function of the FR-MZI thickness for four wavelengths (λ=500, 700, 850 and 1000nm)

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In reverse, in case one has a specific finalized design of an IO FR-MZI, the choice of the appropriate input source becomes straightforward. It can be deduced from Eq. (6) (and taking into account Eqs. (8) and (9)) that the transition region for a waveguide of thickness t is close to

since, for N→n_wg, $\lambda \raisebox{1ex}{$\partial N_i$}\!\left/ \!\raisebox{-1ex}{$\partial \lambda $}\right.\approx 2(N_i-n_{wg})$ (i=s or r), and

Using the approximations of Eqs. (19) and (20) and by employing the effective refractive indices of the two MZI arms for any IO FR-MZI design in question (assuming that the sensing arm is exposed to water for quicker calculations), one can readily set the limits of detection for a planned affinity experiment. From the approximations of Eqs. (19) and (20) it can also be seen that for the entire VIS-NIR range the minimum adlayer thickness that can be detected is given

5. Conclusions

IO Frequency-Resolved MZI overcomes all limitations of conventional Single-Wavelength MZI associated with phase ambiguity and signal fading. Moreover, the need for long SW-MZI for improved sensitivity and low limit of detection is resolved by employing spectral analysis. The IO FR-MZI provides high sensitivities (in terms of detectable spectral shifts per adlayer thickness) in any part of the VIS-NIR spectrum regardless of the design and the specifics of the affinity experiment. This means that any broad-band source within the VIS-NIR region can be used and that the geometry of the interferometer or the targeted application does not affect the selection of the input source. Finally, the performance and the limit of detection of any FR-MZI for any sensing scheme can be predicted by simple design rules as expressed by the set of Eqs. (19)-(21). Inversely, for an optimized optical design well below 1mm in length the limit of detection can be as low as 0.025Å in terms of adlayer effective thickness -in contrast with the standard SW-MZI- allowing for truly miniaturized integrated optical sensors fabricated with high yield with standard microfabrication techniques.