## Abstract

This paper reports on the finding of a critical working point in the sensitivity of hetero-modal interferometric optical sensors using spectral interrogation. At this point the theoretical sensitivity approaches infinity and the practical sensitivity will depend only on the measurement accuracy and noise sources present. If the critical condition is attained at a point of minimal power transfer, a phenomenon of splitting or bifurcation of the minimum dip is expected as sensing occurs. The conditions for attainment of these critical effects are discussed.

©2008 Optical Society of America

## 1. Introduction

Optical evanescent-wave sensors in various forms have been exploited for highly sensitive chemical and biological sensing. Of these sensors, optical interferometric sensors utilize differential phase-based detection for high sensitivity sensing. These sensors require a reference wave to mix with the signal in order to translate phase changes into detectable intensity changes. The reference can be provided in different ways with the most common one being the dual-arm arrangement used in a Mach-Zehnder configuration [1][2]. Both signal and reference may also share a common physical channel, while propagating in different modes: single channel multi mode sensors have been reported in planar waveguide [3–5] and fiber-optic configurations [6–7]. Different detection methods for optical sensors have been used including intensity, phase, coupling angle, and spectral interrogation. Of these methods, spectral interrogation sensors use a broad bandwidth light source or a tunable laser, and detect variations in the sensed material by measuring changes in the wavelength dependence of the transfer power function induced by the sensing process. This method is widely used with many optical sensors configurations such as prism coupler-based SPR sensor [8], grating-based sensors [9], waveguide SPR sensors [4][10], fiber sensors [7][11], and unbalanced Mach-Zehnder based sensors [12].

This paper deals with spectral interrogation of *hetero-modal*, interferometer-based sensors defined as sensors where the propagation paths are realized within different modes, different in the sense that their dispersion properties (i.e. the dependence of the propagation constants on wavelength) are different. An approximate analytical and a numerical approach are presented for the evaluation of the sensitivity of these sensors with a formulation of the critical working point where a divergence in spectral sensitivity is expected. Results show that this working point occurs at the wavelength of the peak in the differential phase curve (defined as the difference in propagation constants of the interfering modes multiplied by the corresponding geometrical lengths) provided that such a peak exist.

The power transfer function of such a hetero-modal sensor is shown in Fig. 1 and examples of physical realizations of such sensors are displayed in Fig. 2. In the bottom graph of Fig. 1, a normalized differential phase curve (divided by *π* for integer normalization) of a sample hetero-modal waveguide structure is shown. The differential phase line is calculated for two values of sensed material refractive index, with a peak around λ=656nm. The top graph in Fig. 1 shows a remarkably enhanced wavelength shift Δλ near the differential phase peak. As seen in the graph, at this critical working point the power transfer exhibits not only a shift as in other sensors, but also a split of the minimum into two. A numerical example for a Mach Zehnder sensor, Fig. 2(A), shows that for nominal wavelength discrimination levels of 0.0025nm [10][11], a theoretical resolution of 10^{-12} RIU (refractive index units) can be reached for sensed material in aqueous environment around λ=656nm. Similar results for a single channel integrated sensor lead to theoretical resolution of 10^{-11} RIU.

## 2. Analytic and numerical examination of hetero-modal interference sensors

#### 2.1 Analytic calculation of the sensitivity and determination of the critical working point

In spectral interrogation sensors, the output power is detected for different values of the source wavelength λ at a given range. Spectral shifts are then detected following small changes in the surrounding parameters that change the mode propagation. For the specific example of measuring a variation in the refractive index of cover bulk media, the sensitivity is defined in units of nm/RIU by:

Where Δn_{c} is the bulk change in the sensed material refractive index (top layer) and Δλ is the wavelength shift at constant output power induced by this change. For an interferometer sensor in which the light propagates at the two paths within different modes, and assuming that the amplitude of the modes is invariant to changes in wavelength at the vicinity of the working point, the peaks (minimum and maximum) in transfer power are given by:

Where the phase difference function Φ was defined. Here *L* is the length of the two arms of the interferometer, λ is the source wavelength, and *η* is the refractive index difference between the propagating modes, namely *η=n _{eff-i}-n_{eff-j}*, which is a function of the wavelength and of the cover layer refractive index

*n*. An odd value for N will correspond to a minimum, and an even value to a maximum. Using implicit function differentiation and assuming

_{c}*∂Φ(λ,n*at the working wavelength λ, the sensitivity (1) can be expressed as a function of

_{c})/∂λ≠0*η, λ*, and

*n*under the constraint of constant phase difference Φ to give:

_{c}Examination of (3) shows the possibility of a divergence in the sensitivity, i.e. the sensitivity can become infinity when the denominator in (3) approaches zero. Explicitly, this condition is reached at a critical working point where the differential phase *Φ(λ,n _{c})* has a peak with respect to wavelength changes. This peak occurs at:

This critical condition defines a *critical wavelength λ _{critical}* for a given value of

*n*. This critical condition (4) is independent on the value of

_{c}*N*, and holds even if the phase difference function (equation 2) does not equal

*Nπ*. If in addition, the phase difference does obey equation (2), namely

*η=λ*then the extremum condition in power transfer will hold for both measurements before and after the change in

_{critical}N/(2L)*n*, and a peak or dip

_{c}*splitting effect*will take place as shown in Fig. 1.

In order to evaluate this effect, explicit approximate expressions for the spectral sensitivity close to the critical point are derived in the following. Since the first derivative at the critical point cancels, the approximation is based on replacing the actual phase difference curve by a parabola with the same curvature. In addition, the phase difference function is assumed to be linear to changes in the refractive index *n _{c}* at the working point vicinity. These two assumptions lead to the definition of the following constants:

Based on these definitions and assumptions, the sensitivity for a working wavelength *λ≠λ _{critical}* can be evaluated using equation (3) to give:

Or in terms of *n _{c} (n_{c}≠_{ncritical})*:

Where *n _{critical}* is linked to

*λ*by the requirement of unchanged reference phase.

_{critical}The sensitivity at the critical working point (*λ=λ _{critical}, n_{c}=n_{critical}*) can be calculated non differentially, using equations (1) and (2), leading to the same results as in equation (7) without the ½ factor (the difference is due to the non differential calculation).

If the phase function undergoes an additional change *Δφ*, due to other mechanisms, the expression for the sensitivity will be:

If *Δφ* has a predictable value, the residual phase will just cause the re-evaluation of *λ _{critical}*. On the other hand, if

*Δφ*has time fluctuations or noise, this noise will set the ultimate limit in sensitivity of the sensor.

Higher order approximations for the spectral sensitivity were also derived leading to similar results.

#### 2.2 Generalization to other sensor configurations

This section remarks that the enhanced sensitivity effect is generic, and can be extended to a diversity of sensor kinds and sensed parameters. The most fundamental aspect of this analysis is the attainment of a critical situation where the phase-difference function *Φ* attains an extremum as a function of the wavelength (equation 4 above or 9 below). At this situation the theoretical limit of the sensitivity to changes in an additional parameter affecting the phase may become infinite. The same effect will therefore take place in other optical interferometric sensors using spectral interrogation, provided the following general equation is fulfilled:

Where *p _{1}, p_{2}, p_{3}*…are additional parameters on which the phase depends. Each of these parameters may be measured with enhanced sensitivity if the condition formulated in equation (9) is fulfilled. Examples of such parameters are: geometrical dimensions, refractive indices, temperature, pressure, and external magnetic or electric fields eventually affecting the interferometer’s phase difference.

#### 2.3 Numerical examples

Several different sensor configurations have been investigated showing a divergence in sensitivity. Three specific examples are depicted in Fig. 2 and are briefly analyzed below. In the first example, a single mode Mach Zehnder interferometer is designed based on SiO_{2} as substrate and cover layer (n=1.457), and Si_{3}N_{4} as guiding layer (n=2.0) with a thickness of 100nm. Differentiation in propagation properties between the two arms is realized by removal of the cover layer at the sensing section, as shown in Fig. 2(A). This configuration is very common for sensing, and is usually operated at the intensity or power interrogation mode. Spectral sensitivity was found by calculating the power transfer through the waveguide for different values of λ (560–700nm) and for two sensed material refractive indexes: n* _{c}*=1.33 and n

*=1.3301 (planar waveguide approximation was used, and a single polarization — TE - was assumed).*

_{c}The transfer power and normalized differential phase *2LΔn _{eff}/λ* (π was omitted to show the interference order N) for a sensing section length of 4.04mm are shown in Fig. 1. Maximum spectral sensitivity is reached at the critical working point when the minimum output power matches the peak of the differential phase line as indicated by (4). Calculated sensitivity, using (1), for this working point is above 200,000 nm/RIU for

*Δn*=10

_{c}^{-4}RIU and above 600,000 nm/RIU for

*Δn*=10

_{c}^{-5}RIU. For sensed material in aqueous environment,

*n*=1.33, calculated values for this waveguide structure are

_{c}*λ*=657nm,

_{critical}*α*=6*10

^{-11}RIU/nm3, and

*β*=0.141. Using these values, equation (7), and quoting from the literature reported wavelength resolutions of 0.0025nm [10][[11], a theoretical resolution of 2*10

^{-12}RIU can be predicted. The divergence in sensitivity is visualized in the graphs of Fig. 3, where the wavelength shift (right) and sensitivity (left) are plotted as a function of the departure from critical conditions. The validity of the parabolic approximation for small values of

*Δn*is also displayed in both graphs.

_{c}Critical conditions may be also attained in a channel waveguide Mach Zehnder interferometer sensor where the sensing section in one arm is wider in the Y-axis (lateral) aspect. Unlike other examples, in this structure, the differential phase line has a minimum instead of a maximum at the critical wavelength.

A second example uses a single integrated optics waveguide structure supporting two guided modes in the sensing section as shown in Fig. 2(B). A comprehensive analysis of such a sensor was reported in reference [3]. Calculated values for this sensor are *λ _{critical}*=616.2 nm, α=1.85*10

^{-9}RIU/nm

^{3}and β=0.119. For these values, using equation (7), and for reported wavelength resolution of 0.0025nm [10][11], a theoretical resolution of 6*10

^{-11}RIU can be predicted.

The third example is based on a free-space (non-waveguided) configuration. It consists of a Michelson interferometer sensor with a stack of layers at each arm, Fig. 2(C). The layers are slightly different at the two arms providing the required non-linear dispersion for the fulfillment of condition (4). The condition can be fulfilled by stacks of 5 repetitions of two 80nm width layers: n_{1}=1.8, n_{2}=2.2 at one arm, and n_{1}=1.81, n_{2}=2.2 at the other arm. Changes in one of the layers types at one of the arms – i.e. *n _{1}*=

*1.8*+

*Δn*– results in a split in the output power and a divergence in sensitivity (the predicated theoretical resolution is similar to one received in the second example above).

## 3. Further considerations

The sensitivity near the critical working point is independent of the length of the sensor (equation 6,7), assuming both arms have the same length. However, if in addition the two modes are required to arrive with a definite phase difference at the end of the sensing section (e.g. Nπ in order to observe the splitting effect), the length of the sensing section needs to be accurate (equation 2). This implies a relatively demanding fabrication constraint on the optical length which can be greatly simplified by active control of the differential phase between the two guided modes (e.g. actively changing the phase at one of the arms of the MZI sensor).

The two interfering arms may have different lengths *L _{1}≠L_{2}*, and the refractive index difference can then be defined as

*η*, with

_{L}=n_{eff}-i - (L_{2}/L_{1})n_{eff-j}*L=L*at the phase difference definition (2) and the same analysis applied using

_{1}*η*instead of

_{L}*η*. The ability to change the length ratio of the interferometer arms adds further flexibility in design parameter that can be used to adjust

*λ*.

_{critical}Sensor calibration is required as the sensitivity of the sensor around the critical working point is not linear. According to equation (6), the sensitivity of the sensor close to the critical point has a functional form of *C*/(*λ _{critical}-λ*) where

*C*is a constant. Two wavelength scans at different values of nc near the critical point will furnish the required calibration constants:

*λ*and

_{critical}*C*.

As final consideration it is noted that the condition for attainment of enhanced sensitivity (Equations 4 and 7), do not depend explicitly on the specific value of the phase difference at *λ _{critical}*. If Equation (2) does not hold at

*λ*, and the critical phase is set to a different value (e.g.

_{critical}*π*/2), the power transmission will still have an extremum point at that phase value, the splitting effect will not occur but the wavelength shift will still be enhanced. This shift will be expressed in the output power as a widening or narrowing of the peak. Working off the minimum transmittance point, (e.g. at higher power levels), enables to perform measurements at a better signal-to noise conditions.

## 4. Discussion and Conclusion

This article presented a wavelength interrogated hetero-modal interferometer sensor. At suitable conditions, this sensor shows sensitivity values much higher than other reported sensors and may be used for applications requiring high sensitivity. The necessary conditions to attain the enhanced sensitivity were discussed, including a peak or dip splitting effect which can be induced by sensing. Approaching the critical point, the sensitivity will be limited only by the presence of noise. The theoretical resolution for all interferometer examples discussed here predict very high values (better than 10^{-11}RIU) for reported wavelength resolution (0.0025nm). Phase modulation, frequency modulation, and numerical extrapolation techniques may be further used to enhance measurement accuracy and increase sensitivity.

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