In this paper we present a novel approach to temperature sensing with optoelectronic devices which relies on the usage of bare silicon as the transducing material. The device is composed by a single mode input waveguide, a MMI region where a number of higher order modes is also allowed to propagate and two output waveguides. The refractive index variation in the MMI section due to temperature shifts induces different phase velocities of the various propagating modes. The position of the input and output waveguides together with the length and width of the MMI section are chosen in order to maximize the sensitivity of the device. Analytical calculations are presented together with BPM simulations aimed to the maximization of the sensitivity of the sensor as a function of its geometries.
© 2003 Optical Society of America
Temperature sensing with optical devices is a very promising research field because of many attractive features common to all-optical sensing schemes, as immunity to electromagnetic interference (EMI) and the hardness in dangerous environments. Many recent works have presented optical sensors made with fiber optic [1–2] and mostly important have appeared those based on Bragg gratings  where the temperature information is encoded into a wavelength shift of the transmitted or reflected Bragg spectrum. This classes of sensors, though highly sensitive and easy to embed within the environment under test calls for specialized readout schemes [4–5] and it cannot be integrated with other electronic circuits needed for calibration and or compensation. On the other hand, all-silicon integrated sensors have many interesting features from their inherent low processing cost to integrability with signal-processing electronics. In a previous paper  we have presented an all-silicon temperature sensor based on a bi-modal Y branch or, in other words, on two-mode interference. We demonstrated through optical, thermal and electronic simulations that the approach, though valid in its theoretical background, did not take into account the possible improvement of performance due to the interference of high order modes and geometry optimization. So the aim of this paper is to investigate numerically the effect of Multi Mode Interference (MMI) on the sensitivity of integrated optical temperature sensors. In the first section of the paper, for completeness sake, we will present some theoretical results of MMI theory in order to clarify the assumptions made in the rest of the paper; afterwards the analytical/numerical approach is presented and finally the results are reported and discussed.
2. Theoretical approach and description of the device
A schematic of the proposed device is reported in Fig. 1. It is composed by a single mode input waveguide (1), a MMI region (2) where the higher order modes are allowed to propagate and two output waveguides (3). The geometries, such as the position of the input and output waveguides, are to be chosen in order to maximize the sensitivity of the device as it will be shown in the next section. All the waveguides are made by silicon-on-silicon technique, that is that a low doped silicon epilayer layer is grown onto a highly doped (or implanted) substrate which acts as a lower cladding due to the refractive index reduction induced by the dopant . Numerical simulations indicate that A substrate doping ND=2E19/cm3 guarantees optical vertical confinement without increasing optical losses which remain confined under 3dB/cm.
The first part of the design concerns the choice of the dimensions for the input/output waveguide and for the active region. Effective Index Method can be exploited  to study the performances of such kind of waveguides and, if particular care in the definition of the geometry of the input waveguide is taken, a single mode operation can be assured. In particular, if the dimensions of the waveguide fulfill the following relationship: where t=weff/Heff, r=heff/Heff, heff=h+q, Heff=H+q, , , nf, ns and nc are refractive indices of the guiding region, the substrate and the air respectively, H and W are the height and width of the optical waveguide and γc,s can be chosen according to the polarization of the propagating light .
Moreover, in order to guarantee an optimal coupling with an input optical fiber, a square optical channel has to be designed. This choice will be good also for both the output waveguides since light has to be coupled again into optical fibers for readout. Regarding the active region, the rib height will be the same of the output and input waveguides while the length and the width of the MMI zone will be parameter to be optimized for our purposes.
Since multi-mode interference is a function of the refractive index of the waveguide, temperature induced refractive variations result in variations of the relative phase shift between the propagating modes and hence of the output at the two exits of the device. The temperature induced refractive index variation will also shift the optical field inside the input and output waveguides but this will not affect the behavior of the sensor since this waveguide allow the propagation of the sole fundamental mode. In order to ensure a very high sensitivity, light coming out the devices is sent to two photodiodes which directly feed the input of a differential amplifier which can be tuned in order to have zero output at the reference temperature and therefore be extremely sensitive to very small temperature variations. Due to the all-silicon nature of the device, this can be realized on the same silicon substrate. In order to quantify the assumptions made so far in the next section we briefly recall the basic principles of MMI devices.
3. Analytical model and calculations
Selt-imaging is the working principle of MMI devices and it has been proposed more than 150 years ago  and its application to slab waveguides can be found in Ref. [10–12] and it can be expressed as follows: “Self-imaging is the property of multi-mode waveguides when the input filed profile is reproduced in single or multiple images which appear periodically along the propagation direction”. In this paragraph we recall the basic principles of MMI theory while we refer to Ref.  for all the explanations and subtleties.
An arbitrary input field (see Fig. 2) Ψ(y,0) totally confined in Wm, can be seen as the superposition of the guided modes inside the waveguides:
where ψν(y) are the waveguide eigenfunctions and, bearing in mind the ortogonality of the eigenmodes, the coefficients cν can be expressed by the overlap integral:
If the input field Ψ(y,0) does not excite radiative modes it can be decomposed in terms of the m guided modes only:
and after a distance z it becomes:
So, in some conditions, the field at the abscissa z will be equal to the input field. Now the relationship between the propagating modes depends on the refractive index of the waveguide. This, on its turn, if an appropriate guiding material is chosen, can be strongly dependent on the temperature, therefore, by carefully optimizing the geometries of the device, a very accurate sensor can be designed. In our case, thermo-optical effect in silicon causes a positive temperature induced refractive index variation according to:
and this variation can be recorded at the output of the device. In what follows some details on the simulations strategy will be given and the results presented.
4. Simulations’ results
In order to correctly foresee the behavior of a temperature optical sensor, both thermal and optical simulations have to be performed. Regarding thermal simulations, state of art Finite Elements analysis using Ansys© has been carried out while, for optical simulations at this point the choice is twofold: we can use one of the widely available commercial optical simulators (i.e., BeamProp, FimmWave, Kymata, etc.) but in this case, an optimization procedure over so many parameters would have been tedious and lengthy, or use analytical considerations and in-house numerical software. This has turned to be an extremely powerful choice in terms of computational load and scalability of the solutions. So a code has been written in order to optimize the sensitivity of the device to the temperature variations. The parameters taken into consideration were the waveguide width w, the position of the input waveguide y0, the position of the two output waveguides y1 and y2 and the device length L; the results obtained are compared to those of our previous proposals [6–14] to show that a multi-mode approach can produce higher sensitivity and flexibility of design. The comparison are made for a given temperature range of 100 °C. We recall that the best results achieved for the previous configuration  was a sensitivity given by:
Let us now compare the performance of a MMI device. If we define I1 and I2 as the light intensity coming out of the two output waveguides, we can define the sensitivity S over a temperature range ΔT as follows:
Therefore the goal of our analysis is to maximize the function S by using a suitable minimization strategy of the function f. This can be achieved by a number of different algorithms (Nelder-Mead, Levemberg-Marquardt), which are of common use in numerical analysis. MATLAB code has been written in order to simulate the behavior or the sensor taking into account the variation of all the aforementioned parameters. In order to avoid some of the problems arising (i.e., local minima) in the minimization of multi-variable functions it has been found convenient to narrow the intervals where the variable could move. In Table 1 the constraints used in all the simulations are resumed.
In Fig. 3 a first result of optical simulations is reported. As it is possible to note, multiple images are formed and different modal content. After an initial minimization procedure it has been found that increasing the waveguide width to guide the 5th mode does not introduce improvement to the sensitivity because of the energy lost at the two output waveguides. We therefore aimed our search to waveguides sustaining 3 to 5 higher order modes.
Finally the minimization of the function f in terms of all the other parameter has been performed and the overall transfer function of the sensor is reported in Fig. 4 where the output of the sensor is reported as a function of temperature and it is compared to a sensor designed to sustain only the first two propagating modes. In both cases it can be noted the good linearity over a broad temperature range while, for the MMI sensor a better sensitivity to temperature variation is obtained.
5. Conclusions and future trends
In this paper we have presented an analysis of an all-optical, all-silicon temperature sensor based on Multi Modal Interference. Numerical simulations on the device have shown that good care in the design of its geometrical feature through an optimization procedure can lead to a better sensitivity respect a bi-modal sensor and good linearity. The strategy presented is not limited by the choice of the host material and can be applied to materials other than silicon provided that the temperature dependence of refractive index is given.
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