In this paper, we investigate the effect of non-uniformities (enlargement of current passage, non-equal surface current densities, etc.) in axial as well as transverse directions of a porous silicon Fabry-Perot (FP) cavity as well as loss nature of bulk silicon on spectral properties of this cavity, even that cavity is created with an anisotropic etching process. Without correct and comprehensive characterization of such cavities by incorporating these non-uniformities and inherent lossy nature of a cavity, detection and identification of biological and chemical molecules by that cavity may yield unpredictable and misleading results. From our simulations, we note the following two key points. First, effects of the refractive index and the thickness of microcavity region of a lossless or lossy FP cavity on resonance wavelength is more prevailing than those of first and last layers. Second, the effect of some small loss inside the FP cavity is not detectable by the measurement of resonance wavelength whereas the same influence is noticeable by the measurement of reflectivity. We carried out some measurements from two different regions on the fabricated cavities to validate our simulation results. From a practical point of view in correct detection and/or identification of lossy biological or chemical vapor by FP cavities, we conclude that not only the measurement of resonance wavelength as well as its shift but also the reflectivity value at the resonance wavelength or some specific wavelengths should be utilized.
©2012 Optical Society of America
In recent years, nanomaterials have been a core focus of nanoscience and nanotechnology - which is an ever-growing multidisciplinary field of study attracting tremendous interest, investment and effort in research and development around the world. Nanoporous materials as a subset of nanostructured materials possess unique surface, structural, and bulk properties that underline their important uses in various fields such as ion exchange, separation, catalysis, sensor, biological molecular isolation and purifications . Nanoporous materials are also of scientific and technological importance because of their vast ability to adsorb and interact with atoms, ions and molecules on their large interior surfaces and in the nanometer sized pore space .
Porous silicon (PSi) is one of the most promising materials for the above discussed tasks due to its very high surface to volume ratio, its integration with surface areas across a chip surface, and its versatile surface chemistry [2,3]. Moreover, it requires only a simple fabrication process, and its optical properties depend directly on morphological structure, making it possible to tune its refractive index just by means of pore diameter and/or the porosity [2,4]. Furthermore, PSi allows production of multilayers with large refractive index contrast, which makes it a promising material for photonic devices like Bragg filters and optical cavities . PSi can be fabricated through an easy and cheap process using electrochemical etching of crystalline silicon. Such a method gives ones the ability of having a wide refractive index contrast within the same material avoiding the problem of inter-diffusion between layers necessary for multilayered filters . Since the refractive index of PSi can be controlled by its porosity, a one-dimensional structure can be produced by periodically or non-periodically altering this parameter .
Photonic crystals (PhCs) are of great interest because they enable strong light concentration along waveguides or in cavities, as well as increasing light-matter interactions . Different kind of PhC devices like microcavities [2,4,7,8], Bragg mirrors , Rugate filters [10,11], and Fabry-Perot (FP) cavities [12–19] are available in the literature. Biosensing principles of these devices are based on the increase of the PSi refractive index due to the binding of biological or chemical molecules to the pore walls manifesting itself as a red-shift of the resonance peak or region in the reflection/transmission spectrum [4,15]. Therefore, an accurate model for analyzing a change in the effective refractive index is crucial. In , Dr. Ouyang et. al. investigated the effect of pore diameter on the refractive index with a simplified effective medium approximation. In addition, Dr. Yang and Dr. Jiang analyzed the effect of capillary condensation of vapors, a topic that has received little attention until recently, in the submicrometer-scale macropores . Furthermore, Dr. Suarez et. al. simulated the effects of silicon-dioxide and polymer on the effective refractive index by varying the pore diameter and porosity . Finally, Dr. Fan and his research team reported that sensing capacity of a PSi FP cavity can be greatly increased by coating a vapor sensitive polymer such as polydimetylsiloxane over the cavity [17–19].
In addition to studies for monitoring a minute change in the effective refractive index, researches on determining the accurate effective refractive index gain importance since these indices are associated with the composition of biological or chemical molecules poured on the PSi structure and since these indices are the key terms for fingerprinting of these molecules. Among these PSi structures, FP cavities [12–14] are much suitable to evidence variations of optical properties when exposed to vapors or immersed into organic solvents , because they are resonant structures with the applied optical wavelength . In the preparation of FP cavities, electrochemical anodic etching process is generally applied due to its superior advantages discussed above [4,5]. In this process, constant but different surface current densities over a specific time (depending on the desired resonance peak) are sequentially applied to crystalline silicon, thus creating essentially identical layers of high or low impedances on one another. As a consequence, the thickness and complex refractive index of each layer with a high (or low) impedance over the whole PSi FP cavity are assumed to be constant in the determination of effective refractive index of each layer after exposure of the cavity to biological or chemical vapor. However, enlargement of current passage (porosity region) of first created layers  as well as the presence of some unpredicted surface irregularities and impurities  may totally result in different properties (thickness and complex refractive index) to two (or more) layers although a constant current over a specific time is applied to these layers. In addition to this axial change in a FP cavity structure, the uniformity of a fabricated FP structure over transverse dimensions also deteriorates due to non-equal surface current densities, impurities inside the structure, and some fabrication tolerances. All these axial and transverse variations, although not much significant one by one, in turn can change the characteristics and spectral properties of PSi FP cavities in varying degrees through alterations in refractive indices and thicknesses of corresponding layers with presumably identical high or low porosity levels. Furthermore, to our best knowledge, researches on the detection of a biological or chemical vapor by PSi FP cavities concentrate on the shift in the measured resonance wavelength with the principal assumption that the fabricated FP cavity is lossless. However, any loss present inside a fabricated FP cavity due to the loss nature of bulk Si may alter cavity spectral properties. As a result, it is for sure that correct characterization of empty (no biological or chemical vapor) PSi FP cavities naturally becomes an important parameter for correct detection and identification applications of biological or chemical molecules.
In this research paper, we perform various simulations to investigate the effect of any change in the thickness and complex refractive index of different layers on the spectral properties (the reflectivity value at resonance wavelength and the value of resonance wavelength) of fabricated PSi FP cavities for their characterization before starting to detection and identification applications. The organization of the remainder of our paper is as follows. First, in Section 2 we give some background for understanding the underlying mechanism in the fabrication of FP cavities and then state the problem in their fabrication. Next, in Section 3, we present the transfer matrix method (TMM) for our simulation analysis, introduce simulation parameters, and thereafter illustrate and discuss the results of spectral property change of two different FP cavities due to layer thickness and/or layer complex refractive index. Then, in Section 4, we show and discuss our measurement results for validation of simulations in Section 3. After, we discuss how the simulation results could be incorporated with accurate detection and identification of biological or chemical molecules by PSi FP cavities, and mention the importance of any loss present inside the FP cavities on this detection and identification in Section 5. Finally, in Section 6 we give highlights of main conclusions in our analysis.
2. Fabrication of PSi FP cavities and statement of the problem
The PSi structures are generally fabricated by electrochemical anodic etching of a silicon wafer in a solution mixture with HF and ethanol [2,14,22]. Since the refractive index of a PSi (approximately) can be controlled by its porosity, one dimensional structure can be produced by periodically altering this parameter via mainly a change in the applied current. Depending chiefly on the time period, it is also possible to create layers of PSi structures with different thicknesses. Among the possible PSi structures, a PSi FP cavity  can be designed or fabricated in the following two ways. In the first way, a top Bragg mirror can be directly placed over a reversed identical bottom Bragg mirror. Adjacent layers of these mirrors form a microcavity region in which most of optical energy resides when the FP cavity is in resonance. In the second way, a defect layer is created between identical top and bottom Bragg mirrors. In this connection, presence of this defect layer forms a microcavity region with the same energy accumulation purpose. Each Bragg mirror has alternating layers of high () and low () relative refractive indices (or low and high porosity levels respectively).
In the above general procedure for the fabrication of PSi FP cavities, it is naturally assumed that fabrication process is ideal. That is, the refractive indices and thicknesses of layers with high (or low) porosity throughout the FP cavity are identical. However, enlargement of current passage (porosity region) of first created layers , the presence of some unpredicted surface irregularities and impurities , non-equal surface current densities, and some fabrication tolerances limit the ideality of any PSi FP fabrication even if it is created with an anisotropic etching process . Although these imperfections one by one seem not so serious, their overall impact should be taken into consideration in the characterization of fabricated PSi FP cavities. To demonstrate such imperfections in the fabrication process, we fabricated some PSi FP cavities with different structural and optical properties. These cavities are formed by using an anisotropic electrochemical anodic etching of a silicon wafer in a solution of HF (%40) and ethanol (%95) by a 1:2 ratio  in our laboratory. For example, Fig. 1(a) shows the refractive index distribution of one of the fabricated FP cavities with 10 periods of identical bottom and top Bragg mirrors (40 layers). Note from Fig. 1(a) that the position of the top Bragg mirror is opposite to that of the bottom one. Refractive indices and thicknesses of the FP cavity layers with low and high porosity levels are , , , and , respectively, and the thickness of the microcavity is where (nm) denotes the resonance wavelength. In the fabrication, current densities of 2 mA/cm2 and 75 mA/cm2 are, respectively, applied for 36.8 s and 9.9 s for the formation of layers with low and high porosity levels. More details on the fabrication process of this cavity can be found in .
Figure 1(b) shows a SEM image of our fabricated PSi FP cavity with nm for demonstration purposes of non-ideality in the fabrication of FP cavities. From Fig. 1(b), we note that the central large horizontal region corresponds to the microcavity region in Fig. 1(a) which has a thickness two times as large as that of the layer with high porosity. In addition, we note that horizontal thin regions just above and just below the microcavity region correspond to layers with low porosity levels. It is seen from Fig. 1(b) that the fabricated FP cavity has some small inhomogeneity in both its axial (vertical) and transverse (horizontal and in/out) directions. Similar observations are also seen from the SEM images in other studies [5,8,15]. Because a SEM image shows cross-sectional (longitudinal and transverse) geometry variations of a structure, and because variations in transverse dimensions of a SEM image are mainly related to applied current density distribution while those in axial dimensions in the same image are chiefly linked to the time the current densities are driven [2,21], we conclude that both the refractive indices and thicknesses of layers with high (or low) porosity concentrations throughout the FP cavity will not be identical. This remark will be elaborated quantitatively in subsection 3.2. This circumstance has motivated us to analyze the effect of any small change in refractive index and thickness of a layer with either high or low porosity level on the spectral properties of FP cavities. Such an analysis plays a vital role in the characterization of these cavities for accurate and correct sensing of biological and chemical molecules, to be discussed in Section 5.
3. Simulation results and discussion
In this section, we present simulation results for monitoring the change in spectral properties of a FP cavity for two different cases: 1) when the cavity is lossless, and 2) when the cavity is lossy. In both cases, for each layer of the cavity we first vary the parameter (or -complex refractive index) and keep constant and then vary the parameter and keep (or ) constant. For these analyses, a theoretical background is needed for realization of simulations as well as for appropriate interpretation of simulation results. In addition, a solid understanding of the simulation parameters when they change and when they do not is a key issue in a simulation analysis. Therefore, this section is divided into four subsections. The first subsection presents the well-known transfer matrix method (TMM) used in the theoretical analysis. The second subsection discusses the parameters and their properties utilized in the simulations. Finally, the last two subsections are for the discussion of simulation results when the cavity is lossless and when it possesses some loss.
3.1. The transfer matrix method
The TMM method can be utilized without loss of accuracy and generality in our theoretical analysis since this method allows an analysis for cascaded networks in a simple manner [20,23]. In addition, it is very diverse and suitably applied to different research fields such as analysis of engineering composite structured materials called metamaterials [24–26] and analysis of microwave networks . Because in this study, we consider the effect of any small change in the refractive index () together with the thickness () of a layer inside a lossy and also lossless FP cavity on the spectral dependence of the cavity, we let be a complex number. In this subsection, we briefly present the TMM method to better discuss the simulation results in subsections 3.3 and 3.4. Here, the analysis is restricted to dielectric materials, but can readily be transformed into an analysis for magnetic materials as well.
To investigate the reflectivity spectrum of a FP cavity with layers (we note that in our analysis, we use the number of layers terminology instead of period of a FP cavity since our main concern is to analyze the effect of refractive index and thickness variations of each layer), for each reciprocal layer or cell, the following TMM elements (it is noted that in some analyses, the elements of a TMM cell are named as the parameters ) can be usedEqs. (1) and (2), it is assumed that the time dependence is in the form of , and the mode of incident wave is the transverse electric (TE) mode to the longitudinal direction of the cell (normal incidence) with linear polarization. In addition, is the relative complex refractive index; is the layer thickness; is the real part of the relative complex refractive index associated with energy storage, and is the imaginary part of the relative complex refractive index linked to power dissipation ; is the phase constant in free-space or vacuum. The negative value of the imaginary part of complies with the electromagnetic properties of a passive medium .
The overall transfer matrix () of a FP cavity composed of multiple layers is obtained by multiplying the transfer matrix of each layer in the order they appear. Once upon computing , the reflectivity can be determined from [20,23]
Because, like any resonant structures such as RLC circuit, microwave cavity , and metamaterials [23–26], the advantage of a PSi FP cavity is its frequency (or wavelength) selectivity, sensing features of a FP cavity (quality factor) can be greatly increased by an increase in the number of its layers. Although important from the measurement point of view, this circumstance in addition to non-linear terms (transcendental and/or complex exponential) in Eqs. (1)-(3) in turn make the mathematical analysis complex. This is one of the main reasons why we perform simulations to analyze the effect of (complex) refractive index and thickness variations on spectral properties of FP cavities.
3.2. Simulation parameters
In our simulations whose results are to be discussed in following subsections, we utilize the structural and optical properties of our fabricated FP cavities as given in Table 1 . The porosity level (filling factor of a porous medium) of layers with and of each cavity in Table 1 can be found as follows. First, thicknesses of layers and are measured from corresponding SEM images of cavities (e.g., the SEM image for the FP Sample 1 in Fig. 1(b)). Then, reflectivity spectrums of the cavities are measured, and the resonance wavelengths () of each cavity are recorded. Next, for each FP cavity, using the following relation28]Fig. 1(b) and as commented in Section 2, such an assumption is not totally correct since and change with corresponding layer to layer and with transverse dimension of the cavity. These changes in and arising from fabrication affect not only all the above steps in the calculation, but also the spectral properties of the FP cavities. Therefore, they must be included in the characterization of FP cavities for sensing applications. Measured thicknesses and calculated refractive indices and porosity levels as well as their variations of layers with high and low porosity concentrations of each FP cavity are presented in Table 1.
In the computation of porosity levels of FP cavities in Table 1, we use and for nm and nm, respectively . The dependences of and of the first FP sample in Table 1 are shown in Fig. 1(a), and those for the second sample in Table 1 are similar to those in Fig. 1(a) except for changes in and values where and . We note from Table 1 that variation of a layer thickness decreases with an increase in that layer thickness.
In our analysis, because has important variations in the analyzed wavelength range (450-1620 nm) , its dispersive character are taken into consideration for correct characterization. Using the data from , we obtained the dependence of and over wavelength as shown in Fig. 2 , and fit that dependence to a polynomial with a degree of 10 whose coefficients are given in Table 2 .
3.3. Simulation results for a lossless cavity
Of the two test samples in Table 1, the first FP sample can be considered a lossless cavity on practical grounds since near nm, becomes real () . Throughout this subsection, our aim is to investigate the effect of any small variation in and of an arbitrary layer in this FP sample on its spectral properties. In the selection of this small variation in this and subsequent subsections, we considered the maximum percentage changes of and of the FP samples in Table 1. Figures 3 and 4 illustrate the effect of and variations of individual layers on the reflectivity value at and the value of of the first FP sample while we keep and constant for the analyzed layer, respectively. In addition, in Table 3 , the also give the maximum percentage changes in resonance wavelength values and reflectivity values at due to maximum percentage changes in and . Furthermore, we illustrate the dependence of whole reflectivity spectrum of this sample for a % change in refractive indices and thicknesses of third and twentieth layers in Fig. 5 .
In Figs. 3 and 4 and herein after, a zero percentage value signifies no change in the corresponding parameter of the analysis, yielding a default value for that parameter, whereas a specific percentage change in that parameter shows a relative change in that parameter. For example, a % change in defines its new value as . While the color bar in Figs. 3(a) and 4(a) denotes the reflectivity value at , the same bar in Figs. 3(b) and 4(b) designates the value of . The same color bars are utilized for the same purpose in our study. In addition, in Figs. 3 and 4 and thereafter, layer 1 denotes the first layer of top Bragg mirror and the layer number designates the first layer of bottom Bragg mirror.
- a) It is seen from Fig. 3(a) that, because of lossless nature of each layer, the effects of percentage change in on the relative change in the reflectivity value at are almost equal. We note that the dependence in Fig. 4(a) is more smoother (approximately constant over a wide region) than that in Fig. 3(a) and also symmetric with respect to no change. This difference comes from the fact that the refractive index of a layer appears in both the argument of trigonometric functions and the ratio in Eq. (2), whereas the layer thickness is present only in trigonometric functions in the same equations.
- b) We note from Fig. 3(b) that, on contrary to the result in (a), any small change in of layers only near to the middle section (resonance region) of the first FP sample is effective for changing the value of . Similarly, the variation in Fig. 4(b) is indistinguishable from that in Fig. 3(b). This similarity and patterns of the dependences in Figs. 3(b) and 4(b) are in complete agreement with the working principles of FP filters: the optical power related to the electric field distribution significantly increases toward the resonance region of a FP cavity , and near the resonance region any change in its value is largely correlated to the change in (or ) and (or ) through the relation in Eq. (4).
- d) The difference between the dependences in Figs. 3(a) and 4(a) is not perceived from Table 3 since presented values in Table 3 illustrate only maximum changes. In addition, from Table 3, we note that while the resonance wavelength negligibly changes with both and , the reflectivity value changes considerably for entries Figs. 3 and 4.
3.4. Simulation results for a lossy cavity
In previous subsection, we obtained the dependences in Figs. 3-5 for the first FP sample (nm) in Table 1. At wavelengths approximately between 1000 and 1620 nm or even higher, this FP sample can be assumed to be lossless, as seen from Fig. 2(b). However, at wavelengths between 450 and 1000 nm, the loss factor value of a bulk Si  cannot be neglected, and its effect should be taken into account for an analysis of spectral properties of FP cavities resonating at lower wavelengths. In this subsection, we will analyze this effect on the dependences in Figs. 3-5 using optical properties of the second FP sample in Table 1.
Because porosity concentrations of and are different, their loss tangent values will also be different. Therefore, this difference should be accounted for correct characterization of the second FP sample in Table 1. Toward this end, the porosity level of and is first found from Eq. (6) and then a direct proportion is made using the loss of bulk Si () at (see Fig. 2(b)) . After simple calculations, we determined and of the second FP sample. Once determining the loss values for and , we started to perform simulations when the determined loss factor values are in effect and when they are assumed to be zero in order to demonstrate the effect of consequences of omitting the loss factor in the analysis. Figures 6 -11 exhibit the same dependences in Figs. 3-5 when the loss tangent of the second sample is included and omitted in the analysis. In addition, Table 3 presents maximum percentage changes in resonance wavelength values and reflectivity values at due to maximum percentage changes in and .
- a) It is noted that the dependence in Fig. 7(a) resembles to that in 6(a) in which there is no loss, except for the following two differences. First, when loss is included into the analysis, the reflectivity value changes considerably increase which can be observed from range values of color bars present at the right sections. While the range of the color bar in Fig. 6(a) is between 0.28 and 0.44, that in Fig. 7(a) is between 0.48 and 0.60. This is an expected result since the reflection coefficient and thus reflectivity value are both metric values for how dissimilar the optical properties of two media are, and thus those values increase by the differences between the optical properties of two media  by the relation
- b) It is noted that a relative change in reflectivity value with respect to for an odd (even) layer number in Figs. 8(a) and 9(a) is lower than that with respect to for the corresponding layer in Figs. 6(a) and 7(a). Its reason parallels with the discussion (a) of subsection 3.3. However, we note from our simulations that a bigger value of loss factor () of layers increases the role of layer thickness as seen from range values of color bars present at the right sections, compared to that of layer refractive index, since then layer thickness becomes effective in reducing the light energy in the FP cavity (). While the range of the color bar in Fig. 8(a) is between 0.34 and 0.40, that in Fig. 9(a) is between 0.54 and 0.58.
- c) We obverse from the dependence in Figs. 6(b)-9(b) that a change in the refractive index and/or thickness value of layers near resonant region (middle layers in the analyzed FP cavity) is prevailing than that of first and last layers, conforming with the expression in Eq. (4) and discussion in subsection 3.3.
- d) We note that the inclusion of loss into the dependences in Figs. 6(b) and 8(b) does not much change their resonant behavior (Figs. 7(b) and 9(b)), since resonance behavior is chiefly associated with energy (a function of the real part of the complex refractive index) accumulation at middle layers of a FP cavity [4,14]. This is different from the discussion (a) of a reflection coefficient based on the discrepancy of optical properties of two layers.
- e) In addition to the conclusion of (c) in subsection 3.3, the presence of loss inside a FP cavity decreases its frequency selectivity and thus the quality  as seen from broadening of the resonance region and increment in the value of reflectivity at resonance wavelength in Figs. 10(a) and 10(b).
- g) There is no one-to-one correspondence between fabrication tolerance and loss of silicon, and measured spectral properties of the second FP sample in Table 1, since there are many multiple points in Figs. 6-9 corresponding to different structural properties but resulting in identical spectral properties. Two such points denoted by black solid circles (P1 and P2) are illustrated in Figs. 6 and 7. This fact is due to non-linear transcendental terms in Eq. (2), periodic nature of FP cavities, and low-loss nature of Si at analyzed wavelengths [30,31]. The same conclusion is also valid for the first FP sample in Table 1, since it is lossless.
Before ending this subsection, it is beneficial to investigate the effect of an increase in loss factor of Si on spectral properties and to monitor variation of reflectivity values over varying loss factor of Si. For example, Figs. 11(a) and 11(b), respectively, show dependences of the reflectivity spectrum and their variances over wavelength for different values for the second FP sample with assumed identical losses () in each layer. It is seen from Fig. 11(a) that while a change in greatly alters reflectivity values at resonance wavelengths of the second FP sample, it does not much vary the resonance wavelength values. In addition, we also note from Fig. 11(a) that the variations in reflectivity values are notable for even small loss values. However, these variations are diverse at different wavelengths over the whole wavelength range. To demonstrate these variations, in Fig. 11(b) we also present the variance of the dependences of reflectivity values in Fig. 11(a). We note from Fig. 11(b) that variation in the reflectivity value increases not only at the resonance wavelength (542 nm) but also at some other wavelengths (approximately at 468 nm and 645 nm), at which tangent of the reflectivity spectrum in Fig. 11(a) increases. Importance of results given in subsections 3.3 and 3.4 will be linked to discussion in Section 5.
4. Experiments and discussion
In this section we present measurement results of the two circular PSi FP samples with a diameter of 5mm in Table 1 to validate our simulation findings in Section 3. The details of the fabrication process and optical properties of the first sample are presented in Section 2 while those of the second sample can be found in . Therefore, they are not repeated for brevity.
We have selected two regions on the surface of each fabricated FP sample, denoted as Region A and Region B, over which their spectral measurements are carried out. The regions are arbitrarily selected from nearly the center of each sample. Figures 12 and 13 show measurement of the smoothed reflectivity data of 6 points taken in Regions A and B of both samples using a manual x-y-z stage. We applied a 5 percent smoothing to obtain a more even reflectivity data and to better illustrate our results. The distance between points in each region is set approximately to 25 to monitor any changes in optical properties of FP samples for this small distance increment and to be within the limit of the measurement region set by the radius (2.5 mm) of FP samples. The fiber optic cable spot is nearly 200. The x-y-z stage utilized in our measurements is manufactured by Thorlabs with a 1 resolution.
We note from Figs. 12 and 13 that not only the resonance wavelengths but also the reflectivity values at resonance wavelengths of reflectivity measurements taken from different points in Regions A and B of both samples are varying in different levels. We think that these differences between resonance wavelengths and reflectivity data at resonance wavelengths may come from different mechanisms such as non-equal surface current densities, impurities inside the cavity, some fabrication tolerances, and some loss present inside the cavity. These mechanisms affect the refractive index and the thickness of layers in FP samples in varying degrees, which in turn alter the reflectivity data and resonance wavelengths.
In Figs. 12 and 13, we observe significant changes in the spectral properties of points in the Regions A and B of both FP samples even though the fiber optic cable spot is greater than the distance of movement (25) within each region. This indicates that more spectral property changes would have essentially occurred if we used a fiber optic cable with a smaller spot size. This is because spectral properties measured by a fiber optic cable indicate a cumulative effect over the region at which the light energy is incident.
In Section 3, we concluded that there is no one-to-one correlation between fabrication tolerance and loss of silicon, and measured spectral properties of our fabricated FP samples. Although these is no such correspondence, variances of resonance wavelength and reflectivity at resonance wavelength measured at closely spaced points over a region, as in our case, could give some information about how well a FP cavity is fabricated. It is evident that a lower variance value means that sample has better structural homogeneities. For example, from Figs. 12 and 13, we note that all the measured resonance wavelengths of regions A and B of both samples lie within the two standard deviations range (indicating a lower variance), which is a widely accepted metric utilized in many statistical and probabilistic problems .
5. On the detection and identification of biological or chemical molecules by porous silicon Fabry-Perot cavities
In Sections 3 and 4, we investigated the effect of changes in the refractive index and the thickness of layers of lossless and lossy FP cavities on their spectral properties for their correct characterization without paying any attention and focusing on linking to sensing applications of unknown chemical or biological molecules by those cavities. In this section, we concentrate on the practical application of the results given in Sections 3 and 4 and discuss concisely how a reliable and accurate detection and identification of chemical or biological molecules by these cavities could be implemented.
It is well-known that when a PSi FP cavity is exposed to a chemical or biological molecule, the effective refractive index of the cavity layers increases depending on a few parameters such as the exposure time, the refractive index of the molecule, and the diameter of the pores. This increase in the effective index makes overall reflectivity spectrum shifts to lower (higher) frequencies (wavelengths), widely known as redshifting. Because an increase in the refractive index of layers due to molecules can be analogically thought as a similar increase due to fabrication parameters as analyzed in Section 3 (no matter the agent is) , we can declare that this shifting becomes considerable when the molecules reach to middle sections of the cavity, as evident from Figs. 3(b), 6(b), and 7(b). More importantly, from Figs. 3(b), 6(b), and 7(b) we can also deduce that it is very difficult to discern whether the molecule exposed to the cavity is lossy just by comparing the measured resonance wavelength. This identification is especially important for fingerprinting of unknown molecules.
In most of the sensing applications, as a general procedure, relative wavelength shift after the cavity is exposed to an unknown chemical or biological molecule is measured for detection and identification of the molecule. It is straightforward that molecules having almost identical refractive index values () will yield similar wavelength shifts, which in turn making difficult to identify these molecules. It was shown in Figs. 10(a) and 10(b) that the presence of loss within an empty FP cavity will decrease its quality factor as evident from broadening of the resonance region. A decrease in quality factor of a FP cavity will accompany with a decrease in frequency or wavelength selectivity of that cavity. As a result, the performance of a lossy cavity in detection and identification of unknown molecules () by using wavelength shifts will decrease as well. Besides, for similar reasons, loss of a molecule () will be another agent in reducing the sensing property of FP cavities.
Enlargement of current passage (porosity region) of first created layers, presence of some unpredicted surface irregularities and impurities, non-equal surface current density distribution, some fabrication tolerances, and loss factor alter in different levels the refractive index and thickness of each layer and thus the spectral characteristics of a porous silicon (PSi) Fabry-Perot (FP) cavity even if it is fabricated by an anisotropic etching process. This circumstance in turn necessitates correct characterization of these cavities before sensing of biological or chemical molecules. To meet this demand, in this study we performed some simulations to analyze the effect of changes in layer refractive index and its thickness on the reflectivity value at resonance wavelength and the value of resonance wavelength of a lossless and lossy empty PSi FP cavities. In our simulations, we utilized the well-known and versatile transfer matrix method to obtain the dependence of reflectivity spectrum over wavelength. We conclude the following specific key results from our simulations: 1) The influence of refractive index (and thickness) on the reflectivity value at resonance wavelength is approximately equal for all layers of the lossless cavity; 2) The same influence of layers near to light illumination is more dominant than that of layers far from the illumination when the cavity has some loss; 3) The effect of refractive index change of an analyzed layer on the reflectivity value at resonance wavelength is higher than the effect of thickness change of the same layer when the cavity is lossless; 4) When the cavity becomes lossy, the effects of changes in both refractive index and thickness of the analyzed layer on the reflectivity value at resonance wavelength are equally important; and 5) Effects of the refractive index and the thickness of middle layers (microcavity region) of a lossless (or lossy) FP cavity on resonance wavelength is more prevailing than those of first and last layers with respect to light illumination. After simulations, we conducted reflectivity measurements of 6 points within arbitrarily selected two regions over the transverse dimension of our fabricated FP cavities (the first one is lossless and the other is lossy) to validate our findings in the simulation analysis. From simulations and measurements, we note two important general results. First, the mechanisms such as non-equal surface current densities, impurities inside the cavity, some fabrication tolerances, and some loss present inside the cavity affect in different degrees the spectral properties of FP cavities and therefore, these mechanisms should be taken into consideration in sensing applications. Second, not only the measurement of the resonance wavelength (or its shift) but also the reflectivity at the resonance wavelength (or at other wavelengths) should be used for correct detection and identification of lossy biological or chemical molecules.
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