Modeling and characterization of deeply etched multilayer resonators under partial coherent excitation via multimode optical fibers

Recently, there has been a resurgence of interest in multimode optical fibers illuminated by a white light source. Largely, in anticipation of many integrated applications in the biomedical domain and spectral sensing benefiting from the broad spectral range and high numerical aperture (NA). Along these lines, the output light from these fibers can be captured by the physics of partially coherent sources. While the Gaussian Schell model has provided a framework for studying partial coherence, to our knowledge, its impact on microstructures remains unexplored. As the sheer complexity arising from the interplay between partial coherence and microstructures transfer function has posed fundamental challenges in deciphering their response. In this work, we introduce a comprehensive numerical model paired with experimental validation to assess the performance of multilayer optical resonators, which are meticulously crafted through high aspect ratio silicon etching under the influence of a partially coherent optical source. The model studies the effects of optical fiber NA, Bragg mirror order, cavity length, and surface roughness of the microstructures on the output of the resonator. The results show that the response under standard multimode fiber (MMF, partial coherent source) has lower insertion loss, more asymmetry versus wavelength, and larger full width at half maximum than the standard single mode fiber (full coherent source). A silicon-on-insulator chip is fabricated using 130 µm deep etching of silicon for Bragg mirrors with 2.25, 3, and 3.25 µm silicon layer widths and a different number of layers. The structures are characterized using a MMF of 62.5 µm core diameter illuminated by an infrared white light source. The theoretical results have been compared with the experimental results and a good agreement has been obtained.

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Introduction
Lately, micro-optical resonators have attracted significant attention in academic and industrial communities.What has motivated these efforts is their ability to confine the light to a certain volume by virtue of a multilayer structure or total internal reflection.Moreover, optical cavities enable myriad applications such as temperature and pressure [1,2], gas sensors [3,4], refractive index sensors [5,6], spectroscopy [7][8][9], and on-chip filters [10], to mention a few.Additionally, multilayer configurations are exploited in liquid crystal display, augmented reality, and virtual reality [11,12].Yet, most of these resonators have a bulky size, precluding their use in many practical applications.A novel type of these resonators, which is based on deeply etched silicon multilayer structures, has been reported in the literature [13][14][15].These microresonators are characterized by a small footprint and compact size, which open the door for many novel applications, especially pertaining to gas sensors [16].Notably, their small footprint enables their integration into photonic integrated circuits, enhancing system performance and enabling dense packing, which is crucial for applications in telecommunications and biomedical sensing.Additionally, these compact resonators overcome the stringent alignment limitations of bulkier designs, facilitating advancements in portable diagnostics and environmental monitoring by enabling the development of miniaturized, highly sensitive devices.These devices are usually used in conjunction with an optical cavity to increase sensitivity, i.e. cavity-enhanced sensing [7,[17][18][19].An example of such a configuration is depicted in figure 1, where the optical power from the input fiber passes through an optical cavity consisting of three silicon/air Bragg mirrors to provide a high-finesse cavity, an optical cavity with high reflectivity mirrors, and the optical power is then coupled to the output fiber.For plane wave excitation of x − y homogenous structure, the transfer matrix method can be utilized to capture the response at the output [20].Moreover, even for sources with a finite beam waist, Fourier optics can be used in conjunction with the transfer matrix to enable the precise modeling of these microstructures.On the other hand, for x − y inhomogeneous layers, rigorous coupled wave analysis can be used to model these multilayer configurations [21].However, the output of different optical structures, such as coated optical fibers and attenuated total reflection spectrometers, substantially varies according to the coherence properties of the optical source [22,23] (i.e.optical fiber type, whether single or multimode).Nevertheless, all the work that has been reported in the literature for deeply-etched microstructures, up to our knowledge, exploits fully coherent sources, i.e. single-mode fiber (SMF) [20].In general, optical sources can be categorized according to their temporal and spatial coherence properties.Temporal coherence properties are related to the optical bandwidth of the source, while spatial coherence properties are related to the amplitude and phase relation between different spatial points of the source.In the case of spatially full coherent sources, such as lasers and SMF, the amplitude and phase relationship between any two points on the optical source is fully deterministic.On the other hand, in the case of incoherent sources, such as thermal sources, there is no phase relation between the different points of the source.In cases involving partially coherent sources, like the light emitted from a multimode fiber (MMF), partial coherence between spatial points on the source is observed [24,25].Notably, this partial coherence arises from a combination of source characteristics and the stochastic variations within the MMF due to random mode coupling caused by bends and perturbations [24].Although the output from the SMF is characterized by full coherence properties, the output of MMF has numerous advantages, such as higher power and broader spectral bandwidth, as it has a wider core radius, and hence, easier coupling.
At this juncture, one may pose the following questions: is it feasible to simulate the output of these resonators when excited by partially coherent sources such as MMFs?If so, how would the output differ when comparing partially coherent excitation to fully coherent excitation?
Our study ventures into uncharted territory by examining the influence of partial coherence on the transfer functions of optical microresonators, an area seldom explored in existing literature.Prior analyses have largely centered on fully coherent sources, overlooking the intricate dynamics introduced by partial coherence.This gap is significant, considering the prevalent use of partially coherent light in various applications.Our work aims to bridge this gap, offering insights that could revolutionize the design and functionality of optical devices, including lab-on-a-chip technologies.In this work, we model and characterize the optical output power of microcavity illuminated by a partially coherent source.The rest of this manuscript is organized as follows.The theoretical background of the multilayer structures and Gaussian Schell model, a theoretical framework used to describe partially coherent light characterized by its Gaussian intensity profile and Gaussian coherence properties, are presented in section 2. A parametric study for the output power from microstructures excited by a partially coherent source is demonstrated in section 3, considering the optical fiber numerical aperture (NA), Bragg mirror order, cavity length, and surface roughness of the microstructure.In section 4, the fabrication steps of the resonators and the resulting chips are elaborated.The experimental characterization of the optical response of the structures using a MMF of 62.5 µm core diameter illuminated by an infrared white light source is given in section 5. Finally, the work conclusion is drawn in section 6.

Theoretical treatment
Interestingly, the serendipitous size matching between microfluidics cavities and the MMF allows them to be used as an excitation source for micro-cavities.Furthermore, the MMF delivers adequate power in a broadband range compared to SMF.Yet, modeling the output power of the MMF is not a trivial task.Since mode mixing, excitation of new modes in the fiber and random phases due to temperature fluctuations and mechanical stress can occur [26].In turn, this leads to a partially coherent output field from MMF.In this section, the elementary source model and Gaussian Schell model have been put forward to model the output of the MMF [27], as elucidated in figure 2(a).In this framework, the partial coherent output field from MMF is decomposed into a summation of a fully coherent output field uncorrelated from each other and with Gaussian distribution intensity.The assumptions of the Gaussian Schell-model are as follows [28]: (1) The source is a statistically homogeneous, stationary, and isotropic random field.(2) The field can be described by its second-order statistical properties, namely the spectral density and the crossspectral density.
(3) The spectral density is a Gaussian function of frequency.(4) The cross-spectral density is a separable function of frequency and position.( 5) The field can be described by a complex degree of coherence, which is a function of position and frequency.
For these reasons, the output field with beam waist (w o ) can be expressed as a summation of coherent sources, each with beam waist γ.Where the beam waist of each coherent source can be given as in [24,29] by: ( Here where θ is the divergence angle of MMF and λ is the operating wavelength, which is related to the NA of the fiber.Additionally, each source propagates independently in the same manner as a fully coherent single-mode source, as depicted in figure 2(b).Next, we exploit the transfer matrix method to predict the performance of multilayer structures [30].In this model, the phase factor encountered after propagation through a layer is given by δ = 2π nlcos (θ i ), where n is the refractive index of the layer, l represents the layer thickness, and θ i is the incident angle.Afterward, by solving the boundary conditions at the interface of each layer of the microstructure, the tangential components of the incident and transmitted electric and magnetic fields can be related by the transfer matrix [20]: where η out is the output medium admittance and is given by η out = n out cos (θ i ) for transverse electric (TE) polarization, while η out = n out /cos (θ i ) for transverse magnetic polarization.Furthermore, the field reflection and transmission can be derived as: where η inc and η out are the incident and output admittance, respectively.However, the transfer matrix method assumes a plane wave incident on the multilayer structure.Thus, Fourier optics can be utilized to decompose an incident Gaussian beam into a summation of plane waves with different incident angles [20].Then, each plane wave is multiplied by the corresponding transfer function.Finally, an inverse Fourier transform is performed to find the output field.Therefore, one can relate the output to the input field by: where k t is the tangential propagation constant given by k 2 t = k 2 x + k 2 y and R fiber stands for the fiber radius.In equation ( 4), E orm and E otm refers to the reflected and transmitted electric field from the microstructure.Moreover, the Fourier transform is defined by FT {f (r, z)} = ´∞ −∞ f (r, z) exp [−j2π f r r] dr while the inverse Fourier transform can be described by stands for the circular function with a radius r o .In the latter equation, t and r denote the reflection and transmission coefficients, respectively.Note that in the case of MMF, only the rays with an angle smaller than the fiber NA angle will be coupled; thus, k c = 2π n fiber √ (1 − cos 2 θ c )/λ, here θ c stands for the maximum angle at which light rays can be coupled in a guided manner in MMF, which could be determined from NA as NA = sin (θ c ).While E i,m,n denotes the incident field of each coherent source, which is assumed to have Gaussian distribution such that: where q (z) = z + iπ γ 2 /λ, and x o and y o are the distances between every two coherent sources in x and y directions, respectively.The output is the incoherent summation of all these sources.Hence, the total reflection and transmission coefficient for the output of MMF can be derived as: where R and T are the total power transmission and reflection coefficient from the multilayer structure.

Simulation study
In this section, a parametric analysis for the model discussed in section 2 will be presented to study the performance of the micro-cavities.A comparison between the output of microstructure excited by coherent source (SMF), partial coherent source (MMF), and plane waves (PW) is given in this section.
The simulated FP cavities compromise a Bragg mirror of d si = 2.325µm, d air = 1.9375µm, cavity length of 1.55µm, N = 5 and excited with TE polarization, unless otherwise mentioned.Both SMF and MMF sources are assumed to have Gaussian field distribution.The source is characterized by a beam waist w o and NA, for SMF and MMF sources, respectively.

Effect of Bragg mirror order
In this subsection, we discuss the influence of the number of silicon layers on the reflectivity of the Bragg mirror excited by PW, SMF, and MMF.Figures 3(a

Effect of beam waist and NA
Here, we assess the effect of spot size and divergence on the performance of the microresonator.Figure 5 depicts the transmission of a Fabry-Perot cavity at the different beam waists.Overall, as the beam waist increases, the transmission increases, and the full width at half maximum (FWHM)

The effect of cavity's length
The effect of the cavity length on the cavity response has also been studied.Figure 7 depicts the cavity response at different cavity lengths.For PW, the insertion loss is unaffected by the cavity length (figures 7(a) and (b)).But for SMF and MMF, as the length increases, the diffraction loss increases, and FWHM decreases (figures 7(c)-(f)) due to divergence.An increase in the length from 1.55 to 3.1 µm can cause an increase in the insertion loss by about 5 and 2.5 dB in the case of SMF and MMF, respectively, as shown in figures 7(d) and (f).Note that the length of the cavity has been taken as multiples of 1.55µm to make the resonance wavelength always at 1.55µm (see (a), (c), and (e)).

Effect of surface roughness
The impact of the surface roughness on the performance of the microcavity has also been studied using the model developed in [31].The surface roughness is considered as an absorbing layer added at each material interface.The thickness of this layer is d r = 2σ, where σ is the surface roughness root mean square (RMS).The refractive index of this layer is given by [20]: Figure 8 depicts the cavity's transmission response at different silicon surface roughness RMS values.Overall, as the surface roughness RMS increases, the loss and FWHM increase, as shown in figures 8(a)-(f).It is interesting to note that the effect of surface roughness is more significant in the case of MMF than SMF (see figures 8(d) and (f)).As the MMF has higher divergence than SMF, and hence, a longer propagation length inside the absorbing layer.At 30 nm surface roughness, an increase in the insertion loss at the peak resonance wavelength was around 10 and 13 dB in the case of SMF and MMF, respectively, as depicted in figures 8(c) and (e).

Chip fabrication
The multilayer resonators are fabricated via a single lithography process that starts with a 500 µm thick silicon wafer.The wafer is first thermally oxidized to form a 1.5 µm thick SiO 2 layer, which serves as a hard mask for the deep reactive ion etching process.The resonator pattern is then transferred to the oxide layer through a photolithography process and plasma etch of the Bosh process in an A601E Alcatel etch tool.The etching process was carried out to achieve an etch depth of 130 µm, hence shaping the resonator profile.A smoothing process was then used to lower the surface roughness of the etched silicon parts inherent to the Bosh process.This smoothing process implies oxidation of the silicon wafer.Thus, a 1 µm thick oxide was thermally grown and then removed with hydrofluoric acid.A final dicing step is then done to get the individual samples.Figure 9(a) illustrates a detailed depiction of the fabrication steps involved in creating microfluidics dies.On the other hand, figures 9(b) and (c) show a scanning electron microscope image of the fabricated structures.In this work, we are interested in the output of the planar mirror cavity, which is highlighted by the dashed red square (see figure 9(b)).In addition, we are also interested in multilayer Bragg mirrors, as shown in figure 9(c).

Experimental results
After the fabrication has been done, the microstructures have been excited using a standard MMF with a core diameter of 62.5 µm, as shown in figures 10(a) and (b).The light source used was an infrared source (Melles GRIOT MGlight-Hal) connected to a standard MMF, which is placed on a five-axis positioner and aligned to the micro-cavity.The output from the micro-cavity is fed to another standard MMF, which is aligned  to the micro-cavity output using another five-axis positioner.Finally, the output has been monitored using the optical spectrum analyzer.

Characterization of deep-etched multilayer resonators
The transmission of various micro-structures has been measured.Figure 11(a) shows the transmission of the fabricated Fabry-Perot cavity using SMF excitation.A microscopic image for a fabricated Fabry-Perot cavity is shown in the inset of figure 11(b), which shows a Fabry-Perot cavity of 200 µm length and a Bragg mirror of d si = 3µm and d air = 5.75µm.Figure 11(c) depicts the transmission response of this cavity using MMFs.Moreover, the suppression ratio of the cavity mode is around 2 dB (figure 11(d)).This low suppression ratio is due to the cavity's long length.The free spectral range of the cavity is around 5 nm.In general, the transmission from MMF exhibits higher insertion loss and lower suppression ratio in full agreement with numerical simulations carried out in section 3. Yet, MMFs can offer larger bandwidth when used in conjunction with incoherent light  sources, as shown in figure 11(c).Next, we assess the effect of increasing the cavity's length experimentally, as shown in figure 12(a).Similarly, a microscopic image for a Fabry-Perot cavity of a length of 326 µm is depicted in the inset of figure 12(a).Here, the Bragg mirror of d si = 2.25µm and d air = 8µm.The transmission response of the cavity's shown in figure 12(a).Note that the suppression ratio of the cavity mode is around 1.5 dB (figure 12(b)).Comparing the previous result with figure 11(b) elaborates that as the cavity's length increases, the suppression ratio decreases and the insertion loss increases in full accord with the numerical model developed in section 3 (see figures 7(e) and (f)).The free spectral range of the cavity is around 3.7 nm.In both cases, the Bragg order N = 3.Note that we have modified the wavelength range in this measurement to accommodate a wider range of results.Moreover, as anticipated, the output of the MMF is able to provide adequate power in a wide spectral range, as shown in figures 11(a) and (c).

Experimental and numerical model comparison
The transmission response for a Bragg mirror of d si = 3µm, d air = 5.75µm and N = 3 is depicted in figure 13.Overall, the suppression ratio is about 10 dB.At the same time, the separation between the transmission peaks is about 100 nm.The minima correspond to the maximum reflection positions.The resonator has a finesse of around 8.3 and a quality factor of 100.The model presented in section 3 is used to simulate the Bragg mirror microstructure shown in figure 13.The simulation results have been compared with the experimental results (see figure 13).Overall, the numerical results are in good agreement with the experimental results over a very broad spectral range.The discrepancy between the simulation and the experimental results is regarded to the uncertainty in the over-etching depth and fiber alignment angle.
In addition to the imperfect Gaussian shape of thermal sources and uncertainties in parameter estimation.To corroborate our results, we utilize Pearson's correlation coefficient (which is defined as r , where n is the number of samples, while x and y represent the respective datasets for which the correlation is being calculated).The obtained value of r as found to be 0.7028 indicating a strong correlation, as illustrated by figure 13.In conclusion, in line with our expectations, the MMF demonstrated an enhanced ability to deliver greater power across a broader spectral range.This capability heralds the advent of numerous innovative applications in the field.

Conclusion
In conclusion, this study lays a foundational understanding of the behavior and performance of multilayer optical resonators under partially coherent excitation, paving the way for a multitude of future research opportunities and practical applications.Looking forward, it becomes crucial to explore the integration of these resonators into more complex photonic systems, examining their potential to enhance optical communication networks and sensor technologies.The advancements in this field could lead to the development of novel, more compact, and energy-efficient photonic devices, significantly impacting various sectors, from healthcare to environmental monitoring.Our model demonstrated that the transmission of FP cavities excited by standard MMF has less insertion loss, higher asymmetry, and higher FWHM than the standard SMF.Yet, the MMF is less susceptible to changes in the cavity length.Finally, the experimental measurements have been obtained.The results were compared with the model, and a good agreement between the model and measurements has been demonstrated.Our research has the potential to lay the groundwork for groundbreaking applications, including lab-on-a-chip technologies, by enabling the miniaturization and integration of optical components.This advancement could significantly enhance diagnostic capabilities, allowing for rapid, on-site analysis in various fields ranging from healthcare to environmental monitoring.

Figure 1 .
Figure 1.Fiber-coupled silicon/air microcavity comprises two Bragg mirrors manufactured on a silicon-on-insulator (SOI) chip for microfluidics applications.Here, d si represents the width of the silicon layer, whereas d air stands for distance of the air gap intervening between the silicon layers.

Figure 2 .
Figure 2. (a) Partial coherent output from MMF decomposed by elementary source mode.(b) The divergence of a single coherent source.Within this representation, xo specifies the separation distance between each two coherent sources, γ stands for the half width at half maximum (HWHM) for each coherent source, Wo represents the HWHM for the output beam from the multimode optical fiber, and θ is the diveregence angle.

Figure 4 .
Figure 4. Reflection from silicon-air Bragg mirror at a different number of silicon layers (Bragg order) and different excitation sources.(a) PW (b) SMF (c) MMF.
) and (b) show a Bragg mirror of one silicon layer, i.e.N = 1 and 2 silicon layers i.e.N = 2, respectively.The effect of the number of silicon layers on Bragg's mirror reflection is depicted in figures 4(a)-(c).Overall, as the number of layers increases, the reflection increases; yet, the increase in the reflection starts to saturate after N = 3.Also, the output from the MMF is smoothed with a lower peak-to-peak ripple, as shown in figure4(c).Yet, the plane wave excitation has the highest reflection coefficient at the same number of the silicon layer.The maximum reflections are around 0.9998, 0.9878, and 0.9711 in the case of PW, SMF, and MMF at N = 4, as shown in figures 4(a)-(c).Note that reflection in the case of multimode excitation (figure4(c)) decreases as wavelength increases since the divergence angle also increases with wavelength, and thus, the loss increases compared to single mode (figure 4(b)) as the divergence effects prevail in the case of MMF.

Figure 5 .
Figure 5. Fabry Perot cavity's spectral characteristics at different beam waist.(a) Spectral response.(b) Transmission at resonance peak and the full width at half maximum versus beam waist.

Figure 6 .
Figure 6.Spectral characteristics of a Fabry-Perot cavity at varying numerical apertures: (a) shows the spectral response, and (b) details the transmission at the resonance peak and full width at half maximum as a function of numerical aperture.

Figure 7 .
Figure 7. FP cavity's transmission response, resonance peak insertion loss, and full-width half maximum at different cavity lengths and different excitation sources.(a) and (b) PW (c) and (d) SMF (e) and (f) MMF.

Figure 8 .
Figure 8. FP cavity's transmission response, resonance peak insertion loss, and full-width half-maximum at different surface roughness RMS and different excitation sources.(a) and (b) PW (c) and (d) SMF while (e) and (f) MMF.

Figure 9 .
Figure 9. (a) Detailed visual representation of the fabrication process and the resulting structures.Scanning electron microscope images of the fabricated structure.(b) Multi layers structures composed of different Fabry-Perot cavities.(c) Enlarged view of a Bragg cavity.

Figure 10 .
Figure 10.(a) The schematic diagram of our experimental setup highlighting the key components.(b) The photograph showcases the experimental lab setup specifically designed for precise measurements of the microfluidics chip.

Figure 11 .
Figure 11.Spectral transmission of Fabry-Perot Cavity with Varied Excitation.The corresponding Fabry-Perot microscopic image is shown in the inset.The resonator mirror consists of two Bragg mirrors of d si = 3.25µm and d air = 11.5µm.(a) Single mode fiber excitation.(b) Zoomed version around 1535 nm.(c) Multimode fiber excitation (d) zoomed version around 1535 nm.

Figure 12 .
Figure 12.(a) The transmission spectrum.The inset shows the corresponding Fabry-Perot microscopic image, which consists of a Bragg mirror of d si = 2.25µm and d air = 8µm, as shown in the inset.(b) Zoomed version around 1535 nm.

Figure 13 .
Figure 13.The simulation and experimental transmission spectrum for a Bragg mirror of N = 3.