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Modeling a grating coupler-based interferometer for far-field-sensing of nanoscale displacements

Federico Galeotti and Andrea Fiore

Open Access Open Access

Abstract

Sensing displacements at the nanoscale is the basis for many metrology applications, in particular atomic-force microscopy. Displacement sensing with nano-optomechanical structures provides interesting opportunities for integration, but it typically features a small dynamic range due to the near-field nature of the sensor-sample interaction. Here, a far-field sensing approach based on a grating coupler is considered and an analytical model used to tune its performance is introduced. The proposed model allows exploiting the full range of design parameters and thereby optimizing resolution and dynamic range. The compact size of the sensor and the possibility of integrating it with an on-chip laser and detector make it very promising for fully-integrated optical sensing systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The precise detection of displacement is fundamental in various types of metrology instruments, such as the atomic force microscope (AFM) [1]. Optical sensing, e.g. based on position-sensitive photodetection or interferometry, is often used for displacement metrology, but mostly based on discrete optics, resulting in relatively bulky and expensive instruments. In the last decades, the developments in photonic integrated circuits (PICs) and the combination with nano-optomechanical sensors have shown potential for compact and mass-manufacturable sensing systems and first steps in this direction have been made [2–7]. However, most of these on-chip optomechanical sensors rely on evanescent coupling between two micro-sized structures, implying a maximum measurable displacement typically smaller than $100\,nm$. Such limitation in dynamic range is a problem for most applications, e.g. in AFM, where the sensing tip must undergo a relatively large travel range of few hundreds of $nm$. In the past, PIC-based displacement sensors working in the far-field have shown limited resolutions [8,9] due to the difficulty of building an efficient interferometer involving both on-chip and off-chip light paths. Recent progress in high-contrast photonic integration platforms and nanophotonics may however offer new opportunities. In particular, the grating coupler (GC) has become a fundamental component in PICs due to its ability to efficiently couple light in and out of the PIC [10].

In particular multiple studies demonstrated GC-based displacement sensing [11–14]. This simple far-field interrogation method is based on the GC ability to efficiently split light off a waveguide to free space and couple the reflection back to the waveguide. By taking advantage of the interference within the GC itself to sense the phase of reflected light, greatly simplifying the interrogation circuit. In this paper, we introduce a simple analytical model for the GC-based interferometer, which allows exploring the regime of resonant enhancement, beyond earlier description of the grating-based sensor [8,15]. By careful choice of the GC design parameters we show that it is possible to design structures for optimum sensitivity and/or dynamic range. Importantly, any reflecting surface can be used as top mirror, allowing the combination of the interrogator with well-developed micromechanical structures such as cantilevers for AFM or photoacoustic spectroscopy.

The sensor, schematically shown in Fig. 1(a), is formed by an input waveguide, a shallow etched GC, a top mirror and an output waveguide, which can all be fabricated within a high-index contrast platform, such as silicon photonics [16,17] or InP membrane on silicon (IMOS) [18]. In first approximation, its working principle is conceptually equivalent to the one of a Mach-Zehnder interferometer where the GC replaces both beamsplitters. In this picture, the incoming light partially is diffracted toward the sensing surface (acting as a mirror), reflected and then coupled back into the output waveguide where it interferes with the light propagating into the grating. The path bouncing off the mirror suffers a phase delay $\Delta =2kz_M$ where $k$ is the wave vector and $z_M$ the distance between mirror and the GC. Interference between light reflected back and light propagating through the grating produces interference fringes, allowing the measurement of small phase shifts [15]. As we show below, the interference can be affected by multiple reflections between grating and sensor, leading to resonant enhancement. As lasers and detectors can be integrated into the platform, this provides a fully-integrated interrogator unit, while the sensor can be either integrated on the same chip, leading to a sensor-grating distance of up to a few $\mu m$, or fabricated separately and packaged together with the interrogator, at distances up to the $mm$ scale.

 figure: Fig. 1.

Fig. 1. (a) Sketch of the grating sensor with fully-integrated laser and photocurrent read-out. The light coming from the integrated laser couples to the waveguide and reaches the grating. The interference between transmission through the grating and reflection from a movable top mirror produces a modulated photocurrent signal in the integrated photodiode. An AFM tip is schematically indicated on the sensing head. (b) The 6-ports model representing the grating sensor. The laser is coupled in from port 2, the output light from port 5 is coupled to the photodetecor, and the mirror-loop is connected through ports 3 and 6. Example of an symmetric (c) and asymmetric (d) shallow etched grating couplers, where $\theta _0$, $h_w$, $h_e$ and $N_s$ are respectively the target scattering angle, the waveguide height, the total etching depth and the number of grating periods. The (e) and (f) panels show the two basic simulation used to assess the efficiency of each output port.

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In order to provide an intuitive picture of the device operation, we provide an analytical description using the scattering matrix formalism and then we discuss a number of practical designs using numerical simulations.

2. Model

The proposed model can be described by a 6-port network, representing the GC, and a delay line, representing the path to the mirror and back. By defining $\mathbf {a}=(a_1\cdots a_6)$ the fields at the input ports and $\mathbf {b}=(b_1\cdots b_6)$ those at the output ports, as depicted in Fig. 1(b), we have that

$$\textbf{b}=\textbf{S}\cdot\textbf{a},\qquad \textbf{S} = \begin{pmatrix} S_{11} & \cdots & S_{16} \\ \vdots & \ddots & \vdots \\ S_{61} & \cdots & S_{66} \end{pmatrix}$$
where $\textbf {S}$ is the scattering matrix. For a lossless grating $\textbf {S}$ is unitary, so $\textbf {S}^\dagger \textbf {S}=\textbf {S}\textbf {S}^\dagger =\textbf {1}$ where $^\dagger$ is the conjugate transpose. As the system is linear and reciprocal, $S_{ij}=S_{ji}$. Additionally, by considering an uniform grating with a vertical plane symmetry plane and that the grating is designed so that $S_{ii}=S_{21}=S_{23}=S_{31}=S_{32}=0$, $\textbf {S}$ can be reduced in two $3\times 3$ submatrices, $\textbf {0}$ and $\textbf {s}$
$$\textbf{S} = \begin{pmatrix} \textbf{0} & \textbf{s} \\ \textbf{s} & \textbf{0} \\ \end{pmatrix},\quad \textbf{s} = \begin{pmatrix} r_\downarrow e^{i\Phi_{r_\downarrow}} & c_\downarrow e^{i\Phi_{c_\downarrow}} & t_\downarrow e^{i\Phi_{t_\downarrow}} \\ c_\downarrow e^{i\Phi_{c_\downarrow}} & c_\rightarrow e^{i\Phi_{c_\rightarrow}} & c_\uparrow e^{i\Phi_{c_\uparrow}} \\ t_\uparrow e^{i\Phi_{t_\uparrow}} & c_\uparrow e^{i\Phi_{c_\uparrow}} & r_\uparrow e^{i\Phi_{r_\uparrow}} \\ \end{pmatrix}$$
where $c_{\uparrow (\downarrow )} e^{i\Phi _{c_{\uparrow (\downarrow )}}}$ corresponds to the coupling between ports 2 to 6 (2 to 4), $r_{\uparrow (\downarrow )}e^{i\Phi _{r_{\uparrow (\downarrow )}}}$ relate to the reflectance from 3 to 6 (1 to 4), $t_{\uparrow (\downarrow )}e^{i\Phi _{t_{\uparrow (\downarrow )}}}$ relate to the transmittance from 3 to 4 (1 to 6), and $c_\rightarrow e^{i\Phi _{c_\rightarrow }}$ is the coupling between ports 2 and 5, as shown schematically in the panels (e) and (f) of Fig. 1. Also $\textbf {s}$ satisfies the condition, $\textbf {s}^\dagger \textbf {s}=\textbf {s}\textbf {s}^\dagger =1$, which is used to retrieve relations between the coefficients in (2) (see the Supplement 1).

2.1 Symmetric grating

In order to obtain insight in the operation of the interferometer and the possible regimes of operation, we first analysed the simplified case of a grating-coupler with an additional horizontal plane of symmetry, as shown in Fig. 1(d). By using this vertical symmetry of the grating we can further simplify $\textbf {s}$ obtaining that $\textbf {s}=\textbf {s}^T$, where $^T$ stands for the transpose matrix, i.e. $c_\uparrow =c_\downarrow \equiv c_\updownarrow$, $r_\uparrow =r_\downarrow \equiv r_\updownarrow$, $t_\uparrow =t_\downarrow \equiv t_\updownarrow$. Now, applying the unitarity condition to this new $\textbf {s}$ we can retrieve the conditions relating the amplitudes and phases in (2), obtaining

$$\begin{array}{l} r_\updownarrow^2+c_\updownarrow^2+t_\updownarrow^2=1 \\ 2c_\updownarrow^2+c_\rightarrow^2=1 \end{array}$$
$$\begin{array}{l} \Phi_{r_\updownarrow}={\pm}\arccos\left(-{\displaystyle\frac{c_\updownarrow^2}{2r_\updownarrow t_\updownarrow}}\right)+\Phi_{t_\updownarrow}\\ \Phi_{c_\rightarrow}={\pm}\arccos\left({\displaystyle\frac{r_\updownarrow^2-c_\rightarrow^2+t_\updownarrow^2}{2c_\rightarrow t_\updownarrow}}\right)+\Phi_{t_\updownarrow}\\ \Phi_{c_\updownarrow}=\frac{1}{2}\left[\Phi_{r_\updownarrow}+\Phi_{c_\rightarrow}\mp\arccos\left({\displaystyle\frac{t_\updownarrow^2-r_\updownarrow^2-c_\rightarrow^2}{2r_\updownarrow c_\rightarrow}}\right)\right] \end{array}$$
Due to those relations we have three free parameters, chosen as the two amplitudes $c_\updownarrow$, $t_\updownarrow$ and the phase $\Phi _{t_\updownarrow }$. The calculations required to achieve (3) and (4) are reported in the Supplement 1.

Finally, by writing the input field as $\mathbf {a}=(0,a_2, a_3, 0, 0, 0)$, where $a_2$, $a_3$ represent the fields coming from the waveguide and the top, respectively, and imposing that $a_3=b_6e^{i\Delta }$, where $\Delta =2kz_M$ is the phase delay in the reflected field, we obtain

$$\begin{pmatrix} b_5\\ a_3e^{{-}i\Delta}\\ \end{pmatrix} = \begin{pmatrix} c_\rightarrow e^{i\Phi_{c_\rightarrow}} & c_\updownarrow e^{i\Phi_{c_\updownarrow}} \\ c_\updownarrow e^{i\Phi_{c_\updownarrow}} & r_\updownarrow e^{i\Phi_{r_\updownarrow}} \\ \end{pmatrix} \begin{pmatrix} a_2\\ a_3\\ \end{pmatrix}$$
and, consequently,
$$T(\Delta) =\left|\frac{b_5}{a_2}\right|^2= c_\rightarrow^2\left|1+\frac{c_\updownarrow^2}{c_\rightarrow}{\displaystyle\frac{e^{i(2\Phi_{c_\updownarrow}-\Phi_{c_\rightarrow}+\Delta)}}{1-r_\updownarrow e^{i(\Phi_{r_\updownarrow}+\Delta)}}}\right|^2$$
This clearly shows that the output of the grating is the result of the interference between two paths, one of which is resonantly-enhanced through multiple reflections between the grating and the top mirror. The grating sensor is therefore equivalent to a Mach-Zehnder interferometer with a Fabry-Perot cavity in one arm. Then, (6) is used to calculate the transmission profile for different values of the coefficients $t^2_\updownarrow$ and $c^2_\updownarrow$. In Fig. 2(a), the visibility of the obtained interference fringes is reported for $c^2_\updownarrow$ and $t^2_\updownarrow$ values for which a physical solution of (3) and (4) is possible. The visibility is maximum along the diagonal $t^2_\updownarrow =c^2_\updownarrow$. The map of Fig. 2(a) is shown for a fixed $\Phi _{t_\updownarrow }=0$, which sets all the phases in the GC coefficients. In Fig. 2(b)-(e), the variation of transmittance with the phase of the reflected field is shown for four representative situations [coloured dots in Fig. 2(a)], three along the diagonal and one in a low-visibility condition. For low vertical coupling, $c^2_\updownarrow \approx 0\,\%$, and high transmittance $t^2_\updownarrow \approx 100\,\%$, the interferometer is very unbalanced and almost all light is transmitted through the grating, resulting in low fringe visibility [orange dot in (a) and orange line in panel (b)]. By increasing $c^2_\updownarrow$ and consequently decreasing $t^2_\updownarrow$ we can increase the visibility, reaching fully constructive/destructive interference at the point $t^2_\updownarrow =c^2_\updownarrow =44\,\%$ (blue dot in panel (a) and blue curve in panel (c)). In this situation the interferometer is equivalent to a Mach-Zehnder interferometer, i.e. the resonant enhancement term in the denominator of (6) is absent. Increasing $r^2_\updownarrow$, i.e. moving along the diagonal towards the origin in Fig. 2(a), produces a second interference term, where both the phases in the numerator and the denominator of (6) play a role, resulting in a more complex change in the transmittance (see magenta dot and panel (d) for the case $t^2_\updownarrow =c^2_\updownarrow =33\,\%$). Finally close to the origin [red dot in panel (a) and curve in panel (e)], $t^2_\updownarrow =c^2_\updownarrow \approx 0\,\%$, the interferometer response is dominated by the Fabry-Perot cavity formed between grating and reflector, resulting in the sharp transmittance dip typical of side-coupled resonators. By tailoring the properties of the grating we can therefore continuously tune the interferometer response from the one of a balanced interferometer to the one of a resonant cavity, while keeping a unity visibility. This allows exploring the full space of sensitivity/dynamic range trade-off allowed by the physics of interference: from a strongly resonant response providing the largest sensitivity $dT/d\Delta$ and smallest dynamic range [Fig. 2(b)], to the typical balanced-interferometer response with relatively low sensitivity and high dynamic range [Fig. 2(d)], and anything in between.

 figure: Fig. 2.

Fig. 2. (a) Visibility of the transmission profiles for every possible combination $c_\updownarrow$, $t_\updownarrow$. The transmission profiles as function of phase delay at the highlighted points are shown in the panels (b),(c),(d) and (e).

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3. Simulations

We now examine the practical implementation of a GC-based sensor and validate the analytical model with numerical simulations. Before defining the GC properties we need to set the working wavelength ($\lambda _0$), refractive indices of the used materials ($n_0$, $n_1$, $n_2$,…) and the suitable out-coupling angle ($\theta _0$). From those the necessary grating period $\Lambda$ is calculated according to [19]

$$\Lambda = \frac{\lambda_0}{n_{eff}-n_0\sin\theta_0}$$
where $n_{eff}=F\,n_f+(1-F)n_e$ is the average effective refractive index over a single period of the grating, $F$ the filling factor and $n_f$, $n_e$ the effective refractive indices of the guided mode in the unetched and etched regions, respectively, as sketched in Fig. 1(c) and 1(d). We assume silicon waveguides, and first consider a symmetric case [waveguides surrounded by air Fig. 1(c)], then a more realistic asymmetric case of silicon on silica [Fig. 1(d)]. We perform two different simulations, both in absence of the mirror, in order to characterize the GC response. The simulations are performed in Lumerical using the finite-difference time-domain (FDTD) method and the designed structure is surrounded by perfectly matched layer (PML) absorbing boundary conditions. The schematics of the two simulations are reported in Fig. 1(e) and 1(f), where the different monitors are indicated. In the first simulation, panel (e), we assess the vertical out-coupling $c^2_\updownarrow$, the transmittance $c^2_\rightarrow$, the emission angle $\theta _0$ and the back-reflection when the light from the input waveguide is incident on the grating. Clearly the back-reflection is not implemented in our model thus it is important to design structures that show a low reflectance. In the presented design the maximum back-reflection amounts to $0.01\,\%$, allowing us to compare the simulation results to our 6-port model, expressed by (6). In the second simulation [Fig. 1(f)], where a plane wave illuminates the GC from top with an inclination angle $\theta _0$, the vertical transmittance $t^2_\updownarrow$ and the reflectance $r^2_\updownarrow$ are derived. The second estimation is affected by some uncertainty due to the assumed plane-wave profile which does not fully match the wave emitted from the GC. However, this has a limited impact on the design process as both initial simulations are only used as starting points. The values used for the comparison with the analytical model are obtained, by fitting the final transmission, once the top mirror is also introduced in the simulation.

Proceeding in this way, we simulated the effect of the mirror, varying its distance from the GC between $3.5\,\mu m$ and $6.5\,\mu m$ for three configurations with three different values of the vertical coupling $c^2_\updownarrow$. The parameters of the three designs are reported in Tab.1. The design process starts by fixing $\theta _0$ and $F$ and then looking for combinations of $h_w$, $h_e$ and $N_s$ which result in the desired values of $c^2_\updownarrow$ and $t^2_\updownarrow$. Evidently there is an interplay between those parameters - for example, we can achieve similar couplings having shallow etching and many periods or deep etching and few periods. The latter case could be used if a more compact design is favourable at the cost of a lower directionality of the grating. The results of the simulations for the case of a symmetric grating are summarized in Fig. 3. Here, three sets of panels (a-c), (d-f) and (g-i) relate to the cases of low (lGC), medium (mGC) and high vertical coupling (hGC), respectively. By comparing the obtained fringes at $\lambda _0$, as visible in the panels (b),(e) and (h) of Fig. 3, a good agreement with our model is achieved, using the values of vertical coupling reported in Table 1. In order to correct for the back reflection, which is not taken into account in the model, the transmittance values from the model in Fig. 3 are normalized to the maximum of the simulated transmittance. The choice of design sets the inevitable trade-off between sensitivity and dynamic range, where dynamic range is defined as the range of displacements for which an approximately constant sensitivity is obtained.

Tables Icon

Table 1. The features of the simulated symmetric or asymmetric configurations reported in Fig. 3–5, labelled as high coupling (hGC), medium coupling (mGC), low coupling (lGC), short GC (s-GC) and far GC (f-GC). The columns report the designed out-coupling angle $\theta _0$, the grating period $\Lambda$, the filling factor $F$, the waveguide height $h_w$, the etching depth $h_e$, the number of periods $N_s$ and the total length $L$ of the GC. The values in the last two columns show the parameters of the model shown in the next figure, for the symmetric (S) designs.

View Table
 figure: Fig. 3.

Fig. 3. Analysis of the three configurations reported in Table 1, with a symmetric geometry and the mirror spacing from $3.5\,\mu m$ to $6.5\,\mu m$. (a-c) Results for the lGC configuration. (a) The transmission map. (b) The cross-section at $\lambda _0$ and the normalized theoretical transmittance (c) The derivative profile given by the fitted model of the cross-section. (d-f) Results for the mGC configuration. (d) The transmission map. (e) The cross-section at $\lambda _0$ and the normalized theoretical transmittance. (f) The derivative profile given by the fitted model of the cross-section. (g-i) Results for for the hGC configuration. (g) The transmission map. (h) The cross-section at $\lambda _0$ and the normalized theoretical transmittance. (i) The derivative profile given by the fitted model of the cross-section.

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 figure: Fig. 4.

Fig. 4. Analysis of the hGC, mGC and lGC configurations reported in Tab. 1, with an asymmetric geometry and the mirror spacing from $3.5\,\mu m$ to $6.5\,\mu m$. The results for the lGC configuration: (a) the transmission map; (b) the cross-section at $\lambda _0$; (c) the derivative profile given directly by the cross-section profile. The results for the mGC configuration: (d) the transmission map; (e) the cross-section at $\lambda _0$; (f) the derivative profile given by the fitted model of the cross-section. The results for for the hGC configuration: (g) the transmission map; (h) the cross-section at $\lambda _0$; (i) the derivative profile given directly by the cross-section profile.

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 figure: Fig. 5.

Fig. 5. Simulation results for the short-GC (s-GC) and far-GC (f-GC) (Table 1), in the asymmetric configuration. (a) The transmission map for the s-GC design. (b) The cross-sections at various $\lambda$, corresponding to the dotted lines in panel (a). The y-axis range for each plot is between 0 and 1, which is shown only in the bottom one. (c) The transmission map for the f-GC design. (d) The cross-sections at various $\lambda$ corresponding to the dotted lines in panel (c). Again, the y-axis range for each plot is between 0 and 1, which is shown only in the bottom one.

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To illustrate this, the derivative profiles of the fitted models are reported in Fig. 3(c)–3(f)–3(i). The design with smallest vertical coupling (lGC) presents the largest displacement sensitivity, as expected from Fig. 2(b), but in a small range of displacements. Depending on the design, regions with different sensitivities and dynamic ranges can be distinguished. For example for Fig. 3(f) we have one region of high derivative, with sensitivity up to $\frac {\partial T}{\partial z_M}=15\,\mu m^{-1}$, in a range of $\Delta z_M=50\,nm$ and a second region with almost constant slope, around $\frac {\partial T}{\partial z_M}=2\,\mu m^{-1}$, for a larger range of $\Delta z_M=500\,nm$. The high-sensitivity regime, not identified in previous designs of GC-based sensors [11–14], is easily understood on the basis of (6) as a result of Fabry-Perot interference between grating and top mirror. Conservatively assuming a bias photocurrent of 100 $\mu A$ in an integrated photodetector, these sensitivities translate into shot-noise-limited displacement imprecision of 2 $fm/\sqrt {Hz}$ and 14 $fm/\sqrt {Hz}$, respectively, showing the potential of this approach for high-resolution metrology. These values are in good agreement with recent estimations based on experimental shot-noise limited noise floors in a similar system [13]. The possibility of continuously tuning the design for the desired sensitivity/dynamic range trade-off with minor structural changes is an important feature of this optomechanical sensor. Indeed, in many practical applications, such as atomic force microscopy, a dynamic range over several 10s to 100s $nm$ is required, which is incompatible with most cavity-based, near-field sensing approaches. We note that the sensitivity and dynamic range are fixed in a simple Mach-Zehnder or Michelson interferometer ($\frac {\partial T}{\partial z_M}\propto \frac {2}{\lambda _0}$, $\Delta z_M\propto \frac {\lambda _0}{2}$), while for a sensor based on a single optomechanical cavity they are tunable as they both scale with the ratio of optomechanical coupling rate $\frac {\lambda }{\partial z_M}$ and the resonance linewidth $\Delta \lambda$ ($\frac {\partial T}{\partial z_M}\propto \frac {1}{\Delta z_M}\propto \frac {1}{\Delta \lambda }\frac {\partial \lambda }{\partial z_M}$). However within a given optomechanical cavity system it is challenging to vary these parameters while keeping high visibility of the resonance.

In order to show that these transmission profiles are not only achievable on purely symmetric waveguides, like in Fig. 1(c), we also performed the same simulations for grating with the same features of Table 1 but built on top of a low-index SiO$_2$ cladding as shown in Fig. 1(d). For this design the the relations (4) among parameters derived for the symmetric case do not hold, but nevertheless the obtained transmittance have similar variations as we compare one by one the three coupling cases. As it could be expected, the presence of the buffer layer introduces some differences in the shape of the profiles with respect to the symmetric case, particularly for the lGC configuration where peaks instead of dips are observed. Overall, the simulations match well the numerical simulations also for the asymmetric case.

Furthermore, we investigated the flexibility of this approach in terms of specific application requirements. We first consider the case of integrated sensing arrays, where a small lateral dimension of the sensor is important to achieve high spatial resolution and/or large parallelism, e.g. for parallel AFM operation [20,21] or photoacoustic imaging [22]. To this aim we studied a much more compact design with a short GC with a total length of 6.5 $\mu m$, labelled as s-GC, whose details are also reported in Table 1. The s-GC is simulated for the asymmetric designs only and the results are shown in Fig. 5(a) and 5(b). The panel (a) shows the transmission map for a mirror set around $5\,\mu m$ and in panel (b) the cross-section at five different $\lambda$ are reported. Due to the limited number of periods the directionality is poorer than previous designs, which provides an almost wavelength-independent response in a range of $100\,nm$, along with a very compact footprint. We note that a large spectral range makes the integration with an on-chip laser easier, as it provides less stringent requirements in terms of wavelength and frequency noise. As a second design problem we consider the case when the mirror is set further away from the GC which is relevant when the displacement of an off-chip target must be measured. The details of the corresponding design, labelled as f-GC, are again reported in Table 1 and they where obtained starting from the mGC configuration and adjusted in order to achieve large visibility of the fringes in this situation. The panels (c) and (d) of Fig. 5 report the transmission map and the cross-sections for the f-GC configuration with the mirror placed around $75\,\mu m$ from the GC. In this case, the vertical Fabry-Perot has a smaller free spectral range, resulting in a stronger wavelength dependence. Nevertheless, remote sensing of the displacement can be realized in a relatively wide spectral range. Due to the large distance between mirror and GC, the optimal visibility is achieved at almost vertical emission corresponding to a $\lambda \approx 1.6\,\mu m$. We expect that sensing at substantially larger distances in the 100s $\mu m$ to $mm$ range can be also achieved by using shallow gratings which produce collimated output beams [23].

4. Conclusion

A theoretical model illustrating the working principle of a compact and versatile displacement sensor is presented. Differently from other types of integrated displacement sensors [2,6,24], which make use of a near-field coupling to sense displacement, this sensor employs a far-field coupling between the GC and the mirror allowing a remote displacement measurement. The developed model allows identifying a resonant regime of operation with high sensitivity, and provides insight on the trade-off between sensitivity and dynamic range. In order to prove its versatility, a number of practical designs are presented for different application cases. The small required footprint and the possibility to customize the sensitivity, dynamic range and spectral range according to the application requirements, is encouraging for the development of an integrated displacement sensors for metrology. By combining this optomechanical sensing approach with the recent developments in photonic integration [18,25,26], we envisage compact nano-optomechanical displacement sensors, with fully-integrated optical read-out, adequate for massively parallel scanning for high-throughput semiconductor metrology applications.

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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Figures (5)
Fig. 1.
Fig. 1. (a) Sketch of the grating sensor with fully-integrated laser and photocurrent read-out. The light coming from the integrated laser couples to the waveguide and reaches the grating. The interference between transmission through the grating and reflection from a movable top mirror produces a modulated photocurrent signal in the integrated photodiode. An AFM tip is schematically indicated on the sensing head. (b) The 6-ports model representing the grating sensor. The laser is coupled in from port 2, the output light from port 5 is coupled to the photodetecor, and the mirror-loop is connected through ports 3 and 6. Example of an symmetric (c) and asymmetric (d) shallow etched grating couplers, where $\theta _0$ , $h_w$ , $h_e$ and $N_s$ are respectively the target scattering angle, the waveguide height, the total etching depth and the number of grating periods. The (e) and (f) panels show the two basic simulation used to assess the efficiency of each output port.

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Fig. 2.
Fig. 2. (a) Visibility of the transmission profiles for every possible combination $c_\updownarrow$ , $t_\updownarrow$ . The transmission profiles as function of phase delay at the highlighted points are shown in the panels (b),(c),(d) and (e).

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Fig. 3.
Fig. 3. Analysis of the three configurations reported in Table 1, with a symmetric geometry and the mirror spacing from $3.5\,\mu m$ to $6.5\,\mu m$ . (a-c) Results for the lGC configuration. (a) The transmission map. (b) The cross-section at $\lambda _0$ and the normalized theoretical transmittance (c) The derivative profile given by the fitted model of the cross-section. (d-f) Results for the mGC configuration. (d) The transmission map. (e) The cross-section at $\lambda _0$ and the normalized theoretical transmittance. (f) The derivative profile given by the fitted model of the cross-section. (g-i) Results for for the hGC configuration. (g) The transmission map. (h) The cross-section at $\lambda _0$ and the normalized theoretical transmittance. (i) The derivative profile given by the fitted model of the cross-section.

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Fig. 4.
Fig. 4. Analysis of the hGC, mGC and lGC configurations reported in Tab. 1, with an asymmetric geometry and the mirror spacing from $3.5\,\mu m$ to $6.5\,\mu m$ . The results for the lGC configuration: (a) the transmission map; (b) the cross-section at $\lambda _0$ ; (c) the derivative profile given directly by the cross-section profile. The results for the mGC configuration: (d) the transmission map; (e) the cross-section at $\lambda _0$ ; (f) the derivative profile given by the fitted model of the cross-section. The results for for the hGC configuration: (g) the transmission map; (h) the cross-section at $\lambda _0$ ; (i) the derivative profile given directly by the cross-section profile.

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Fig. 5.
Fig. 5. Simulation results for the short-GC (s-GC) and far-GC (f-GC) (Table 1), in the asymmetric configuration. (a) The transmission map for the s-GC design. (b) The cross-sections at various $\lambda$ , corresponding to the dotted lines in panel (a). The y-axis range for each plot is between 0 and 1, which is shown only in the bottom one. (c) The transmission map for the f-GC design. (d) The cross-sections at various $\lambda$ corresponding to the dotted lines in panel (c). Again, the y-axis range for each plot is between 0 and 1, which is shown only in the bottom one.

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Tables (1)

Tables Icon

Table 1. The features of the simulated symmetric or asymmetric configurations reported in Fig. 3–5, labelled as high coupling (hGC), medium coupling (mGC), low coupling (lGC), short GC (s-GC) and far GC (f-GC). The columns report the designed out-coupling angle $\theta _0$ , the grating period $\Lambda$ , the filling factor $F$ , the waveguide height $h_w$ , the etching depth $h_e$ , the number of periods $N_s$ and the total length $L$ of the GC. The values in the last two columns show the parameters of the model shown in the next figure, for the symmetric (S) designs.

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Equations (7)

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$$\textbf{b}=\textbf{S}\cdot\textbf{a},\qquad \textbf{S} = \begin{pmatrix} S_{11} & \cdots & S_{16} \\ \vdots & \ddots & \vdots \\ S_{61} & \cdots & S_{66} \end{pmatrix}$$
$$\textbf{S} = \begin{pmatrix} \textbf{0} & \textbf{s} \\ \textbf{s} & \textbf{0} \\ \end{pmatrix},\quad \textbf{s} = \begin{pmatrix} r_\downarrow e^{i\Phi_{r_\downarrow}} & c_\downarrow e^{i\Phi_{c_\downarrow}} & t_\downarrow e^{i\Phi_{t_\downarrow}} \\ c_\downarrow e^{i\Phi_{c_\downarrow}} & c_\rightarrow e^{i\Phi_{c_\rightarrow}} & c_\uparrow e^{i\Phi_{c_\uparrow}} \\ t_\uparrow e^{i\Phi_{t_\uparrow}} & c_\uparrow e^{i\Phi_{c_\uparrow}} & r_\uparrow e^{i\Phi_{r_\uparrow}} \\ \end{pmatrix}$$
$$\begin{array}{l} r_\updownarrow^2+c_\updownarrow^2+t_\updownarrow^2=1 \\ 2c_\updownarrow^2+c_\rightarrow^2=1 \end{array}$$
$$\begin{array}{l} \Phi_{r_\updownarrow}={\pm}\arccos\left(-{\displaystyle\frac{c_\updownarrow^2}{2r_\updownarrow t_\updownarrow}}\right)+\Phi_{t_\updownarrow}\\ \Phi_{c_\rightarrow}={\pm}\arccos\left({\displaystyle\frac{r_\updownarrow^2-c_\rightarrow^2+t_\updownarrow^2}{2c_\rightarrow t_\updownarrow}}\right)+\Phi_{t_\updownarrow}\\ \Phi_{c_\updownarrow}=\frac{1}{2}\left[\Phi_{r_\updownarrow}+\Phi_{c_\rightarrow}\mp\arccos\left({\displaystyle\frac{t_\updownarrow^2-r_\updownarrow^2-c_\rightarrow^2}{2r_\updownarrow c_\rightarrow}}\right)\right] \end{array}$$
$$\begin{pmatrix} b_5\\ a_3e^{{-}i\Delta}\\ \end{pmatrix} = \begin{pmatrix} c_\rightarrow e^{i\Phi_{c_\rightarrow}} & c_\updownarrow e^{i\Phi_{c_\updownarrow}} \\ c_\updownarrow e^{i\Phi_{c_\updownarrow}} & r_\updownarrow e^{i\Phi_{r_\updownarrow}} \\ \end{pmatrix} \begin{pmatrix} a_2\\ a_3\\ \end{pmatrix}$$
$$T(\Delta) =\left|\frac{b_5}{a_2}\right|^2= c_\rightarrow^2\left|1+\frac{c_\updownarrow^2}{c_\rightarrow}{\displaystyle\frac{e^{i(2\Phi_{c_\updownarrow}-\Phi_{c_\rightarrow}+\Delta)}}{1-r_\updownarrow e^{i(\Phi_{r_\updownarrow}+\Delta)}}}\right|^2$$
$$\Lambda = \frac{\lambda_0}{n_{eff}-n_0\sin\theta_0}$$
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