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The extreme mode converter consists of two stages: stage 1 is an expander that converts the photonic waveguide mode into the slab mode and stage 2 is an apodized grating that couples out the slab mode into free space. Figure 1a provides a not-to-scale schematic diagram and a scanning electron microscopy (SEM) image of the chip structure with a color overlay to illustrate the mode transformation. To achieve the extreme mode conversion, the expander in stage 1 is designed to attain a wide 1D-Gaussian intensity profile with a flat wavefront in the slab. The grating in stage 2 is apodized by varying both the grating period and the duty cycle to realize a 2D-Gaussian beam in free space with a large beam waist (w0 > 100 µm). Specifically, the period is varied such that the phase of the out-coupling beam is designed to be flat to achieve a high-quality beam collimation. Figure 1b shows a microscope image for the fabricated converter, including specified coordinates and angles. θinc is the incident angle between the slab mode βslab and the grating vector ${\textstyle{{2\pi } \over \Lambda }}$ (y-axis). The azimuthal angle ϕ is also defined with respect to the grating lines (x-axis). Figure 1c illustrates an application in which a chip with two extreme mode converters is integrated to interrogate a gas-filled cavity in a compact optical system. To demonstrate this concept experimentally, we place a mirror above the chip surface at the plane of the mode overlap from the two converters to characterize the grating-mirror-grating coupling. The devices are designed to operate at the free-space wavelength of λ0 = 780 nm and silicon nitride (SiN) is chosen as the photonics guiding material to minimize the losses at this wavelength1.
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Stage 1 is a mode expander that converts the photonic waveguide mode to the slab mode with a flat wavefront and a 1D-Gaussian lateral power density distribution in the x-direction (the expander part in Fig. 1a). The basic principle for the mode expander is evanescent coupling. The coupling strength between the waveguide and the slab depends on the gap size g between them and the gap profile g(z′) can be designed to form a Gaussian intensity distribution in the slab (along the waveguide direction of the wave propagation, which is denoted as z′). First, to quantify the coupling strength between the waveguide and the slab, a finite element method (FEM) is used to evaluate numerically the complex effective refractive index (neff) as a function of g. A commercial FEM solver in the frequency domain is used to calculate the waveguide cross-sectional mode profile and its effective index. The power from the waveguide couples to the slab and is radiated in-plane, which results in an imaginary component of the index, thereby accounting for the mode power decay along the expander. To model this within a finite geometrical domain and to avoid the complications of using perfectly matched layers (PMLs) within an eigenmode calculation, we choose to introduce optical losses into the slab material instead. The 1 µm wide part of the slab that is closest to the waveguide is modeled as a perfect (lossless) dielectric and further away from the waveguide the slab material optical loss is increased adiabatically (Supplementary Figure S1). This ensures that there is no reflection of the slab mode back toward the waveguide from either within the slab or the domain boundary (Fig. 2a inset, Supplementary Figure S1, and Supplementary movie 1). The numerical results are independent of the choice of the loss profile if the loss is introduced at least 1 µm away from the gap and increased from 0 gradually over a distance of several micrometers.
Fig. 2 Stage 1: photonic waveguide mode to 1D-Gaussian slab mode conversion via evanescent coupling.
The numerically calculated effective refractive index (neff) of the photonic waveguide mode as a function of the gap size g between the waveguide and the slab: a The real part of neff (the dashed orange line: limiting value for g > 500 nm; inset: cross-sections of the schematic diagram and the FEM domain with the computed TE mode profile) and b the imaginary part of neff (the dashed orange line: fitting curve; text: fitting parameters). The height h0 and the width wwg of the waveguide are 250 nm and 300 nm, respectivelyFigures 2a and 2b show the real and imaginary parts, respectively, of the simulated neff. The insets in Fig. 2a show a schematic diagram of the simulation domain and the mode profile of the fundamental transverse-electric (TE0) mode. All the devices in this paper are designed based on the TE0 mode. The thickness of the SiN layer (waveguide and slab), which is denoted as h0, is set to 250 nm and the waveguide width, which is denoted as wwg, is set to 300 nm. All elements are made of SiN and clad with SiO2 (upper: > 1 µm, lower: ≈2.9 µm). The refractive indices of SiN and SiO2 are 2.01 and 1.45, respectively, at λ0=780 nm.
Re(neff) corresponds to the propagation constant β = Re(neff)β0 of the waveguide mode (β0 = 2π/λ0 is the free-space wavenumber) and determines the tilt-angle θtilt of the slab-mode direction of propagation relative to the waveguide. For the chosen value of h0, the effective index of the 1D TE slab mode is calculated to be 1.79 and the tilt angle between the waveguide and slab modes can be estimated by θtilt = cos−1(Re(neff)/1.79). In Fig. 2a, Re(neff) approaches 1.578 for g > 500 nm. This indicates that the evanescent-coupled slab mode would have the same tilt angle, namely, θtilt = cos−1(1.578/1.79) = 28.18°, for gap sizes that exceed 500 nm. In other words, the evanescent-coupled waves in the slab will be collimated if g(z′) > 500 nm. In Fig. 2b, Im(neff) corresponds to the power loss of the evanescent coupling and decreases approximately exponentially as the gap size increases. Using this data and assuming adiabatic variation of the gap profile, g(z′) can be designed for the desired power distribution along z′. The phase at the slab boundary is the same as the phase in the waveguide and increases linearly along z′ for a constant value of Re(neff). For g(z′) > 500 nm, the variation in gap size does not shift θtilt and only affects the power distribution. Smaller gaps can be employed; however, variation in Re(neff) may need to be compensated. The slab boundary and the waveguide can be appropriately curved to achieve the desired wavefront for the slab mode. In our design, the slab edge is straight, thereby creating a flat wavefront for the collimated slab mode.
The following procedures are used to design g(z′) and achieve the correct Gaussian intensity profile of the slab mode. The optical power in the waveguide, which is denoted as P(z′), can be expressed as
$$ \frac{{dP\left( {z' } \right)}}{{dz' }} = - P\left( {z' } \right)\alpha \left( {z' } \right) $$ (1) where α(z′) is the loss coefficient, which can be expressed as $\alpha \left({z' } \right) = \frac{{4\pi }}{{\lambda _0}}{\rm{Im}}\left({n_{{\rm{eff}}}} \right)$. The initial power in the waveguide is P(−∞) = 1. The power density in the slab $\frac{{dP_{\rm s}\left({z' } \right)}}{{dz' }}$ forms a Gaussian mode with a beam waist of w and can be represented as:
$$\frac{{dP_{\rm s}\left( {z' } \right)}}{{dz' }} = C\exp\left( { - \frac{{2z' ^2}}{{w^2}}} \right) $$ (2) The coefficient, which is expressed as $C = \frac{1}{w}\sqrt {\frac{2}{\pi }}$, is obtained by setting the total power to 1 and integrating: ${\int}_{-\infty }^{\infty} {\frac{{dP_{\rm s}\left({z' } \right)}}{{dz' }}dz' = 1.}$ Per energy conservation, the total power in the waveguide and the slab should be equal to 1, i.e., P(z′) + Ps(z′) = 1, and the loss in the waveguide should be equal to the coupling power of the slab at that segment, i.e., dPs(z′) = −dP(z′). Rewriting these two conditions, the following equations are obtained:
$$ P(z' ) = 1 - {\int}_{-\infty}^{z'} {\frac{1}{w}\sqrt {\frac{2}{\pi }} \exp \left( {\frac{{ - 2\zeta ^2}}{{w^2}}} \right)d\zeta } $$ (3) $$ \frac{1}{w}\sqrt {\frac{2}{\pi }} \exp \left( {\frac{{ - 2z^{' 2}}}{{w^2}}} \right) = P\left( {z' } \right)\frac{{4\pi }}{{\lambda _0}}\rm{Im} \left( {n_{{\rm{eff}}}} \right) $$ (4) Solving these two equations and using the relation ${\rm{Im}}\left({n_{{\rm{eff}}}} \right) = a{\exp} \left(-bg(z') \right), $ where a and b are the fitting coefficients from Fig. 2b, the gap profile g(z′) is:
$$g(z' ) = \frac{1}{b}\ln \left\{ {\frac{1}{a}\frac{{\lambda _0}}{{\sqrt 2 \pi ^{3/2}w}}\frac{{\exp \left( { - 2z' /w^2} \right)}}{{1 - {\rm{erf}}\left( {\sqrt 2 z' /w} \right)}}} \right\} $$ (5) The beam waist w is of the Gaussian distribution along the waveguide direction. The actual beam waist w0 of the resulting 1D-Gaussian slab mode, which is normal to its direction of propagation in the slab, is obtained as w0 = w sin(θtilt). In our design, the beam waist is set to w0 = $100\sqrt 2$ µm.
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Stage 2 is an optimized apodized grating with a spatially varying duty cycle and period that outcouples the 1D Gaussian slab mode into the 2D free-space Gaussian. The grating lines are straight and parallel. The slab mode is collimated so that the phase is invariant along the grating lines, while the intensity is varied only gradually. Therefore, to create the Stage 2 out-coupler, it is sufficient to solve a 2D problem with translational invariance along the grating lines to create a collimated Gaussian profile in the plane that is normal to the grating lines (in the y-direction).
For a collimated Gaussian output, further simplifications could have been applied by leveraging the slow variation of the intensity and phase across the grating. However, the 2D TE scattering problem from the slab mode into the free space can be quickly and accurately solved for the ≈300 µm grating using a commercial finite element frequency domain solver. This makes it possible to apply a more general numerical optimization technique to solve the inverse problem of finding a grating design that optimizes the coupling between the input slab mode and any arbitrary prescribed free-space mode (i.e., the "inverse design"). By adding a geometric deformation field into the finite element problem setup, we could employ an efficient gradient-based optimization technique to optimize the values of the 14 scalar parameters that define the device geometry and the Gaussian beam, including the spatially varying period and duty cycle, as described in the Materials and Methods section.
Figure 3a shows a schematic diagram of the resulting apodized grating in the yz-plane with spatially varying period Λ(y) and grating width wg(y). The duty cycle is defined as 1-wg(y)/Λ(y). The thicknesses of the layers are h0 = 250 nm, hg = 85 nm, hs = 2.9 µm, and hu = 2.8 µm. Figures 3b and 3c show the optimization results for the grating period and duty cycle, respectively. The insets in each figure show the optimized polynomial coefficients. Figure 3d shows the numerically simulated out-coupling power flow (blue), its Gaussian fit (black dash-dot line), and the wavefront phase error (orange) when the geometric parameters of Figs. 3b and 3c are used. Figure 3e is the magnified view of the simulated electric field profile (Ex). The resulting optimal out-coupling angle of the Gaussian is 2.2° in free space and the wavefront error is < 2π/20 rad over the beam. The power distribution fits well with the Gaussian profile; however, the beam full width at half maximum (FWHM) is ≈103 µm, which is ≈15% lower than the desired outcome. The port width is numerically forced to FWHM = 117 µm by an added term in the cost function and a constraint and a small coupling penalty is associated with this width mismatch. We speculate that the optimization algorithm is balancing this mismatch loss with additional losses (increased wavefront error and loss into the substrate) that are associated with extending the Gaussian or, alternatively, that the parameters of the optimization algorithm are not selected perfectly, thereby resulting in a small residual error. The calculated Gaussian port coupling is 68 %; the power that is flowing down into the substrate is 26 %; the slab mode reflection and transmission are negligible. The remaining power accounts for the mode mismatch with the Gaussian port.
Fig. 3 Stage 2: 1D-Gaussian slab mode to 2D-Gaussian beam conversion.
a A schematic diagram of the apodized grating with the following geometric parameters: h0 = 250 nm, hg = 85 nm, hs = 2.9 µm, and hu = 2.8 µm. The grating period Λ(y) and grating width wg(y) are apodized. The numerically optimized grating b period Λ(y) and c duty cycle 1-wg(y)/Λ(y) (insets: optimized polynomial coefficients). d FEM results of the out-coupled beam: power flow (blue), Gaussian fit (black dash-dot line), and wavefront phase error (orange). Dashed lines indicate the upper and lower bounds of the phase error, which are within ± 2π/40. e The electric field profile (Ex) within a portion of the FEM simulation domainDue to the constructive/destructive interference with the downward-outcoupled light that is back-reflected up from the Si wafer surface, the numerical simulation demonstrates a periodically varying upward coupling efficiency from ≈67 % for the optimum oxide thickness down to ≈35 % for ≈150 nm thicker or thinner oxide. As the destructive interference decreases the out-coupled intensity, the optical power in the slab mode propagates further along the grating in the y-direction, thereby resulting in significant widening of the out-coupled Gaussian beam. Experimentally, it is likely that some oxide thickness variation existed between runs, which contributed to the observed Gaussian beam width variation in the y-direction.
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To demonstrate the extreme mode converter that was designed in Section 2, we have fabricated and tested the devices, as described in the Materials and Methods section.
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To characterize the mode profiles of the converted Gaussian beam, microscope images of the beam on the grating were captured. Figure 4a shows a microscope image of the converted Gaussian beam. The inset shows the direction of the gratings: the grating lines are parallel to the x-axis and perpendicular to the y-axis. The slab mode is incident from the top of the image. Figures 4b and 4c show the normalized powers of the beam that are integrated along the y- and x-directions and projected to the x- and y-axes, respectively. The dashed blue lines are the data and the red lines are the reference Gaussian curves. The FWHM and the beam waist, namely, w0, of each projection are also shown in each figure. The power distributions of the beam fit well with the Gaussian curves within the grating area and the beam waist on both the x- and y-axes is w0 ≈160 µm, which is reasonably close to the design target value (w0 $= 100\sqrt 2$ = 141.4 µm). The fabrication imperfections and the index differences between the model and the real materials may have caused these errors. In Fig. 4c, the Gaussian shape is cut at the beginning part of the gratings; this is due to the minimum feature size limit of the grating (20 nm) in the fabrication, which is similar to the FEM result in Fig. 3d.
Fig. 4 Gaussian mode profile on a grating.
a A microscope image of a converted Gaussian beam on a 300 µm×300 µm grating. The grating lines are parallel to the x-axis (as schematically indicated). The scale is calibrated based on the known physical size of the grating. b, c are the projected images of a, which show the Gaussian mode profiles along the x- and y-axes, respectively (blue dashed lines: data; red solid lines: fitting curves). The full width at half maximum (FWHM) and the beam waist w0 are shown in each figure. d Measured mode profiles (projected on x-axis) for various gap sizes that have constant ± 40 nm variations on the gap profile g(z′) (blue: g - 40 nm, orange: g, and yellow: g + 40 nm). e Numerically calculated (ODE) mode profiles that are similar to d. The uncertainties in the characterized beam waist are approximately ± 1 µm, as determined by the Gaussian fitUniformly increasing or decreasing g(z′) shifts the center position of the slab mode and the Gaussian mode in the grating (along the x-axis). It also affects w0 and the FWHM in the x-direction. Figures 4d and 4e show the measured and numerically calculated intensity mode profiles that are projected to the x-axis. For the numerical calculation, Eq. 1 is solved via the ordinary differential equation (ODE) solver with the gap profile g(z′) of Eq. 5. The orange line is the original design with the g(z′) and the blue and yellow lines are cases in which the gap is uniformly decreased or increased, respectively, by 40 nm, i.e., g(z′)−40 nm and g(z′)+40 nm. In both the experimental and numerical cases, the narrower gap size shifts the center position to the left (closer to the beginning of the evanescent coupler). A narrower gap increases the coupling, thereby increasing the slab mode intensity at the beginning of the coupler; hence, less light remains in the waveguide to couple out toward the end, thereby decreasing the slab mode intensity there. The opposite is observed for a wider gap. Furthermore, the smaller gap size yields a narrower w0; again, opposite is observed for the larger gap size. In the numerical results, g(z′)+40 nm yields w0 = 164.0 µm, which is similar to the beam waist from the experiment (w0 ≈168 µm ± 1 µm); this gap variation is a possible reason for the ≈15 % width difference between the designed w0 = 141.4 µm and the experiments.
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To check the beam collimation of the converted Gaussian mode, its far-field intensity is measured as a function of the angle (Fourier space) by capturing the back focal-plane (BFP) images (Fig. 5a; Materials and Methods). A movable variable-diameter circular aperture was placed in the image plane of the microscope, before the beam splitter, thereby enabling full or spatially selective evaluation of the far-field light, i.e., of light that is coming from the whole grating or any part of the grating that is selected by the aperture. Figure 5b shows a real image at the grating and the blue line indicates the outline of the aperture with a diameter of ≈250 µm. Figure 5c shows the angular (Fourier) space images at the BFP; the colored lines represent the grid (one degree/line) of the polar angle θ. The gray spots are the actual beam images from each device. All spots are of small size and near diffraction-limited; otherwise, the spots would spread broadly over wider angles. The white dot serves as a guide for the eye, with a diameter that corresponds to the ideal diffraction-limited beam's FWHM. For each device set, we have two identical mode converters with opposite orientations (mode converter 1 is rotated 180° relative to mode converter 2); thus, in Fig. 5c, the upper four gratings are in opposite orientations relative to the lower four gratings. Device rot0 is a device whose grating lines are nominally orthogonal to the incident slab mode, which is designed to have a tilt angle of θtilt = 28.18° relative to the waveguide. Devices rot2, rot3, and rot4 are devices with additional rotations of 2°, 3°, and 4°, respectively, of the grating relative to the incident slab mode. We have rotated the gratings to engineer the Gaussian beam polar angle and direction. One aim is to maximize the modal overlap between the two beams at a certain height such that a flat mirror can be used to couple a beam from one device into the other, as illustrated in Fig. 1c. A detailed analysis follows in Section 4.
Fig. 5 Back focal plane (BFP) measurement.
a The BFP measurement setup for characterizing the beam profile in both angular (Fourier) space and real space. b Real-space image of the converted Gaussian beam (blue line: image plane aperture with a diameter of 250 µm). c A BFP composite image of the Gaussian beams for various grating rotational angles relative to the incident slab mode (rot0: 90° − θtilt = 61.82° rotation from the expander waveguide, rot2 = rot0 + 2°, rot3 = rot0+3°, and rot4 = rot0+4°). The BFP image X (Y)-axes are oriented along (normal to) the grating lines in b. Two identical devices that are rotated 180° are measured to establish the origin (surface-normal). The white dashed lines are the theoretical out-coupling angles that are obtained via Eqs. 7 and 8. A white dot indicates the ideal diffraction-limited beam's full width at half maximum (FWHM)Figure 6a shows a magnified view of the beam spots (rot0) from top to bottom with image-plane aperture diameters of d ≈ 50 µm to 250 µm (left: grating 1, right: grating 2). As the aperture size increases in real space, the spot size in the Fourier domain decreases accordingly. Figures 6b–6e show cross-sections of the normalized powers through the center of the spots (blue dots: data, red lines: fitting curves). Figures 6b and 6c show the cross-sections along θx for d ≈ 50 µm and d ≈ 250 µm, respectively, and Figs. 6d and 6e show the cross-sections along θy for d ≈ 50 µm and d ≈ 250 µm, respectively. The Gaussian distribution in the x-direction (or θx) relates to the mode expansion from the waveguide to a slab mode (stage 1), while the Gaussian distribution in the y direction (or θy) is formed by the apodized grating during stage 2; they are formed by the two independent stages. More importantly, for a large aperture of d ≈ 250 µm, as in Figs. 6c and 6e, the measured angular FWHMs are close to the expected FWHM of a diffraction-limited beam, namely, FWHM=$\frac{{2\sqrt {2{\ln}2} \lambda _0}}{{w_0\pi }} = 0.2094^ \circ$ for w0=160 μm, and demonstrate a good beam collimation.
Fig. 6 Near-diffraction-limited Gaussian beam.
a Magnified back-focal-plane (BFP) images from mode converters 1 and 2 for various aperture diameters from 50–250 µm. Clipping the beam by the aperture increases diffraction. b–e Normalized powers of the BFP images of (a) at the center of each axis: (b, c) along θx and (d, e) along θy (blue circles: data; red lines: fitting Gaussian curves). (b) and (d) are cases in which the aperture diameter is d ≈ 50 µm, while (c) and (e) are cases in which d ≈ 250 µm. The uncertainty in the characterized FWHM is approximately ±0.005°, as determined by the Gaussian fit -
To integrate and optically couple the photonic chips with other systems in free space, the extreme mode converter can be used as a building block for establishing optical coupling between the systems. To explore such an engineering opportunity, we have placed two couplers with opposite orientations on the same photonic chip and arranged their locations and outcoupling angles (polar and azimuthal) to create a large beam overlap several millimeters above the chip. Finally, a flat mirror is introduced at that location and grating-to-grating coupling experiments are conducted to quantify the coupling of the generated beam back to the photonic system. Using this setup, we can also evaluate experimentally the overall mode converter efficiency and account for the mode mismatch loss between identical mode converters.
Figure 7a shows a microscope image of the two extreme mode converters facing in opposite directions with a center-to-center separation distance of ≈475 µm. In Fig. 1b, the incident angle θinc is defined for the angle between the slab mode βslab and the grating vector Λ (y-axis). For devices rot0, rot2, rot3, and rot4, the designed θinc=0°, 2°, 3°, and 4°, respectively. The tilt angle θtilt is fixed at the nominal value of 28.18° and the gratings are rotated to adjust θinc. In addition, the x- and y-axes are referenced to the grating, not a global frame. The azimuthal angle φ and the polar angle θ are defined with respect to the grating lines (x-axis) and the surface normal (z-axis), respectively (inset of Fig. 7a). To verify the propagation direction of the slab mode (θtilt) after stage 1 (expander), in a separate experiment, a nominally identical device was co-fabricated, in which a series of holes in front of the grating were deliberately introduced to serve as scatterers for the slab mode. Figure 7b shows a microscope image of such a device (rot0) with TE0 input. The scatterers form shadows in the slab mode along its propagation direction, which are made visible by the grating. As expected, the angles of the shadows are close to θinc ≈ 0° and the long, uniform-contrast shadows qualitatively demonstrate satisfactory collimation of the slab mode. This further validates the performance of the mode expander.
Fig. 7 Out-coupling angles and modal overlap.
a A microscope image of the two extreme mode converters. The inset shows the definitions of the polar (θ) and azimuthal (φ) angles. b A magnified converter image with scatterers that are embedded into the slab. c Polar (θ, blue) and azimuthal (φ, orange) angles of the out-coupled beam as functions of incident angle θinc. The solid lines are analytical calculations and the points with error bars are the measured angles. The error bars indicate the one-standard-deviation uncertainties that are propagated from the beam center estimates in the images. d Microscope images of the two out-coupling beams at various distances z (left: rot2; right: rot4). The red circles indicate the z-positions of the maximum overlap. e 3D stack images of d. f Measured modal overlap percentages of the two out-coupling beams as a function of z (blue: rot2 and orange: rot4)The polar θ and azimuthal φ angles of the out-coupling beam can be engineered by varying the incident angle θinc via grating rotation. Due to the momentum conservation in the grating plane, the out-coupling beam angles should follow
$$ k\sin \theta \sin \phi = \beta _y - \frac{{2\pi m}}{{\rm{\Lambda }}}\quad \left( {m} {\rm : integer} \right) $$ (6a) $$ k\sin \theta \cos \phi = \beta _y $$ (6b) where k = 2π/λ, βy = βslab cos(θinc), βx = βslab sin(θinc), Λ is the effective grating period, and the propagation constant of the slab mode can be represented as $\beta _{{\rm{slab}}} = \frac{{2{\rm{ \mathsf{ π} }}}}{{\lambda _0}}n_{{\rm{eff}}}$. Rewriting Eq. 6 for θ and φ,
$$\sin \theta = \sqrt {\left( {n_{{\rm{eff}}}\sin \theta _{{\rm{inc}}}} \right)^2 + \left( {n_{{\rm{eff}}}\cos \theta _{{\rm{inc}}} - \frac{{m\lambda_0 }}{\Lambda }} \right)^2} $$ (7) $$\tan \phi = \frac{{\cos \theta _{{\rm{inc}}} - \frac{{m\lambda _0}}{{n_{{\rm{eff}}}\Lambda }}}}{{\sin \theta _{{\rm{inc}}}}} $$ (8) To avoid losing light into multiple diffraction orders and to create a near-vertical Gaussian beam for θinc = 0, m = 1 is chosen and $1 \gg n_{{\rm{eff}}}\left({\frac{{\lambda _0}}{{\Lambda n_{{\rm{eff}}}}} - 1} \right) > \, 0$. The blue and orange lines in Fig. 7c show the polar (θ) and azimuthal (φ) angles as functions of θinc, which follow Eqs. 7 and 8, respectively. For the parameters, neff = 1.79 and Λ = 425 nm are assumed. The points represent the experimentally measured angles for θinc=0°, 2°, 4°, and 6° and match the analytical estimates. Figure 7d shows the captured images of the rot2 (left) and rot4 (right) devices from this set at various z-positions. For each device, the z-positions of the maximum modal overlap are marked with red circles (≈4 mm for rot2 and ≈2 mm for rot6). Figure 7e shows the 3D stack images of Fig. 7d, the azimuthal angles and the mode overlap positions.
We have also experimentally estimated the overlap percentage of the two beams from each converter. Each of the beam images was captured at various heights and the following equation was used to extract the modal overlap:
$$ {\rm{Overlap}}\left( z \right) = \frac{{{\int} {\sqrt {I_{1}\left(z\right)I_{2} \left(z\right)} dA} }}{{\sqrt {{\int} {I_{1} \left(z\right)dA} {\int} {I_{2} \left(z\right)dA} } }} \times 100\,\,{{\% }} $$ (9) where I1(z) and I2(z) are the beam intensity images from each device, captured separately. The intensity measurement is not phase sensitive and this equation provides an upper bound on the expected mode coupling loss by assuming perfect phase matching between the two beams, such as perfect collimation with beam waists at the mirror location. The blue line in Fig. 7f represents the measured overlap percentage of rot2 and the orange line in Fig. 7f represents that of rot4. As expected from Fig. 7f, the z-positions of the maximum overlap for rot2 and rot4 are ≈4.5 mm and ≈2 mm, respectively. The maximum overlap percentage for rot4 exceeds 90 %, thereby fulfilling a necessary condition for low-loss grating-to-grating coupling, in which the radiating beam that is produced by one mode converter is reflected back to the chip (and a second converter) by a mirror that is placed at this position.
Figure 8a shows the experimental setup for the grating-to-grating coupling that is used to characterize the single-mode coupling efficiency of the mode converter in a realistic application scenario. The photonic chip is glued to a fiber array for input/output coupling and a mirror is placed ≈2 mm above the chip, thereby resulting in maximum overlap for this rot4 device. The inset image shows the top view of the chip with the glued fiber array. Figure 8b shows a schematic diagram of the entire device structure with key loss parameters: PF: the fiber-to-chip edge-coupling loss; PC: the connecting waveguide and extreme mode converter loss; and PM: the mode-mismatch loss between the two beams. The typical loss for the edge-coupling is PF = −3 dB ± 0.5 dB, which was determined via multiple measurements using short waveguide loop-back structures that were connected to the inverse-tapered couplers. Here and below, the uncertainties are measured and the statistical uncertainties are propagated one standard deviation. Measuring the outgoing Gaussian beam power and assuming the abovementioned value for the edge-coupling, the loss for the extreme mode converter, including the SiN waveguide, stage 1 mode expander and stage 2 grating coupler, is PC = −4.5 dB ± 0.5 dB, which corresponds to an overall experimental fiber to Gaussian beam loss of approximately PF + PC = −7.5 dB ± 0.5 dB for each device. Using the setup in Fig. 8a, the light is successfully coupled back to the chip. By subtracting the independently measured fiber-to-Gaussian-beam losses for each of the two mode converters from the total fiber-to-fiber loss, the excess loss due to the mode-mismatch between the two beams is experimentally determined to be PM = −2.5 dB ± 1.0 dB, which results from a combination of lateral and angular beam misalignments and wavefront errors.