1. Introduction

The design rules for a grating that couples light from a single mode fiber into a single mode waveguide are well understood [1,2]. The size, period and depth of the grating can be adjusted to accept radiation of a given wavelength, direction and cross-section. If the radiation is coming from an arbitrary direction or from many directions simultaneously, then the coupling into a single mode waveguide is fundamentally limited. This limit is apparent from the fact that combining power from incoherent modes is not possible. Otherwise, it would be possible to concentrate thermal radiation beyond the black body limit, which cannot be achieved [3–5]. Nevertheless, coupling between two multimode systems can be efficient, as the example of coupling from a multimode fiber into free space illustrates.

Another important example is the coupling from integrated photonics platforms to and from optical fibers. In [6] the excitation of several fiber modes by higher order modes in planar waveguides with finite widths is discussed. A 2D grating is excited from two opposing sides via guided waveguide modes and, depending on the mode profiles and their phases, coupling into different fiber modes can be realized. Besides waveguides with finite widths, a slab system can be employed that provides guidance of modes in arbitrary directions within the 2D plane. To the best of our knowledge, a multimode coupler between a waveguide slab and a multimode fiber has not yet been optimized. Here, we introduce a grating coupler that allows extraction of multimode signals from a 2D system into 3D space or, conversely, the excitation of modes in the 2D system from free space. We use a silicon slab coupling into a multimode optical fiber as a model system. Simulation and experimental results are compared and the limitations of multimode coupling are summarized.

We follow a three-step approach for the grating design.

To exemplify the importance of multimode couplers, we refer to the concept of a 2D integrating cell that we have introduced previously [8]. Such a cell realizes a large optical path length in a very small area, which is important for most sensing applications, but it is necessarily a multimode system. Multimode grating couplers are relevant for either the excitation or detection of light in such a system.

2. Coupling system

An overview of the different coupling mechanisms between various single mode and multimode systems is given in Appendix A. Here, we treat the coupling between a multimode slab waveguide and multimode optical fibers as shown in Fig. 1. Our model system is a silicon-on-insulator slab with a silicon thickness of 220 nm and an oxide thickness of 3 µm, which supports a single TE mode for a wavelength between 1.5 and 1.6 µm. Multimode fibers with a core diameter of $d_{\text{f}}=200 \text{µm}$ are employed and, for simplicity, the area of the grating (indicated by the blue dashed line in Fig. 1 is made equal to the cross-section of the fiber core. The numerical aperture of the input fiber is $NA=0.22$, whereas the output fiber has a numerical aperture of $NA=0.39$. The difference in the numerical apertures was chosen in order to model a typical experimental system where excitation is realized with a single mode fiber-based ASE (Amplified Spontaneous Emission) source. The fiber of the ASE source has a numerical aperture of $NA=0.22$ and is connected to the input multimode fiber which maintains the single mode nature, but has a larger diameter, so essentially acts like a beam expander, which we have confirmed experimentally.

 

Fig. 1 Schematic setup for the coupling between a multimode fiber and a silicon slab (input) and vice versa (output). Enlarged image of one of the grating couplers is presented. The facets of two multimode fibers are positioned close to the two gratings. An optical signal is introduced by an optical fiber with $NA=0.22$ (determined by the excitation with a single mode fiber-based ASE source) and extracted by a multimode fiber with $NA=0.39$.

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Following the proposed three-step approach, we develop a grating coupler for coupling between the multimode fibers and the silicon slab.

 

Fig. 2 (a) Scattering element based on a ring-shaped air hole. (b) Wavevector distribution for a ring-shaped scatterer with an inner radius of ${\text{r}}_1=510 \text{nm}$, an outer radius of ${\text{r}}_2=730 \text{nm}$ and a height of $\text{h}=70 \text{nm}$, obtained via a spatial Fourier transformation of the simulated electric field in a plane 1 µm above the silicon surface at a wavelength of 1.55 µm after excitation of a guided TE mode propagating from left to right in x-direction. The black circle includes all k-vectors that can be coupled from the slab into air. The blue and gray dashed lines indicate the k-vectors which are accepted by an optical fiber with $\text{NA}=0.39$ and $\text{NA}=0.22$ respectively.

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In order to assess the out-of-plane scattering angle, we used a 3D electromagnetic solver in time domain [11]. Regarding the specific shape of each element, we considered both disk and ring shaped elements, and found that rings exhibit a better angular selectivity. For more detail of the simulation, please see Appendix C. Figure 2(a) shows the design of a canonical structure, with the radii $r_1$ and $r_2$ as the key parameters. We then varied $r_1$ and $r_2$ in order to optimize the scattering angle.

The best directionality was observed for an inner radius of $r_1=510 \text{nm}$ and an outer radius $r_2=730 \text{nm}$. The k-vector distribution of the up-scattered signal is shown in Fig. 2(b). By integrating the power scattered into the upper and lower plane, we observe that 24% of power is scattered in the upward direction. The black circle represents all k-vectors that can scatter from the silicon slab into free space, while the blue and gray-dashed lines represent the k-vectors that can be coupled into an optical fiber with $\text{NA}=0.39$ and $\text{NA}=0.22$ respectively. Clearly, the k-vector components outside of the blue/gray circle cannot couple to the fiber and are lost. We note that the broad k-vector distribution we observe is a fundamental limit of the circular scattering element and. It should be mentioned that varying the etching depth will influence the scattering angle but it will also not help to improve the directionality of the symmetric scattering element. Therefore, we decided to fix the etching depth to 70 nm and adjust the scattering angle by a proper choice of radii of the ring-shaped scatterers. We discuss a possible improvement of the directionality by subwavelength gratings in Section 4. The percentage of power scattered into the acceptance angle of the output fiber ($\text{NA}=0.39$) is 39%. Given the 24% scattering efficiency in the upward direction, the total efficiency of coupling into the fiber is $0.24\text{ x }0.39=9.3\text{ \%}$ or −10.3 dB.

Similarly, we calculate the coupling efficiency of the in-coupling grating, which is limited by the numerical aperture of the input multimode fiber ($\text{NA}=0.22$). For backward excitation, the percentage of power upscattered by the grating into the acceptance angle of the input fiber is thus given by 9.5%, resulting in a loss of −16.4 dB. Due to reciprocity, we assume that the same loss is present for the excitation of the grating from the fiber.

 

Fig. 3 (a) Top view of the simulation volume used to estimate the scattering behavior of an array of ring-shaped scattering elements. The red triangle represents an electromagnetic dipole that is positioned inside the silicon slab in order to excite a broadband TE signal. The blue lines indicate the electric boundaries which are acting as mirrors and thus define a square array of ring-shaped holes with lattice constant a. (b) Cross section of the structure. (c) Energy decay over time for a scattering element as shown in Fig. 3(a) and a lattice constant of 2.5 µm, 3.5 µm and 4.5 µm after excitation with the electromagnetic dipole. The gray dashed line indicates the target energy decay.

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As a next step, the scattering strength of the structure is adjusted via the density of scatterers to match its decay length $s_{\text{decay}}$ to the size of the grating. This approach is similar to the “critical coupling” condition and it assumes that the decay length is measured along a straight line and in-plane scattering is negligible. We assume that the effective propagation path of the guided signal through the grating is equal to the average distance between two points on the circumference of the grating given by $s_0=2d/\pi =127 \text{µm}$ [8]. The density of scatterers is adjusted by placing the scatterers onto a square grid and adjusting the lattice constant a. The decay length is then determined via the time-dependence of the electromagnetic energy decay of a dipole excited in the slab. We use electric boundary conditions (Fig. 3(a) and 3(b)), excite a broadband TE polarized signal between wavelengths 1.5 µm and 1.6 µm and determine the decay length via the group velocity of the guided mode $s_{\text{decay}}=c\tau /n_{\text{g}}$.

We varied the lattice constant between 2.5 µm and 4.5 µm and observed the corresponding energy decays (Fig. 3(c)). Since the target propagation path is 127 µm, the target energy decay is −2.7 dB/ps by using a group velocity of ${\text{n}}_{\text{g}}=3.75$ (determined via the dispersion of the mode). We note that the lattice constant of $\text{a}=3.5 \text{µm}$ closely matches the requirement.

3. Experimental results

Using e-beam lithography and reactive ion etching, input and output gratings with the design parameters determined above were prepared on an SOI wafer (Fig. 4(a)). Since the multimode grating excites light in all directions in the slab, the power decays as $\text{d}/(2\text{πR})$ between two gratings placed a distance $R$ apart. We verified this relationship using the cutback technique (see Appendix D). In the following, the distance between input and output grating is fixed to $\text{R}=3 \text{mm}$, resulting in a spreading loss of −20 dB. This spreading loss is subtracted from the measured loss to consider coupling losses only.

 

Fig. 4 (a) Optical micrograph of a grating with 200 µm diameter consisting of ring-shaped scattering elements. Inset: SEM image of six scattering elements. The rings have an inner radius of ${\text{r}}_1=510 \text{nm}$ and an outer radius of ${\text{r}}_2=730 \text{nm}$ and are arranged in a square lattice with a lattice constant of $\text{a}=3.5 \text{µm}$. (b) Measured transmission using pairs of multimode couplers in a setup as shown in Fig. 1 normalized to the emission spectrum of the ASE source where losses caused by intensity reduction in the silicon slab are already subtracted. Maximal transmission is observed for a lattice constant of 3.5 µm at a wavelength of 1585 nm resulting in a coupling efficiency between two couplers of −34 dB.

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The measurement follows the geometry of Fig. 1, with the facets of two multimode fibers being positioned vertically above the gratings. Figure 4(b) shows the normalized transmission measured for three pairs of multimode couplers with lattice constants 3.5, 4.0 and 4.5 µm. Maximal transmission is observed for a pair of multimode couplers with a lattice constant of 3.5 µm at a wavelength of 1585 nm. This result is in good agreement with the simulation (Fig. 3), which predicts an optimal case close to $a=3.5 \text{µm}$, and we observe the best transmission of −34 dB for that lattice constant.

4. Discussion

To enable a comparison between simulation and experiment, all losses of the system need to be considered. Table 1 provides the corresponding simulation values.

Table 1. Summary of all losses expected for a grating with 3.5 µm lattice constant

View Table

The Fresnel loss arises from the reflectivity at the fiber end faces and is −0.1 dB at the input and output respectively. The angular mismatch is described in detail in Section 2. The transmission loss is a direct consequence of the 1/e point of the critical coupling condition, which means that some light is transmitted, corresponding to a loss of −2 dB.

For a coupling setup as shown in Fig. 1, additional losses occur due to the spreading of the excited input signal to all directions in the silicon slab. As discussed in Appendix D the transmission to the output grating is inversely proportional to the distance between both gratings. We have calculated this spreading loss out and considered only coupling loss in Fig. 4(b) and Table 1.

This analysis allows to identify possible improvements according to the three design steps. I. The numbers of modes in the different systems are limiting the maximal coupling efficiency to −2.3 dB (compare Section 2). An improvement is possible by reducing the diameter of the input coupler and input multimode fiber. II. Regarding the angular selectivity of the scattering elements, multi-ring-shaped holes could be considered, thereby transforming the scatterers from near-point sources to grating-like sources, which are well known to have a better directionality. The difference between a point source and grating-like source is already apparent from the fact that the ring geometry we chose has a higher directionality than the disk geometry. Naturally, a grating is designed for propagation along the grating vector and not for off-axis propagation; this directionality is already apparent from the asymmetry of the scattering observed in Fig. 2(b). A solution would be to use sub-wavelength scattering elements with good directionality described by Krasnok et al. [12] that also exhibit good directionality. In terms of improving the out-of-plane scattering efficiency, improvements could be achieved by introducing a back reflector into the substrate similar to Taillaert et al. [10]. III. Regarding the scattering strength of the overall structure, the simple critical coupling-based approach we used is responsible for the 2dB transmission loss. An alternative would be to apodize the density of scatterers in order to more closely match the fiber mode [13] which can completely eliminate the transmission loss in the case of a 1D coupler. Some improvement will also be possible in the 2D case, but it will not be as effective due to the circular symmetry.

Consequently, the losses for a multimode coupler could be as low as a few dB overall, which is an interesting result that clearly justifies further research in this area.

5. Conclusions

We have presented important design considerations for a multimode grating coupler that interfaces between multimode fibers and a multimode waveguide slab. We used a model system of an array of ring-shaped holes that have been shallow-etched into an SOI slab. This approach allows for a separate optimization of the scattering angle via the shape of the scatterers and the coupling strength of the overall coupler by adjusting the lattice constant of the scattering elements within the coupling area. As an example, the coupling between a silicon slab and multimode optical fibers was experimentally demonstrated. The measured transmission loss of −34 dB is in very good agreement with simulation results and is mainly caused by a mismatch of the scattering angle range of the grating and the acceptance angle range of the multimode fibers. We highlight that further improvement is possible by optimizing the angular selectivity and distribution of the scattering elements and by implementing a back reflector. The methodology we describe and the insight gained from this work can also be used to optimize the coupling between spatially incoherent radiation such as LED light or a thermal source into and out of an optical chip.

Appendix A coupling between multimode systems

The coupling efficiency between two systems is defined as the ratio between the power coupled into the receiving system and the power originating from the emitting system. Both systems can either be single mode or multimode. Figure 5 summarizes all possible combinations for coupling between two systems either being single mode or multimode. The number of modes in the emitting and receiving systems is the key metric for the maximal coupling efficiency. In case of suboptimal coupling designs additional losses can occur due to back reflection into the emitting system or scattering into other modes.

 

Fig. 5 Coupling between multimode systems. ${\text{M}}_1$ defines the number of modes occupied in the emitting system and ${\text{M}}_2$ the number of available modes in the receiving system. (a) The power in a single mode system can be coupled into another single mode system without loss. (b) Coupling from a single mode to a multimode system is also possible with 100% efficiency. (c) If all modes in a multimode system contribute equally to the emitted power, the maximal power that can be coupled into a single mode system is given by the power carried by one mode, the remainder being reflected or radiated at the coupling interface. (d) Considering two multimode systems with the same number of modes, coupling without loss is possible in analogy to (a). (e) If the available number of modes in the receiving system is larger than the number of occupied modes in the emitting system all power can be coupled in analogy to (b). (f) Coupling from one multimode system to another will necessarily incur additional loss if the number of modes in the receiving system is smaller than the number of modes in the emitting system. If the input power is distributed equally among all modes in the emitting system the maximal power that can be coupled is given by the ratio of the number of modes in the receiving system to the number of modes in the emitting system.

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Appendix B coupling between a slab waveguide and free space

A 2D waveguide slab and free space represent two particular multimode systems. In free space arbitrary propagation angles are allowed, whereas in the 2D slab the propagation of optical signals is restricted to two dimensions. Thin waveguide slabs can effectively be considered as 2D systems if only one transversal mode is allowed and propagation along the two extended waveguide dimensions is realized by index guiding. By structuring the surface of the waveguide slab scattering centers are created and the guided modes become leaky and can interact with the free space. Figure 6(a) shows an example of a grating structure that is included into the surface of a slab to realize coupling between the guided slab modes and propagating free space modes. The blue dashed line encircles the scattering structure and thereby defines the coupling area.

As discussed in Fig. 5 the numbers of modes available for the coupling process in both systems limit the coupling efficiency. Here, they are defined by the number of modes $M_{3\text{D}}$ that are required to describe the spatially incoherent radiation incident on the coupling area from free-space and the number of modes $M_{2\text{D}}$ that are required to describe the radiation that goes into the slab at the circumference of the coupling area.

A signal from free space can enter the grating via the coupling area. In some cases, the angle range of incident radiation is limited. In the following, the case of free space radiation coming within a cone with a maximal angle $\theta$ with respect to the vertical axis and incident on the coupling area is considered. The number of modes required to describe this excitation is equal to the number of modes in a cylindrical mirror waveguide with circular cross section multiplied with ${\sin }^2\theta$. Thus, the number of modes per polarization is given by [9]:

(3)$$M_{3D}=\frac{\pi }{4}\cdot \frac{A}{{\left({\lambda }_0/2n\right)}^2}\cdot {\sin }^2\theta$$

with ${\lambda }_0$ being the vacuum wavelength, $A$ the coupling area and $n$ the refractive index of the 3D medium from which the radiation comes.

 

Fig. 6 (a) Silicon slab including a grating coupler for coupling from free space into the slab. The grating consists of several scattering elements that are periodically arranged in the coupling area with a lattice constant much larger than the wavelength of the guided signal. The signal incident from free space is spatially incoherent. The number of modes entering the grating from free space ${\text{M}}_{3\text{D}}$ and thus contributing to the coupling process is proportional to the coupling area. On the other hand, ${\text{M}}_{2\text{D}}$ modes are required to describe radiation exiting the coupling area via the circumference line into the silicon slab. (b) The excited radiation leaves the grating via the circumference of the coupling area. If the radius of the coupling area is much larger than the wavelength, in order to model the problem, we can cut the circumference of the grating area and unfold it to a two-dimensional mirror waveguide. The number of available modes is then proportional to the circumference of the coupling area. (c) The coupling from 3D free space to the 2D slab as sketched in (a) can thus be described as the coupling between two multimode waveguides.

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The maximal number of modes in a 2D system that can be excited by the coupling process is estimated by the number of modes that can enter or exit the circumference of the coupling area in horizontal direction. For a grating diameter much larger than the wavelength we can neglect the curvature and can imagine to cut the circumference of the coupling area and unfold it to a rectangular waveguide with a width equal to the circumference of the grating [8]. Figure 6(b) shows the procedure for a circular coupling area with a diameter $d$ much larger than the wavelength which is unfolded to a straight line with length $S=\pi d$. The maximal number of modes ${\text{M}}_{2\text{D}}$ that is required to describe radiation going through this line is equal to the number of modes in the mirror waveguide with the same width [14]

(4)$$M_{2D}=\frac{S}{{\lambda }_0/\left(2n_{eff}\right)},$$

where ${\lambda }_0$ is the vacuum wavelength, $n_{\text{eff}}$ is the effective refractive index of the slab mode and $S$ represents the circumference of the coupling area.

From Eqs. (3) and (4) we find that the number of modes in the 3D system that can contribute to the coupling is determined by the grating area $A$, whereas the number of modes contributing from the 2D system is proportional to the circumference of the grating $S$. It is therefore possible to adjust the coupling in both directions by the size of the grating. If coupling in both directions is required the maximal number of modes contributing in the 2D and 3D system should be matched ($M_{2\text{D}}=M_{3\text{D}}$).

Appendix C design parameters

The radii ${\text{r}}_1$ and ${\text{r}}_2$ of the scattering element can be adjusted in order to optimize the scattering angle range. The wavevector distribution of the up scattered signal after excitation of a guided TE mode propagating in x-direction is shown in Fig. 7 for three different parameter settings. For our coupling system the wavevector distribution should be improved for angles that are accepted by the input multimode fiber with $\text{NA}=0.22$ (indicated by the gray dashed lines in Fig. 7. The design parameters considered in Fig. 7(a) and 7(c) result in maxima of the wavevector distribution outside of the gray circle and a large amount of the scattered signal cannot be coupled to the multimode fiber. The optimal case is achieved for radii of ${\text{r}}_1=510 \text{nm}$ and ${\text{r}}_2=730 \text{nm}$ as shown in Fig. 7(b). The maximum of the wavevector distribution is located within the gray circle.

 

Fig. 7 Wavevector distribution 1 µm on top of the silicon surface for three different parameter settings at 1.55µm wavelength. The inner radius ${\text{r}}_1=510 \text{nm}$ of the scattering element is fixed for all cases, the outer radius is (a) ${\text{r}}_2=630 \text{nm}$, (b) ${\text{r}}_2=730 \text{nm}$, (c) ${\text{r}}_2=830 \text{nm}$. The black circle includes all k-vectors that can be coupled from the slab into air. The blue and gray dashed lines indicate the k-vectors which are accepted by an optical fiber with $NA=0.39$ and $NA=0.22$ respectively.

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It is obvious that the angle distribution in x-direction can be adjusted by the radii of the scattering element whereas a broad wavevector distribution in y-direction is observed for all parameter settings.

Appendix D Spreading loss in the slab between two grating couplers

To measure the coupling efficiency, the output grating is positioned at a distance $\text{R}$ from the input grating. The input multimode coupler in the waveguide slab is a circular source emitting in two dimensions. For this reason, it is assumed that the power coming from the input grating is homogeneously distributed on the circumference $2\text{πR}$ and only a portion $\text{d}/2\text{πR}$ reaches the output grating if $\text{d}$ is the diameter of the coupling area (see Fig. 8(a) for comparison). To verify this statement, pairs of multimode couplers with various relative distances $\text{R}$ have been tested. Figure 8(b) shows the measured transmission between two multimode couplers with a lattice constant $\text{a}=5.5 \text{µm}$ separated by a distance $\text{R}$ on a logarithmic scale. The solid line indicates the fitting curve

(5)$$T\left(R\right)=T_0\cdot \frac{d}{2\pi R}$$

with ${\text{T}}_0=6.3\cdot {10}^{-6}$ (52 dB) being the efficiency due to in- and outcoupling. The deviations between the measured transmission and the fitting curve are below 1 dB. Therefore, the transmission loss of −20 dB for a distance of $\text{R}=3 \text{mm}$ between input and output grating in the manuscript is assumed as a good approximation.

 

Fig. 8 (a) The input coupler is a circular source emitting in the plane. Thus, the average intensity decreases inversely proportional to the distance $\text{R}$ from the center of the input coupler. (b) Measured transmission for multimode couplers arranged at a distance $\text{R}$. The solid line indicates the theoretical transmission by assuming the input coupler as a circular source exciting isotropic radiation.

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Funding

Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 392323616 and the Hamburg University of Technology (TUHH) in the funding programme *Open Access Publishing*.

Acknowledgments

The authors acknowledge the support from CST, Darmstadt, Germany, with their Microwave Studio software [11] and would like to thank Thomas F. Krauss from the University of York, United Kingdom, for providing his clean room facilities for the preparation of samples and numerous inspiring discussions.

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