Fig. S1 shows the spectral dependence of the transmission for PhC cells with different sizes and a fixed wavelength resolution of 2 nm. The black curve indicates the result for a small hexagonal cell with 4.5 µm side length. Strong resonances occur and averaging over a broad wavelength range is necessary. Increasing the side length by factor 10 (blue curve) does already reduce the wavelength dependence fluctuations of the measured transmission. For a side length of 450 µm (red curve) the transmission curves are increasingly flattened with fluctuation amplitudes below 3 dB. Thus, our approach describing the average transmission as a function of reflectivity and cell size becomes better applicable for larger cells. For the evaluation of our measurements it is important to consider the transmission for small cells where according to equation (1) waveguide losses are dominant and larger cells where loss occurs mainly due to vertical scattering at the PhC boundaries. Thus, the results shown in Fig. 5 where averaged over a large bandwidth of 70 nm in order to obtain representative averaging even for the smaller cells.
For applications as, for example, gas sensing, optical path lengths of several centimeters are required. This can be realized by PhC cells with diameters of 1 mm. One example is the detection of gases which exhibit absorption bands that are typically spread over a wavelength range of several tens of nanometers. In this case, the average transmission coefficient can be well defined.
S2. Coupling from a multimode cell into a single mode waveguide
The number of modes in the multimode waveguide as shown in Fig. S2(a) is defined by the width of the waveguide divided by half the wavelength.1 We assume that the power incident on the cell boundary Pcellis equally distributed among M incoherent propagating modes. Hence, each mode carries the same power P0:
As discussed in Ref. 37 of our manuscript combining the power of M incoherent modes into a single mode port can never exceed the power of any of these input modes. In this case, the maximum power that can be coupled from the multimode slab waveguide with incoherently excited modes into a single mode coupling port waveguide is again P0.
The case of a hexagonal shaped integrating cell as shown in Figure S2(b) is a bit more complex. Light propagates in a random direction inside the cell. To estimate the number of propagating modes impinging on the PhC boundary we can imagine to cut the PhC boundaries as indicated by the dashed line in Figure S2(b) and unfold it to a rectangular waveguide with a width Sc being the circumference of the former hexagon. Analog to the system discussed before the number of modes M can now be estimated to be
where a is the lattice constant of the PhC lattice, N is the number of lattice constants per side length of the hexagon and λ is the effective wavelength of the slab mode. This estimation is valid if the circumference of the hexagonal cell is much larger than the wavelength. It follows that the coupling loss to a single mode waveguide is inversely proportional to the side length of the hexagonal PhC cell.
S3. Comment on the accuracy of measurement results in Figure 5:
Fig. S3 shows the design for the chip with tapered integrating cells. Cells with 11 different sizes are arranged close to each other with two waveguides used for in- and out-coupling of an optical signal. As a reference two singlemode strip waveguides where positioned on the top and bottom side of the chip (indicated in red in Fig. S3). Grating couplers are used in the experiments to couple a signal from a single mode fiber into the waveguides and vice versa. Unfortunately, the grating parameters vary over the chip area. It is assumed that an inconsistency during the fabrication process is responsible for the variation in the grating parameters. Gratings that are in close distance to each other are still comparable but for gratings at larger distances a deviation in the parameters was observed. To reduce the influence of the grating couplers in our evaluation the transmission for each integrating cell was normalized to the transmission of the closest strip waveguide. In this way, we can assume accurate results for the smallest cells that are close to the lower reference waveguide and for the largest cell that was positioned close to the upper reference waveguide. For cells with medium sizes the distances to both reference waveguides were quite large resulting in inaccurate normalization of their transmission. Thus, the transmission results indicated by * in Figure 5 are connected to cells with large distances to the reference waveguides which results in a deviation from the fitting curve. A much smaller deviation in the grating parameters was observed in case of the chips with uniform boundaries.