[IEEE 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Munich, Germany (2019.6.23-2019.6.27)] 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC) - Determining the Waveguide Profile using the Overlap Integral

摘要： The determination of the index pro?le in guiding structures is a central problem in applied photonics, ranging from optical ?bers to femtosecond-written waveguides. A non-destructive and relatively easy method consists in the measurement of the index pro?le by measuring the transmitted ?eld. From the transmitted ?eld, the refractive index pro?le is computed by direct inversion of the Helmholtz equation. This technique is called near-?eld method. Here we present a new near-?eld method based upon the inversion of the overlap integral. From the waveguide theory, the power coupled to the m-th mode with pro?le ψm(x) from an input Ein(x) is am = (cid:2) Ein(x)ψ ? (x)dx. If the input Ein(x) is shifted by an amount x0, the overlap am(x0) is the convolution between the pro?le of the input beam Ein and the mode pro?le ψm. The convolution operator implicitly ?lters out the noise, but without creating distorsions or artifacts. An experimental measurement of the transmitted ?eld Eexp will be Nη, where η is a white Gaussian noise and N the noise amplitude. In the previous formulae P is the overall power given by |am(x0)|2. If both amplitude and phase of the transmitted ?eld are simultaneously measured, the overlap am can be inverted for any guided mode m. If only the intensity is measured, the overlap can be inverted only if the waveguide is monomodal. An example of the reconstructed mode in the case of intensity-only measurements is plotted in Fig. 1(a) for ψ0 = cosh(x/w) with w/λ = 4 and N/P = 1 × 10?4. The retrieved ?eld is very close to the exact one, with appreciable differences only for |x/λ | > 10. Important to stress, this range is larger than what it is achievable with a single direct measurement of the transmitted ?eld. The next step is to invert the Helmholtz equation, that is, to compute the second derivative of the retrieved mode. Direct application of the inversion protocol strongly enhances the noise, above all on the tails of the mode. The net result is the appearance of several fake oscillations, even where the retrieval of the mode is good. The problem can be overcome by ?tting the mode tails with a decaying exponential, in accordance with the waveguide theory. The reconstructed guide pro?les shown in Fig. 1(b) do not present arti?cial oscillations on the tails, showing a 10% error with respect to the original waveguide, the error depending slightly on the original signal-to-noise ratio N/P.