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Abstract
Laser-written waveguides in glass have many potential applications as photonic devices. However, there is little knowledge of the actual profile of the usually asymmetric refractive index (RI) change across the femtosecond (fs) laser–written waveguides. We show, here, a new nondestructive method to measure any symmetric or asymmetric two-dimensional RI profile of fs laser–written waveguides in transparent materials. The method is also suitable for the measurement of the RI profile of any other type of waveguide. A Mach-Zehnder interferometer is used to obtain the phase shift of light propagating transversely through the RI-modified region. A genetic algorithm is then used to determine the matching cross-sectional RI profile based on the known waveguide shape and dimensions. A validation of the method with the comparison to a RNF measurement of the industry-standard SMF-28 is presented, as well as a demonstration of its versatility with measurements on fs laser–written waveguides.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser written waveguide-based devices and sensors are being engineered for a growing number of applications [1–7]. To properly design such devices, knowledge of the refractive index change resulting from the laser interaction in glass is critical. The RI profile can be measured by techniques that have been developed for optical fibers but those are often inaccurate, unsuitable or unpractical for use with laser written waveguides. For instance, the commonly used RNF (refracted near-field) technique [8–10] is destructive since it requires the waveguide's face to be exposed by a cleave. Acquiring the RI profile of fibers or waveguides which are non-uniform along their propagation axis requires making multiple cleaves at different position, making the method time consuming and impractical. It is also known that cleaving the fiber can release mechanical stresses or induce defects such as cracks, effectively modifying the RI profile at the surface [11]. In the context of laser written waveguides where perfect perpendicular cleaves are very difficult to make, the surface must be polished. The polishing process induces mechanical stresses in the material, effectively altering a RNF measurement. Other RI profiling methods mainly demonstrated for optical fibers involve measuring the phase shift across the waveguide either by analyzing the fringe deformation on an interferogram produced by the sample [12–15], using the transport of intensity equation [16,17], using an N-step phase-shifting interferometry algorithm [18,19] or using deconvolution phase microscopy [20]. Optical fiber-based RI profiling techniques already described in the literature either suppose a very specific RI profile [12], an axial symmetry [15–17,19,21] or require the fiber to be rotated around its propagation axis to acquire many-angle tomographic images [13,14,18,20,22] to reconstruct the RI profile from the phase shift. However, laser written waveguides rarely possess axial symmetry, making the profile reconstruction very inaccurate at best. Turning the glass slab around the waveguide axis for tomographic analysis is either not possible or not practical for laser written waveguides. A widely used technique to estimate the RI change profile of a laser written waveguide is to measure its numerical aperture (NA) and use the following equation, derived from the case of a step-index fiber: NA = (ncore2 – nclad2)1/2 [1]. Although it gives an idea of a waveguide's ability to guide light, the use of this equation is in no way justified given the complex profile of laser written waveguides, not even remotely close to a step-index profile and lacking axial symmetry. Another technique is to measure the reflectivity of the sample’s top surface where a waveguide has been inscribed [23]. A step-index profile is again assumed and the result is thus only an approximate value for the average RI, not a true two-dimensional RI profile. Others have tried to adapt the tomography technique to glass samples by rotating the beam around it [24]. Angles up to only 36° were achieved, after which the induced distortion in the image becomes significant. Given the lack of rotation angle and distortion, only the optical path length (OPL) for a given angle is obtained, from which the ellipticity of the waveguide is deduced by again assuming a step-index RI profile. A true two-dimensional RI profiling technique is described in [25]. DHM is done on the waveguide’s end-face (light is sent along the waveguide) to measure the OPL difference, and thus the RI, at every point on the end-face. However, the main drawback of the technique is the necessity to use a very thin sample to keep the OPL difference below 2π, meaning that the method is destructive, assumes that the waveguide is constant along its propagation axis and requires a very lengthy sample preparation process.
We present here a new, straightforward, non-destructive method for RI profile measurements applicable to waveguides of arbitrary shapes. The method is divided into two steps. First, an interferogram of the sample is obtained from which it is possible to calculate the phase shift induced by the presence of the waveguide. Since angular tomography is not possible, information on the waveguide's shape and dimensions are required and can be obtained from reflection microscopy of the waveguide's end-facet. Although a cleaved and (at least roughly) polished end-facet is needed, the problems this brings to the RNF technique are not applicable here since the measurement is not done on the end-facet, but anywhere further along the waveguide. As a second step, using the shape and dimensions as well as the measured phase image, a genetic algorithm is used to reconstruct the RI profile that matches the measured phase shift with no restriction regarding axial symmetry.
The main limitation of our technique, which arises from the absence of tomography, is the indiscernibility of two identical features at different depths in the measurement. The individual contributions to the phase shift of two features cannot be separated if both have the same width in the direction perpendicular to the light beam. Given the very specific conditions necessary for this limitation to manifest itself, it is not an issue for the overwhelming majority of laser written waveguides.
2. Experimental method
2.1 Measurement setup
The Mach-Zehnder interferometer (MZI) setup used for the phase measurement is shown in Fig. 1(a). A HeNe laser at a wavelength of 632.8 nm is launched into a SMF-28 optical fiber. The light is split into the two arms of the interferometer using a 50:50 fiber coupler (C' in Fig. 1(a)). One arm features a piezoelectric actuator (A) to stretch the fiber and induce a phase delay. Each arm includes a fiber collimator (C1,2), a half-wave plate (HW1,2) and a polarizer (P1,2). Polarizers and wave plates are adjusted to maximize interference fringes visibility. The sample (S) is placed just before the objective with the beam of the interferometer passing through the sample transversally (i.e. the waveguide's propagation axis is orthogonal to the beam). The reference arm features an identical objective to match the sample arm to producea flat phase reference. The beams are recombined by a beam-splitter and an image (the interferogram) is formed on the CMOS camera using the final lens (L).
Fig. 1 (a) Mach-Zehnder interferometer setup used to measure the phase shift produced by the waveguide. A piezoelectric actuator is used to induce a phase delay in one arm. The polarizers and wave plates are adjusted to maximize the fringe visibility. An image of the sample is formed at the camera by the objective and lens. (b) Setup used to obtain an image of the waveguide's cross-section. The polished end-facet of the sample is illuminated by a white light source coming from an angle.
The focal plane of the objective in the sample arm (O1) is placed just under the waveguide sample so that the light going through it is collimated. Also, the focal plane is close to the waveguide to avoid any long-range deformation of the wavefront caused by the waveguide. Given the very small RI change of typical laser written waveguides (~10−3), the lens effect caused by the waveguide is negligible in near field. Test measurements have shown that the resulting phase image is independent of the exact focal plane position if the said plane is close enough to the waveguide (within ∼10 µm).
As the light propagates through the waveguide perpendicularly, its phase is more or less shifted depending on the position relative to the waveguide (see Fig. 2(b)):
In practice, φ(x,z) is measured using the interferometric setup of Fig. 1(a). The image seen on the camera is the intensity fringe pattern determined as the cosine of the phase of Eq. (1) plus any reference phase variation due to misalignment and lens imperfections. To precisely measure the phase shift, the phase delay is scanned using the piezoelectric actuator and a video of fringe displacement is acquired. As the phase delay is increased, the intensity at each pixel of the camera follows a sine function (see Fig. 3). A simple Fourier transform of the cosine functionallows the relative phase φ for each pixel to be calculated by taking the phase of the dominant frequency of the very narrow spectrum.Fig. 2 (a) Schematic cross-section of the sample used for the validation of the method and (b) the expected phase shift of light propagating downwards through the sample from the top. The fiber is immersed in index-matching liquid and placed between glass plates. Secondary fibers are used for support. The phase shifts more or less, depending on the optical path length for light at a given x position.
Fig. 3 (a) Phase image after reconstruction for an SMF-28 fiber. (b) Measured light intensities at camera pixels A and B versus phase delay. (c) Result of the FFT at pixel A. The measured phase for the given pixel is the phase of the dominant frequency.
2.2 Genetic algorithm
To recover the RI information from the transverse phase image, a general knowledge of the waveguide's shape and dimensions is required. A cross-section image is thus acquired bysimple camera inspection with angled illumination of the waveguide's end-facet, as shown in Fig. 1(b). Such an image shows the relevant RI structures, but cannot resolve the actual value for the RI change profile. A genetic algorithm programmed in Matlab is used to find the RI distribution that matches the measured phase shift. The genetic algorithm takes as an input a two-dimensional model of the profile based on the cross-section microscopy as well as starting values and bounds for the parameters (see Eqs. (4) and (5) in the next section for a list of parameters). The model's various parameters dictating the different features' dimensions and RI change are then optimized by the algorithm as to minimize the difference between the phase shift produced by the modeled RI profile (calculated using Eq. (1)) and the one measured. The fitness function used in the genetic algorithm to compare the different solutions is simply the sum of squared differences.
The optimization process follows a standard method for genetic algorithms. First, random variations of the parameters of a random starting initial solution (within specified bounds) are generated (first generation “children” in the genetic analogy). The solutions with the best fitness scores as well as their parameters (or “DNA”) are stored and will act as “parents” on which further variations are generated to form the subsequent generation.
The variation of the parameters is performed by two means: crossovers and random mutations. In the case of crossovers, children share some unchanged portion of each parent’s DNA, thus preventing important genes from disappearing from the gene pool. Random mutations (small random variations of single parameters) are also used to improve diversity in the gene pool. The amplitude of these mutations is reduced progressively as newer generations stop showing improvements. The algorithm is stopped when all parameters have converged, with the solution thus not evolving anymore. Calculations are split on multiple CPU cores and high-end GPUs to allow for fast iterations, producing a solution (the two-dimensional RI profile) within seconds.
An example of the evolution of the DNA of the best solution of each generation when modeling the SMF-28 fiber’s RI profile is shown in Fig. 4 as well as its fitness score and its resulting RI profile (see next section for parameter descriptions and explanations). The parameters dictating the geometry of the fiber quickly converge because of their narrow bounds (given the information obtained from microscopy of the cross-section) and their importance on the resulting phase shift (and hence fitness score) whereas the RI levels need more time to converge to the correct value given their wider bounds.
Fig. 4 (a) Evolution of some of the parameters used to model the SMF-28 fiber using the genetic algorithm and the corresponding error. Parameters dictating the geometry converge very fast while fine-tuning of the RI levels takes more iterations. (b) Evolution of the RI profile calculated by the genetic algorithm.
3. Results
3.1 SMF-28 optical fiber
The method was validated by performing the measurement on an SMF-28 optical fiber, a waveguide for which the parameters are well known. Figure 2(a) shows a cross-section of the sample setup used for the validation of the method. The fiber is surrounded by an index-matching liquid and placed between a microscope slide and a cover slip. Other fibers are used as supports for the cover slip to ensure the entry surface is not tilted. The model based on the cross-section shown in Fig. 5(a) and used to find the RI distribution is the following function of the fiber radius r:
where H is the Heaviside function and Δni, ri, and wi are the parameters to optimize using the algorithm, namely, the RI differences of the core, cladding and central dip, the radii of the core and cladding, and the width of the transition from the core to the cladding and of the central dip, respectively. A complementary error function is used to model the core to cladding transition since the step-index nature of the SMF-28 fiber is unknown a priori, unlike the cladding-liquid interface, hence the Heaviside function for this region.Fig. 5 (a) Cross-section of the SMF-28 fiber obtained with the setup of Fig. 1(b). (b) RI change profile reconstructed by the genetic algorithm for the SMF-28 fiber based on the measured phase shift and the known shape and dimensions from Fig. 5(a). (c) Measured and modeled phase shifts along cross-section B of Fig. 5(a). (d) Reconstructed RI change profile and result of the RNF measurement along cross-section A of Fig. 5(b).
The reconstructed profile resulting from the genetic algorithm is shown in Figs. 5(b) and 5(d). Figure 5(c) shows how closely the phase shift produced by the reconstructed profile matches the measurement. Given the RI of pure silica at 632.8 nm of 1.45702 [26], the RI change of 5.4∙10−3 at the core corresponds to 0.37% which is close to the value of 0.36% given by Corning [27]. The values for the core and clad diameters are also a very close match at 8.6 µm (full width at half maximum) and 124.6 µm versus the values of 8.2 µm and 125.0 ± 0.7 µm given by Corning [27]. To validate the method, the RI profile of the SMF-28 was also measured using the RNF method. The result is shown in Fig. 5(d). Both measurements are in close agreement. The tests performed on the SMF-28 fiber let us confirm that the method is indeed valid and quite precise, at least for simple waveguides.
3.2 Weakly asymmetric waveguide
Let us now focus our attention on laser written waveguides. The first waveguide on which the method is demonstrated was written with a femtosecond laser (250 fs pulse length) operating at a wavelength of 515 nm, an average power of 200 mW, a repetition rate of 606 kHz, focused by an oil-immersion objective of 1.25 NA and by translating the sample at a speed of 7 mm/s. The waveguide was written in Corning’s Gorilla Glass 3 (n = 1.5127 at 532.8 nm [28]). Figure 6(a) shows a cross-section of the resulting waveguide. Although it is nearly circularly symmetric, not considering the loss of symmetry for y < 0 would yield other methods to produce an erroneous RI profile. The discrepancy would be even more obvious for most typical waveguides which show a stronger asymmetry.
Fig. 6 (a) Cross-section of the weakly asymmetric laser written waveguide obtained with the setup of Fig. 1(b). (b) RI change profile reconstructed by the genetic algorithm for the weakly asymmetric laser written waveguide based on the measured phase image and the known shape and dimensions from Fig. 6(a). (c) Measured and modeled phase shifts along cross-section B of Fig. 6(b). (d) Reconstructed RI change along cross-section A of Fig. 6(b).
The model of the waveguide used in the algorithm is, of course, fairly more complicated than for the case of the step-index fiber presented earlier. Given the asymmetry, the model is now a function of both the radius and the angle, as stated by Eq. (4):
The complementary error function of amplitude Δnout is again used to model the outer region while Gaussians of amplitude Δni and width wi are added (or subtracted) at radial positions ri to represent the structures observed by reflected light microscopy. Most radii and widths are now function of the angle θ to represent correctly the stretching along the y axis for some of the structures.The results for the measurement on the laser written waveguide are shown in Fig. 6. The measured and modeled phase shifts are again in very close agreement, as are the reconstructed RI profile and the cross-section microscopy.
3.3 Strongly asymmetric waveguide
The method is also demonstrated on a strongly asymmetric waveguide written in the same glass material. This waveguide (shown in Fig. 7(a)) was written with the same femtosecond laser (250 fs pulse length, operating at a wavelength of 515 nm and a repetition rate of 606 kHz) but at a higher average power of 400 mW, focused by an objective of 0.65 NA and with a translation speed of only 0.5 mm/s, resulting in a much higher fluence. The outer region has an aspect ratio of about 1.25 while the region closer to center has an aspect ratio of ~2.5 (in addition to being shaped closer to a droplet than an ellipse). Assuming axial symmetry would yield a RI profile nowhere near reality.
Fig. 7 (a) Cross-section of the strongly asymmetric laser written waveguide obtained with the setup of Fig. 1(b). (b) RI change profile reconstructed by the genetic algorithm for the strongly asymmetric laser written waveguide based on the measured phase image and the known shape and dimensions from Fig. 7(a). (c) Measured and modeled phase shifts along cross-section C of Fig. 7(b). (d) Reconstructed RI change along cross-sections A and B of Fig. 7(b).
The waveguide’s model used in the algorithm is very similar to the one used for the previous waveguide (see Eq. (4)). The only difference is that the parameter dictating the radius of the outer region rout is now a function the angle (rout(θ)) since the outer region is not circularly symmetric anymore.
The strong asymmetry is not a problem for this method as the results shown in Fig. 7 indicate. The modeled phase shift is again a near perfect match to the measured phase shift.
4. Discussion
The three tests presented show how well the method works for optical fibers and asymmetric waveguides. The limitations of the method are, however, not entirely clear from these tests. To evaluate the spatial resolution, a sample with resolution targets made of sharp ~0.75 µm deep square grooves of various widths was fabricated by applying photoresist to a 1 mm thick glass slab followed by exposure with an e-beam before etching. Another sample with identical patterns was also made with a chromium coating allowing us to compare the phase measurement’s spatial resolution to regular transmission microscopy (without the interference of the reference arm). Figure 8 shows the picture of the smallest 1 µm wide groove pattern for both transmission microscopy and the measured phase shift. It is clear from these picture that the spatial resolution is slightly better than 1 µm (0.92 µm by the 10-90 criteria).
Fig. 8 Resulting image of the 1 µm resolution target for (a) transmitted light amplitude without interference and (b) the calculated phase shift. The pattern is clearly resolved, indicating a sub-micron spatial resolution.
Typical phase measurements show a noise level of 0.01 radian. How this translates to noise in the RI profile reconstruction depends on the structure studied, more specifically its size and RI change, as stated by Eq. (1). For a typical laser written waveguide with a diameter of about 30 µm, this phase noise corresponds to an uncertainty of ± 10−4 in the RI. Of course, for a waveguide with twice the diameter, the noise in the RI reconstruction is halved.
At a wavelength of 632.8 nm, the phase noise of 0.01 radian corresponds to an optical path difference of 1 nm. As such, for a feature to be distinguishable on the phase measurement, its optical path difference must be greater than this value. For example, the dip in RI at the center of the SMF-28 optical fiber can barely be observed on the phase measurement since its optical path difference is only of about 2.6 nm.
Due to the nature of the method, providing a value for the precision in the RI profile reconstruction proves to be rather difficult. Still, multiple solutions for the same waveguide were found using the algorithm and all agree within 10−4 refractive index units (RIU). Of course, this does not include errors in the modeling, which are negligible for waveguides of simple geometry such as those presented. Small differences in the geometry between the front-facet and location of measurement can be accounted by the genetic algorithm. However, if those differences are significant such that the model used cannot properly represent the evolving structure within its acceptance bounds, then this method is limited. Either the measurement should be repeated closer to the end-facet or the sample should be recut closer to the desired measurement position. Nonetheless, we believe that our technique compares favorably with the different measurement methods for RI profiling [21], in that it allows for measurement of waveguides within seconds with little preparation of the sample and with the added advantage of being able to measure asymmetric waveguides, unlike previously documented methods.
5. Conclusion
A new non-destructive interferometric method is presented to quickly measure the two-dimensional RI profile of laser written waveguides for which the shape is known. The phase shift of a cross-propagating wavefront is first measured. Then, the use of a genetic algorithm allows for the reconstruction of the RI profile of any arbitrarily asymmetric waveguide without the need for tomography, a first to our knowledge. The method was validated by comparison to RNF measurements on the SMF-28 optical fiber and demonstrated for asymmetric fs laser written waveguides. Its repeatability was shown to be within ± 10−4 RIU while its spatial resolution was shown to be sub-micron, making the method comparable to expensive commercially available optical fiber profilers based either on the RNF method or an interferometric method. This method's versatility and ease of use makes it an invaluable tool to better understand laser induced modification of transparent materials.
Funding
Natural Sciences and Engineering Research Council of Canada (NSERC).
Acknowledgments
We thank Nvidia for their donation of two Pascal Titan X GPUs used in the calculations. The authors also thank Mr. Jules Gauthier for the fabrication of the resolution targets as well as Mikaël Leduc and Simon Bolduc-Beaudoin for their help with the RNF measurements.
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