A comparative study of quantitative phase imaging techniques for refractometry of optical waveguides is presented. Three techniques were examined: a method based on the transport-of-intensity equation, quadri-wave lateral shearing interferometry and digital holographic microscopy. The refractive index profile of a SMF-28 optical fiber was thoroughly characterized and served as a gold standard to assess the accuracy and precision of the phase imaging methods. Optical waveguides were inscribed in an Eagle2000 glass chip using a femtosecond laser and used to evaluate the sensitivity limit of these phase imaging approaches. It is shown that all three techniques provide accurate, repeatable and sensitive refractive index measurements. Using these phase imaging methods, we report a comprehensive map of the photosensitivity to femtosecond pulses of Eagle2000 glass. Finally, the reported data suggests that the phase imaging techniques are suited to be used as precise and non-destructive refractive index shift measuring tools to study and control the inscription process of optical waveguides.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
From micrometer-scale integrated silicon-on-insulator optical circuits to long-distance optical fiber network spanning hundreds of kilometers, most photonic devices and systems rely on low-loss optical waveguides. As such, the refractive index profile of optical waveguides is the key design parameter of integrated photonic devices because it determines, among other properties, the insertion losses and propagating modes. Therefore, an accurate refractive index profiling method is of paramount importance to the development, optimization and eventually quality monitoring of mass-produced photonic devices.
The measurement of the refractive index profile of optical waveguides, generally relies either on refracted near-field (RNF) [1,2] or quantitative phase imaging (QPI) . RNF is the state-of-the-art in the optical fiber industry because of its relatively low-cost and ability to resolve abrupt changes in the refractive index profile . However, this method presents a few practical drawbacks. The index profile cannot be obtained unless one of the end-face of the volumetric optical material is accessible. Moreover, if the index profile has to be measured at different points along the same optical waveguide, it must be broken repeatedly. On the other hand, QPI makes it possible to quantify the phase shift that light experiences when passing through a transparent object. As long as the refractive index of the phase object remains close to the one of the surrounding medium, in the present case the optical waveguide and the host material, the phase shift is proportional to optical path length (OPL). As such, it is only possible to get a precise measurement of either the thickness or the refractive index assuming the other is known. Fortunately, it is generally possible to measure accurately the thickness of the object with a complementary technique, for example with scanning electron microscopy (SEM). Therefore, QPI is well suited to identify, from OPL measurements, refractive index alterations of optical waveguides and has the advantage of being non-destructive. However, the provided measurement corresponds to an integrated refractive index difference over the optical path [4–6] and hence, the measurement is not suited for 3D refractive index distribution of optical waveguides.
Among a few others, a non-interferometric QPI technique developed by Barty et al.  based on the transport-of-intensity equation (TIE), or closely related to it, proved to be the most employed to date in the field of optical fibers [7–10] and photo-induced waveguides [11–13]. Over the past 10 years, a significant number of novel interferometric QPI methods have been developed [4,14–19]. Among those, the quadri-wave lateral shearing interferometry (QWLSI) and digital holographic microscopy (DHM) techniques have proven records of accomplishment in materials science [20–25]. However, to our knowledge none was transposed to the measurement of the refractive index difference of optical waveguides in a non-destructive manner up to now. In fact, there has been a single report of DHM for strain-induced birefringence measurements in optical fibers, but based on a polarimetric approach .
Recently, femtosecond laser direct inscription emerged as an efficient method for the fabrication of integrated photonic devices. This technique allows 3D micro-structuring inside the volume of transparent optical materials by translating a focused fs-laser beam below the surface, inducing a permanent alteration of the refractive index along the trajectory of the beam . As such, refractive index modifications can be shaped with submicron precision and the index contrast with the surrounding medium can be precisely adjusted which provides an ideal testbed for the evaluation of refractive index measurements methods.
In this paper, we present a thorough investigation of refractive index measurements of optical waveguides using three different QPI methods, namely TIE, QWLSI and DHM. First, the accuracy and the precision of the QPI approaches were assessed by measuring the refractive index profile of a standard telecommunication fiber (SMF-28). Then, the three QPI techniques were used to characterize the refractive index contrast of photo-induced optical waveguides. Precise control over the photo-induced index contrast enabled for the formation of weak modifications, which allowed determining the sensitivity of the QPI approaches. Finally, the QPI methods were used to produce a comprehensive map of the photosensitivity of a glassy optical material to fs-laser direct inscription. Specifically, the effect of the laser power and the number of successive passes of the beam on the induced index contrast in Eagle2000 glass was studied.
2. Experimental methods
Two different samples were selected for this study. The first one is a widespread telecommunication optical fiber (SMF-28, Corning), chosen because of its standardized geometry and refractive index profile. The second sample is a borosilicate glass chip (Eagle2000, Corning) containing multiple photo-induced optical waveguides with uniform refractive index shift distributions.
2.1.1 Optical fiber
The Corning SMF-28 fiber is produced via the modified chemical vapor deposition (MCVD) process and is composed of a 125-µm diameter fused silica cladding with an 8.2 µm diameter germanium-doped core. The core is known to exhibit a step-index profile and a refractive index difference of 5.2 × 10−3 with respect to the cladding [28,29]. Two segments of about 1 cm long optical fiber stripped of the polymer coating were immersed in a refractive index matching liquid (n = 1.458 at 589.3 nm, Cargille Laboratories). The assembly was mounted between two coverslips and separated by a 120-µm thick imaging spacer (SecureSeal SS1X20, Grace Bio-Labs). Three images of the fiber segments have been acquired along their length from a top-view perspective for each QPI technique.
2.1.2 Photo-induced waveguides
Owing to its strong photosensitivity, mechanical resistance and thermal stability, Corning Eagle2000 is a proven host for photonic devices fabricated using femtosecond laser inscription and operating in the visible and near infrared spectra. Recently, low-loss high-index waveguides as well as photonic circuits were formed in this boro-aluminosilicate glass [30–32]. For this work, homogeneous waveguides were inscribed in the bulk of a 1.1-mm thick Eagle2000 sample using a femtosecond laser system (RegA, Coherent). The system was operated at a wavelength of 790 nm and a repetition rate of 250 kHz. The temporal FWHM of the pulses was measured to be 60 fs at the laser output and estimated at 85 fs on the sample. The beam was focused 100 µm beneath the surface of a glass sample using a 50X-microscope objective (f = 4 mm, 0.55 NA, Edmunds Optics). The sample was translated at a speed varying between 5 and 20 mm/s, across the focal point, perpendicular to the laser beam using motorized mechanical stages (XML210, GTS30V, Newport). A cylindrical lens telescope was used to produce an astigmatic beam and shape the focal volume as to form uniform waveguides with nearly circular cross-sections .
Overall, 61 waveguides were photo-inscribed in the sample. Waveguides were divided between 12 series, inscribed using increasing laser power and number of passes. Series were separated by a distance of around 100 µm and waveguides were set apart by a distance of about 50 µm. The complete set of experimental parameters for every waveguide is detailed in Fig. 4 and particular care has been taken to ensure the formation of uniform waveguides. For each series and waveguide, a phase image has been acquired using the three QPI methods. All images were taken from a top-view perspective.
2.2 QPI methods
2.2.1 TIE (Transport-of-intensity equation)
The TIE method yield a quantitative phase image of a transparent object using a pair of defocused bright-field images taken on both sides of a single in-focus image. The intensity distribution is directly obtained from the in-focus image while the phase information is inferred by the pair of images obtained by defocusing the microscope slightly in both positive and negative directions . The phase is then retrieved by a Taylor series expansion of the difference between the defocused images. This expansion is usually valid up to the 3rd order . Bright-field intensity images were obtained using an inverted microscope (IX71, Olympus) and two different microscope objectives. An oil-immersed 100X-objective (LMPLFLN, Olympus) with a working distance of 0.17 mm and a NA of 1.3 was used to retrieve the refractive index profile of the optical fiber. For the measurement of the refractive index contrast of the photo-induced waveguides, a 50X-objective (UPLANFL LWD, Olympus) with a working distance of 2.1 mm and a NA of 0.5 was used. Samples were illuminated in a Köhler illumination scheme by an incoherent white-light source (U-LH100L-3, Olympus). Series of defocused images were obtained by moving the sample along the vertical axis with a piezoelectric translation stage (C-Focus, Mad City Labs). A band-pass filter (FL514.5-10, Thorlabs) centered at 514.5 nm with a FWHM of 10 nm was inserted in the light path for all the measurements. Intensity images were captured with a 12-bit CCD camera (QICAM Fast, QImaging) recording at 10 frames per second. Each intensity image is an average of 10 single-shot images and the corresponding phase image was retrieved using a proprietary software (QPm, Iatia). The camera resolution is 1392 × 1080 pixels and the sensor has a pixel size of 4.65 × 4.65 µm. An intensity image of the object at the focus along with three pairs of defocused images at ± 0.5, ± 1 and ± 2 µm and two pairs of defocused images at ± 4 and ± 8 µm were taken for the case of the optical fiber and the photo-induced waveguides respectively. Phase images were retrieved for each pair using the above-mentioned software. The best image in terms of contrast and noise was chosen and used for refractive index determination of both samples.
2.2.2 QWLSI (Quadri-wave lateral shearing interferometry)
To realize quantitative phase images, QWLSI uses a wave-front sensor placed in the image plane of a microscope. QWLSI is based on the interference of four replicas of an incident wave front that is distorted by the observed sample. The replicas are created by a modified Hartmann mask which is a bi-dimensional diffraction grating [35,36]. The superposition of those four replicas creates, after a small propagation, an interferogram that is recorded by a CCD camera. A thorough explanation of the working principle of QWLSI can be found in . The imaging setup was constituted of the same microscope, objectives and light source described in Section 2.2.1. The QWLSI sensor (SID4Bio, Phasics) was mounted on the camera port of the microscope with a C-mount adapter. The 12-bit CCD camera recording at four frames per second gives 400 × 300 phase and intensity measurement points with a mask of a lateral pitch of 29.6 µm. Each phase image is an average of five single-shot images of the object at the focus, each being the result of the reconstruction from an interferogram performed by the proprietary software (SID4Bio, Phasics). At the beginning of each imaging session, a reference image in a clean and homogenous region of the sample has been recorded. A band-pass filter (FB550-10, Thorlabs) centered at 550 nm with a FWHM of 10 nm was inserted in the light path for all the measurements.
2.2.3 DHM (Digital holographic microscopy)
A digital holographic microscope (T-1003, Lyncée Tec) based on a Mach-Zehnder interferometer was used. Within the microscope, the beam produced by a 666-nm laser diode is separated in two, forming an object wave and a reference wave. The object wave, diffracted by the sample, interferes with the reference wave producing an off-axis hologram recorded by a digital camera. The microscope objectives magnification is 63X and 40X with free working distances of 160 and 400 µm and NA of 1.3 and 0.80 (HC PL FLUOTAR, Leica) for the measurements of the optical fiber and photo-induced waveguides respectively. The camera is equipped with a monochrome 2.3 megapixels Sony IMX174 CMOS sensor with a pixel size of 5.86 µm and recording at 162 frames per second (Grasshopper3, FLIR). The off-axis geometry, meaning that holograms are recorded with the object and reference waves having a small angle between their directions of propagation allows reconstructing quantitative phase image from a single recorded hologram. A more detailed description of the experimental setup can be found in . The reconstruction of the original image from the hologram is computed numerically using a proprietary software (Koala, Lyncée Tec). The reconstruction algorithm consists in a simulation of the illumination of the recorded hologram by a digital reference wave followed by a numerical correction of the wave-front modifications induced by the optical components, the microscope objective, and the off-axis geometry as well as background flattening .
2.3 RNF (Refracted near-field)
An optical fiber analyzer (NR-9200, EXFO) based on the RNF technique  was used to recover the refractive index profile of the core of the SMF-28 optical fiber. Raster scanning of the fiber end-face yielded such profiles with a resolution of 0.2 µm at a wavelength of 657.6 nm.
2.4 SEM (Scanning electron microscopy)
The end-faces of the SMF-28 optical fiber and photo-induced waveguides were observed using SEM (Quanta 3D FEG, FEI). The fiber was cleaved and the end-faces of the glass chip were cut and polished. The cross-sections of the fiber core and waveguides were examined with a SEM using a 1000X-magnification or higher.
2.5. Image processing for quantification
Image processing was performed using an in-house software implemented in Matlab (R2016a, MathWorks). The purpose of the image processing procedure is to obtain leveled phase profiles, which can be readily quantified and interpreted. A procedure similar to the one demonstrated by Kouskousis et al.  has been applied to all QPI techniques and both samples. The image processing procedure of an optical fiber core is depicted in Fig. 1. For the sake of clarity, the following explanation refers to the case of an optical fiber, but remains valid for the case of a waveguide. Typical quantitative phase images of the optical fiber for TIE, QWLSI and DHM are shown in Figs. 1(a)-1(c) respectively. Representative single-line profiles, corresponding to the white lines in Figs. 1(a)-1(c), are shown in Figs. 1(d)-1(f). Phase images were averaged line-by-line along the axis of the fiber to obtain a one-dimensional mean phase profile. Next, the beginning position and the end location of the fiber core inside the mean profile were found using a 2nd order derivative treatment. The data points corresponding to the fiber core were then temporarily removed from the mean profile and the remaining was considered as the baseline. Afterward, the baseline was fitted using a 7th order polynomial and subtracted in order to level the mean phase profile. The portion of the profile corresponding to the core was repositioned on the baseline by applying the same correction, leading to the leveled phase profiles shown in Figs. 1(g)-1(i). For comparison purposes, leveled phase profiles have also been expressed in terms of OPL by normalizing with the nominal wavelength of each QPI approach. Within this new framework, the relative phase shift or the optical path difference corresponds directly to the value of the y-axis.
The procedure to infer the refractive index profile from the phase profile was different for the fiber and the photo-inscribed waveguides. For the fiber, the axial symmetry of the core allows the refractive index profile (∆n(r)) to be retrieved directly by performing an inverse Abel transform using the equation:4–6]. The thickness was determined by SEM measurements, as described in Section 2.4, and values are summarized in Fig. 4. One should note that dispersive effects could arise since a different central wavelength was used for the three QPI methods. However, it is well established that the dispersion curves of the core and cladding of an optical fiber are very similar, and consequently, the dispersion of the ∆n(r) is generally negligible over a relatively small wavelength range like the one used in this study (514.5-666 nm). Moreover, it has been shown recently that the total dispersion in photo-inscribed waveguides is dominated by material dispersion and the dispersion ofcan be neglected .
3. Results and discussion
First, we evaluated the accuracy of the three QPI methods by measuring the ∆n(r) of the SMF-28 optical fiber. Typical phase images of the optical fiber for TIE, QWLSI and DHM are shown in Figs. 1(a)-1(c) respectively. TIE and QWLSI, to a lesser extent, show phase images with an asymmetric background. Noteworthy, in this particular application, the DHM phase profile requires hardly any leveling and the QWLSI profile would be flattened using only a 1st order polynomial. However, the TIE phase profile necessitates minimally a 7th order polynomial because of the multiple inflection points, as shown in Figs. 1(d)-1(f).
Also, phase images obtained with DHM display an inherent coherent noise in contrast to TIE and QWLSI images which are intrinsically smoother, as shown in Figs. 1(a)-1(f). This is because DHM uses a coherent light source, whereas the TIE and QWLSI utilize an incoherent white-light source. Indeed, the off-axis recording geometry of DHM requires a certain coherence length to obtain contrasted interference fringes. Intuitively, the presence of this coherent noise would be thought to diminish the sensitivity of DHM compared to the other two approaches. However, we see that, in practice, this is not the case as line-by-line averaging can substantially decrease the noise level while preserving the accuracy and resolution of the measurement, as shown in Figs. 1(g)-1(i). In addition, it should be mentioned that in this specific case DHM benefits more from the spatial averaging process than TIE and QWLSI. Nevertheless, among the three QPI techniques DHM has been used with the lowest integration time, 0.3 s, compared to 1 s and 1.25 s for TIE and QWLSI respectively. These integration times are influenced by the acquisition rates of the different cameras used and were chosen on a qualitative basis to yield sufficiently contrasted phase images. At longer integration time, similar to TIE and QWLSI, temporal averaging would have reduced the coherent noise of DHM .
Otherwise, the three QPI methods provide quasi-identical leveled phase profiles, as shown in Figs. 1(g)-1(i). As expected, we can see that all leveled phase profiles exhibit a minute depression with a typical magnitude of around 1° in the middle of the fiber core. This central index dip is typical for optical fibers with germanium-doped cores, which undergo an uncompensated collapse during the MCVD process . Note that neither TIE nor QWLSI could resolve adequately the core dip with a 50X-microscope objective whereas DHM succeeded at 40X (data not shown).
Next, SEM images of the fiber cross-section are shown in Figs. 2(a)-2(c). From these images, the diameter of the elliptically shaped central index dip is evaluated to be (600 × 700) ± 10 nm. Leveled phase profiles were processed using the inverse Abel transform and the corresponding ∆n(r) were retrieved and are plotted in Figs. 2(d)-2(e). Solid lines represent the average refractive index profiles of the acquisitions whereas the shaded areas denote their standard deviation. Results are compared with profiles obtained with the RNF technique and by Yablon  using phase-shifting interferometry (PSI) at a wavelength of 675 nm. PSI measurement shows an index decrease down to 1.95 × 10−3 in the region of the central dip. Using all QPI methods, the depression was adequately resolved since values ranging from 3.25 × 10−3 to 3.50 × 10−3 are recorded, but still less pronounced than with PSI. In this regard, QPI along with the RNF measurements are close to each other. Overall, the refractive index profile measurement with all the QPI approaches are consistent. However, the size of the dip was slightly overestimated, ~1 µm at FWHM, exposing the limit of the spatial resolution of the QPI techniques. At this point, it should also be noted that the exact magnitude of the index depression is not known.
Besides, all QPI methods yield a highly accurate measurement of the ∆n(r) of the optical fiber. The edges of the fiber core are clearly defined and a diameter of 8.2 µm at FWHM is measured within a 1% margin for all cases. Relatively flat plateaus, which peak at 5.2 × 10−3, are observed. The average values of the top of the plateaus obtained with all QPI methods is ~5.1 × 10−3, which represent an error inferior to 2% compared to the RNF measurement. However, the uncertainty range is considerably larger for TIE (standard deviation of 3.8% at r = 2 µm) than QWLSI and DHM (standard deviation of 1.4% and 1.5% at r = 2 µm respectively).
In TIE, the phase information is retrieved by a simple mathematical algorithm computed on a group of adequately defocused pairs of bright-field images. It is known from prior literature that the calculated phase depends on the defocus step relative to size of the sample  and on the precision of the position of the in-focus image. While the position of the in-focus image can be precisely adjusted, the a priori knowledge of the sample morphology illustrates one of the limitations of TIE and can explain the increased uncertainty. Furthermore, our experimental apparatus necessitates post-processing of the acquired bright-field images using the proprietary software to generate the corresponding phase images. Accordingly, problems with the measurement are revealed only at the end of a relatively lengthy process. It should be noted that TIE can be implemented in an automated imaging system that acquire a stack of bright-field images and generate the phase image in quasi real-time . Still, TIE operation is intrinsically slower than QWLSI and DHM since a minimum of three images must be acquired to produce the phase image.
In addition, the sharp features of the plateau that are apparent on the RNF measurement are not preserved in TIE, QWLSI and PSI. In these cases, sharp ripples are smoothed and translated to a slow wave-like variation across the plateau. However, the DHM ∆n(r) fits more closely the RNF measurement in terms of undulation and straightness.
Quantitative phase images of a representative waveguide (800 mW, 20 passes) obtained with TIE, QWLSI and DHM are shown in Figs. 3(a)-3(c). The image processing procedure was successfully applied to all phase images. As for the case of the optical fiber, the DHM phase profile requires almost no leveling while the other QPI methods require higher order polynomials, as shown in Figs. 3(d)-3(f). Still, DHM has an inherent coherent noise compared to the other techniques. The three leveled phase profiles of the waveguide show similar qualitative characteristics, as shown in Figs. 3(g)-3(i). Indeed, profiles reveal asymmetric waveguides with an indentation on the right side, which appears slightly more pronounced using DHM.
Next, bright-field images of five typical waveguides across different series are shown in Fig. 4(a). The measurements have been obtained from a cut and polished end-face of the glass chip in a side-view perspective with the apparatus described in Section 2.2.1. This set of images shows that the waveguides are homogenous across the entire range of inscription parameters used in this study. Thefor each QPI technique and every waveguide was determined using the method described in Section 2.5. Theof waveguides inscribed with increasing power and successive passes of the laser beam were measured and are presented in Figs. 4(b)-4(c). The QPI measurements show an increase of thealong the first 6 series, from 0.2 × 10−3 to 2 × 10−3, as shown in Fig. 4(b). The refractive index increase is accompanied by a slow augmentation of the size of the waveguide from 5.4 µm, which correspond roughly to the size of the focal volume of the fs-laser beam, up to 8.5 µm. It can be seen that the TIE method could not quantify the waveguides inscribed at power of 350 mW or less, namely from series I to IV inclusively. As explained earlier, in TIE, the position of the object center must be precisely determined, and pairs of adequately defocused bright-field images must be taken in order to yield accurate phase images. Identifying the position of the center of the phase object or simply the object itself can prove to be difficult, especially when it exhibits a small. In fact, waveguides inscribed with alower than ~0.7 × 10−3 are practically invisible under bright-field imaging thus preventing from locating the phase objects and precluding the possibility of generating a phase image. Moreover, it was possible to obtain a good measurement of the first four waveguides of series V only because the last waveguides were visible and preceding ones were inscribed in predetermined location. The in-focus position of the center of the first visible waveguide of the series was determined and the sample was then moved laterally to the location of the previous waveguides without changing the focal plane of the microscope.
Overall, all three QPI measurements appear to be in quite good agreement. TIE and QWLSI suggest slightly higherin comparison to DHM in the few cases when discrepancies were observed. DHM and QWLSI were able to provide an adequate measurement of waveguides exhibiting aas low as 2.0 × 10−4 which corresponds approximately to the sensitivity limit of both methods. Indeed,inferior to this value becomes difficult to distinguish from the coherent noise in the case of DHM and the tilted background for QWLSI.
In a second experiment, the three QPI methods were used to evaluate the effect of successive laser passes on the photo-inducedin Eagle2000 glass (series VII to XII, shown in Fig. 4(c)). As expected, all QPI methods are still in good agreement. Waveguides display quasi-linear increases in theas a function of the number of passes. For the set of experimental parameters used for series VII to XII, the photo-inducedspan from 0.6 × 10−3 to 3.0 × 10−3. From our experiment, we can see that for a given laser power the photo-inducedcan be fine-tuned by modifying the number of laser passes.
A few interesting results stand out. First, we observe that the photo-inducedvariation as function of laser passes changes with laser power. Indeed, at high laser power (e.g. series XII, 800 mW), the photo-inducedundergoes only a modest increase, passing from 1.6 × 10−3 to 1.9 × 10−3 while increasing the number of passes from 1 to 20. In opposition, theincreases dramatically for inscriptions made using moderate laser power (e.g. series VII and VIII, at 425 and 450 mW respectively), passing from 0.6 × 10−3 to 2.8 × 10−3 for the same number of laser passes. Secondly, we show that a largercan be reached using moderate rather than high laser power. Although this may seem misleading, thermal diffusion out of the focal volume is limited for low laser power and, as such, theis confined over a smaller volume. It is evidenced clearly by series IX and X as a sharp decrease of theis observed when the laser power passes from 500 to 600 mW and the waveguides size increases from 6.9 to 9.7 to 10.2-13 µm. Under this moderate laser power condition, increasing the number of passes of the laser beam allows for an increase of theof photo-induced waveguides while retaining a relatively small diameter. This phenomenon can be used advantageously to form small optical waveguides exhibiting a strong, which may prove difficult to inscribe otherwise. Indeed, varying other irradiation parameters such as laser power, repetition rate or translation speed can yield waveguides with a strongbut most of the time, at the expense of a much larger diameter.
In addition, it is seen that the photo-induceddoes not reach a saturation point under the current inscription parameters. This is consistent with researches from other groups in which up to 8.0 × 10−3 were successfully induced in Eagle2000 glass . In future work, it would be interesting to repeat a similar experiment but to increase the number of passes further to reach the saturation point. Nevertheless, we presented, to the best of our knowledge, one of the most thorough and accurate measurement of theof waveguides inscribed using a femtosecond laser. Indeed, even if the photosensitivity of several glasses to femtosecond pulses was evaluated [12,44–46], only partial information is available and the complete mapping of theas a function of laser parameters is still lacking for most materials. More specifically, the variation of theinduced by successive laser passes, was previously left unexplored.
In summary, the QPI methods all yield reliable and comparable refractive index measurements and have the advantage of being non-destructive and well suited to identify index variations along the axis of optical waveguides. However, these techniques measure the integrated refractive index differences along the optical path through the sample. While being truly quantitative, the integrated refractive index lacks the spatial context, which might become an issue in the case of irregular transverse index variations. In fact, the QPI approaches as such do not provide a refractive index profile of the cross-section of the object, contrary to RNF. Note that such a measure could be realized based on a QPI technique but this would require either cleaving or dicing and polishing the sample down to a thickness of a few tenths of microns to directly access the cross-section of the phase object. Obviously, this is a tedious and destructive approach which is undesirable, especially in the case of waveguides . An alternative way to obtain a non-destructive 3D refractive index distribution based on a QPI technique, would be to use optical diffraction tomography but this comes at the price of more involved imaging setups and reconstruction processes [48–52].
In conclusion, we presented a comparative study of QPI techniques for refractometry of optical waveguides. It is seen that the three QPI methods studied, namely TIE, QWLSI and DHM, all yielded accurate, reliable and mutually consistent refractive index measurements. Indeed, the QPI approaches were able to produce successfully high-resolution quantitative phase images of a Corning SMF-28 optical fiber. The submicron index depression in the center of the core was adequately resolved and the ∆n(r) recovered from the phase images are in good agreement with RNF and PSI benchmark measurements. The step-index measured using the QPI techniques all fall within a 2% margin of the expected value of 5.2 × 10−3. In addition, the QPI methods were applied to theof photo-induced waveguides in Eagle2000 glass using a femtosecond laser. All three QPI approaches produced precise and consistent measurement of the induced. DHM and QWLSI proved to be highly sensitive asas low as 2.0 × 10−4 were successfully quantified. On the other hand, the sensitivity of the TIE method was limited to about 5.0 × 10−4 mostly by its inability to locate the waveguides and the in-focus plane in a bright-field configuration. Furthermore, a detailed map of the photosensitivity of Corning Eagle2000 glass to femtosecond laser pulses is reported. More specifically, the effect of the number of successive laser passes on theof photo-induced waveguides was studied. It is shown that varying the number of passes allow for the control over the inducedof waveguides while inducing only a small variation of the diameter. Finally, the reported data suggests that the QPI methods are suited to be used as precise and non-destructive refractive index shift measuring tools for quality control during mass production of novel photo-induced photonic devices.
Natural Sciences and Engineering Research Council of Canada (NSERC) (IRCPJ469414-13); Canada Excellence Research Chairs (CERC) program; Canada Foundation for Innovation (CFI) (37422, 34265).
The authors wish to thank Stéphane Gagnon for his help with the RNF and SEM measurements. The authors gratefully acknowledge clarifications provided by Pierre Bon on methodological aspects of QWLSI.
Pierre Marquet declares the presence of a potential conflict of interest as a co-founder of Lyncée Tec, a company that commercializes digital holographic microscopes. However, the study has been performed independently of Lyncée Tec in academic laboratories.
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