1. Introduction

Most of the characterization methods of optical waveguides in nominal operation focus on the output or input side of the waveguide, in order to analyze the transmitted or backscattered field in power, polarization, or spectrally. Nevertheless, an optical characterization technique allowing the simultaneous spatial and spectral measurements of the guided light along an optical waveguide in nominal operation remains a challenging but crucial experimental result for the engineering of waveguides (measurement of local losses, defects, modal behavior), or for comparison with theoretical predictions of field propagation. Moreover, for centimetric or millimetric waveguides, the required spatial resolution is in the micrometric range as it allows the accurate measurement of the spatial evolution of the guided light, i.e., waveguide loss evolution, localization of hot points, scattering or defects points, or even spatial mode beating. Furthermore, high spectral and spatial resolutions would allow the analysis of the evolution of the Raman spectrum all along a waveguide in linear and nonlinear scattering regimes, or the generation of spectral components along waveguides by nonlinear effects (four wave mixing, second harmonic generation, etc.).

Several techniques are already used for longitudinal optical characterization of an optical waveguide. The cut-back technique is used to characterize global attenuation of an optical waveguide in nominal operation, or the spectral evolution of the field [1]. However, apart from its destructive method, it is more dedicated to single-mode than multi-mode waveguides, and it cannot be used for sub-micrometric size waveguides or nanofibers for example. On the other hand, different Rayleigh or Brillouin reflectometry and derivative techniques (optical time-domain reflectometry OTDR, optical frequency-domain reflectometry OFDR [2], Brillouin optical time-domain reflectometry BOTDR [3]) are also used for long waveguides such as standard optical fibers with different spatial resolution. Among these techniques, the state-of-the-art Optical Backscatter Reflectometer (OBR) allows micrometric resolution along waveguides [2,4]. Nevertheless, the waveguide is not in its nominal operation conditions and is limited to infrared telecom wavelength. Moreover, all these reflectometry-based techniques cannot spectrally characterize the waveguide in the nonlinear regime. The system based on an OTDR system developed in [5] offers a spectral resolution at the expanse of a spatial resolution of few meters. For high spatial resolution, near-field techniques have been used for nanometric spatial resolution. The technique based on a SNOM tip as a near-field probe sensor has been developed for supercontinuum generation measurements [6]. However, this technique is invasive as the detected mode is influenced by the presence of the tip, and is more dedicated for micrometric waveguides [7]. Another technique have been used to measure the evanescent field and propagating modes in an optical nanofiber by using another nanofiber as a near-field probe [8]. Otherwise, longitudinal measurements in far field configuration of right-angle Rayleigh scattering along a nanofiber with an EMCCD camera have been perfectly realized, however without any spectral discrimination [9,10].

In the present paper, a confocal micro-spectrometer is used to analyze the spatial evolution of the light propagation along an optical waveguide. The principle is based on the detection of the right-angle radiated field by Rayleigh scattering (RS) outside the waveguide [11]. Besides being able to observe the (1D) spatial evolution of the radiated field along the waveguide optical axis, this technique can also be extended by mapping in two dimensions (2D) a plane containing the waveguide. At the expense of acquisition time depending on the various parameters previously chosen for the measurement (number of acquisition points along the waveguide and the exposure time of the spectrometer detector), the technique presents the advantage to be non-destructive, non-invasive, with immunity to waveguide vibrations and to present simultaneously a micrometric spatial resolution and a spectral resolution. Moreover, the use of a cooled-CCD detector offer a high signal-to-noise ratio.

The potentialities of the technique is investigated in the case of an optical stretched fiber also called optical nanofiber (ONF) [12,13]. A nanofiber presents a large interest due to its specific properties, i.e., the high intensity of the confined field, and a strong evanescent field in interaction with the external medium [14]. Then, applications in both linear and nonlinear regimes go from atom trapping in the evanescent field [15–18] to supercontinuum generation [19–21], efficient evanescent Raman scattering [22], evanescent Kerr effect [23], Brillouin spectroscopy [24], or chemical or biological sensor [25] and provides advances in quantum information technologies [26–28]. Therefore, optical characterization along an ONF is highly useful for different input power and wavelength, which are not allowed by current systems. More precisely, the present work concerns the characterization of the optical field along a sub-micrometric diameter fiber, homogeneous along several centimeters of the optical fiber, and pumped in the visible wavelength range. The entire waveguide is mapped with a micrometric spatial resolution and is compared to the state-of-the-art OBR system (OBR 4600 from Luna). This approach is evaluated to observe the beating of higher-order modes (HOMs), the leaking of the non-guided modes, called thereafter radiation modes, from the surface along the taper region, and the transition from core–cladding to cladding–air guidance for different modes. Furthermore, the linear losses and the localization of defects along an optical waveguide in operation are also analyzed.

The use of a micro-spectrometer also allows the spectral analysis of the RS signal. Therefore, the coupling of the spatial and spectral data represents a significant advantage over other techniques. To demonstrate this potentiality, we have analyzed the generation of a Raman cascade along a nanofiber working in nonlinear regime. The longitudinal evolution of the intensities of the first three Raman Stokes orders is in good agreement with the theory, showing that our technique can provide quantitative data without any numerical post-processing.

2. Experimental setup

The experimental setup is illustrated in Fig. 1(a). It consists of a confocal Raman micro-spectrometer (Monovista, S-$\&$-I GmbH) centered on the Rayleigh scattering (RS) detection (obtained by removing the Notch filter of the spectrometer). The system is equipped with a high-precision motorized stage for longitudinal $X$ and transverse $Y$ displacement of the waveguide under study and a microscope objective translation $Z$, all with a step size and reproducibility better than $<100$ nm. This system allows the generation of an accurate recordable 3D spatial trajectory point-by-point all along the waveguide. Therefore, after having recorded the trajectory corresponding to the waveguide, the system allows us to measure a $1D-X$ RS trace along the $X$ waveguide axis or $1D-Y-Z$ RS trace perpendicular to the waveguide, or a $2D-XY-XZ-YZ$ maps in the $XY$ or $XZ$ planes containing the waveguide or in the $YZ$ plane perpendicular to the waveguide. The RS of the guided light in the optical waveguide medium which exits to the outside is collected in the $Z$ right-angle to the propagation direction $X$ (Fig. 1(a)) with a x10 microscope objective and recorded through the spectrometer. The use of a x10 long working distance objective is motivated by the size of the closed transparent box which protects the fiber (Fig. 1(a)). The monochromator is equipped with three distinct diffraction gratings (300, 1500 and 2400 lines/mm), and a back-illuminated cooled CCD detector (-$85^{\circ }$C) leading to a low detection threshold and a high signal-to-noise ratio (SNR). Alternatively, an imaging system composed of a CCD camera can image the waveguide for positioning the waveguide at the optical axis of the RS detection system, and also for rough fiber diameter measurements. The fiber is pumped with a power stabilized cw laser diode with a wavelength of 653 nm (Fig. 1(a)). The input laser power is typically between 2 and 3 mW in order to limit the thermal effects or to avoid the ONF breaking. The laser power is simultaneously monitored in the transmitted power during the measurements without significant variation of the output power.

 

Fig. 1. (a) Schematic experimental setup. PD: photodetector, ES: entrance slit of the spectrometer, (b) geometrical data of the nanofibers, ONF: optical nanofiber, (c) evolution of the spatial resolution versus the entrance slit width.

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The entrance slit width of the confocal spectrometer have been ranging from about 30 $\mu$m to 300 $\mu$m for our measurements which leads to a spatial resolution of about 5 $\mathrm{\mu}$m up to 35 $\mu$m (Fig. 1(c)) for the microscope objective of magnification $\times 10$. This resolution has been estimated by a $Y$-scan of an ONF with a diameter of 690 nm which can be considered therefore as a punctual object. The asymptotic value of 5 $\mu$m corresponds to the spatial transfer function of the confocal spectrometer. Note that we can use other objectives $\times 50$ and $\times 100$ to improve this spatial resolution. Furthermore, the exposure time is in general case equal to about 0.5 s per spectrum. The duration of a complete scan depends on the number of recorded points and exposure time.

Even if the method can be used for any type of optical waveguide, this work focuses on ONF obtained by heating and pulling a portion of a standard optical fiber up to obtain a fiber portion with a diameter lower than 1 $\mu$m and homogeneous up to ten centimeters in length [12,13]. ONF study with our method represents a challenge because these waveguides are not perfectly straight, and a 3D trajectory have to be done accurately to follow very precisely the ONF. For that purpose, we first roughly locate the optical fiber with the CCD imaging system for the $X$, $Y$ and $Z$ positions, and thereafter optimize the scattering signal by adjusting precisely the $Y$ and $Z$ positions with the minimal entrance slit width in order to have the highest spatial resolution and thus precision.

The 3D spatial trajectory which follows the waveguide is made in two steps. The first step corresponds to the record of the spatial position (x,y,z) of the nanofiber point by point with a step size of about 10 mm by optimizing the Rayleigh signal and with an entrance slit of about 30 $\mu$m in order to have the best spatial resolution. The second step corresponds to the interpolation between these points in order to obtain a trajectory with the desired number of points.

In terms of data acquisition and treatment, the RS data were recorded in the same experimental conditions. It is noteworthy that no normalization or correction of intensity point by point has been applied for the different 1D-X or 2D-XY scans. The raw data are given, and, therefore, the evolution of the intensity of the signal represents exactly the evolution of the scattering process all along the waveguides.

Two ONFs are presented in this work. The first one is manufactured from a standard single-mode fiber at telecom wavelength (SMFG652D) called thereafter SMFG652D-ONF. The optical losses measured just after manufacturing were measured between $-1.5$ and $-4$ dB at $532$ nm of wavelength, strongly depending on the injection of the laser beam in this multi-mode fiber. Therefore, this high level of losses is certainly due to the multi-mode property of the fiber in the visible range conjugated with the modal filtering by the taper part of the fiber, leading to optical losses. The second ONF is fabricated with a standard single-mode fiber in visible range (SM450) called thereafter SM450-ONF. The optical losses measured just after manufacturing were equal to about $-0.2$ dB at $532$ nm of wavelength. This low level of losses ensures that the transitions are adiabatic at that wavelength. Just after manufacturing, ONF are placed inside closed transparent plexiglass boxes in order to protect them from external air vibrations and to reduce any kind of contamination by dust for example [29].

The two nanofibers have been manufactured with the exact same drawing parameters, so they should have similar geometrical characteristics. The diameters of two SMFG652D-ONFs measured with a scanning electron microscope (SEM) are reported in Fig. 2. The minimal diameter is equal to $690$ nm with 2$\%$ of precision for the both ONF. The lengths of the both ONFs are equal to $20\pm 0.5$ mm and each transition tapers have a length of about 48 mm long (Fig. 1(b)). The relative difference of the diameters along the fibers is lower than $3\%$, indicating that both fibers have very similar geometrical parameters in the following of the study.

 

Fig. 2. Evolution of the diameters for two different SMFG652D-ONFs along the longitudinal dimension $X$.

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3. Physics considerations

A tapered fiber consists of three different parts (see Fig. 1(b)): i) the standard fiber for which the light is guided in the doped silica core of refractive index $n_{co}$ and diameter $d_{co}$ surrounded by a silica cladding with a lower refractive index $n_{clad}$ and higher diameter $d_{clad}$ than for the core, ii) the two transition parts from the standard fiber to the ONF with decreasing external diameter $d_{clad}$, and iii) the ONF itself for which the light is guided in the silica cladding surrounded by air with a homogeneous diameter lower than $d_{clad}<1~\mu$m along several centimeters.

With a physics point of view, different processes occur along the whole tapered fiber. For what concerns Rayleigh scattering signal measured by the confocal spectrometer, it depends first on the scattering efficiency of the different media, i.e., core, cladding and surface of the fiber, which leads to specific profile when the cladding diameter evolves [4]. It also depends on the field intensity guided in the different parts of the fiber (core, cladding), according to the fiber diameter.

When the diameter decreases along the transition 1 (Fig. 1(b)), the spatial localization of the modes in the taper changes progressively from the core to the silica cladding. This change occurs when the effective index of a mode guided in the core reaches the silica cladding one. Thus, this change depends on the fiber diameter, the wavelength, and differs according to the optical modes of the fiber. This process leads to a decrease of the Rayleigh scattering intensity, because the contribution of the core to the scattering decreases, whereas the cladding contribution is still weak due to the large spreading of the mode and thus of its weak field intensity. The cladding scattering increases thereafter due to the decrease of the fiber diameter and thus of the field intensity. This intensity dip is specific to each mode and allows thus its identification.

If the transitions are adiabatic, each mode is conserved all along the transition without any energy coupling to other modes, up to the cladding cutoff diameter for which some high order modes are no more guided by the fiber and are radiated in the surrounding medium. This latter process should lead to the increase of the scattering signal along the first transition of the taper, due to the high scattering efficiency of the interface glass-air of the fiber of the light that leaks outside the fiber.

If several modes are guided in the transition 1, mode beating can occur as the effective index of each mode differently evolves. This leads to intensity oscillation along the transition and consequently oscillation of the Rayleigh scattering signal. This observation allows again the mode identification.

The reader can refer to the appendix for the numerical simulations applied to the specific case of the fibers used concerning the effective index evolution, the cutoff diameters of modes, and mode beating frequencies evolution in function of the cladding diameter.

Prior to RS traces measured by the confocal micro-spectrometer, the backscatter reflectometry were measured using LUNA OBR 4600. This system, based on swept-wavelength coherent interferometry, uses a central wavelength of 1566 nm with 88 nm of sweeping bandwidth, and a longitudinal spatial resolution down to 10 $\mu$m. The OBR trace can only be measured for the SMFG652D-ONF fiber, as the SM450-ONF fiber presents a too small core diameter for telecom wavelength guiding. Even if the techniques are intrinsically different, the back-scattered light detected by the OBR system and the light scattered outside the fiber in the right-angle could be compared. The scattered power from an elementary isotropic emitter can be decomposed in several parts. For a three-layer model (core-cladding-air) with an emitter localized in the core fiber, one part of the scattered radiation is guided again in the backward or forward core modes (the backward part corresponds to the OBR signal), one part is coupled to the backward or forward cladding modes, and the rest of the power exits the fiber at different radiation angles, a fraction of which is collected by the numerical aperture of the microscope objective of the spectrometer. For a two-layer model (where the core no longer guides and the light is now extended into the cladding), only two parts are concerned, the cladding (detected by the OBR system for previous core modes) and exit parts. As a first approach, the conservation of the scattered light energy by an emitter leading to consider that these parts can be considered as proportional and should give the same scattering signal, as long as the number of modes is the same, and the guiding process are conserved. This hypothesis will be confirmed by our measurements.

4. Experimental results and discussions

4.1 OBR / RS comparison for single-mode fibers

Figure 3(a) shows the backscatter trace from the OBR system at $\lambda =1566$ nm for the SMFG652D-ONF (blue curve), and the scattering trace from our experimental setup obtained for the SM450-ONF with the cw laser source at $\lambda =653$ nm of wavelength (red curve). Therefore, we compare two different fibers however working in the single-mode regime at their respective working wavelengths in the unstretched part. Then, if the transitions are adiabatic, the fundamental mode would not be coupled to other higher order modes. It is noteworthy that the OBR trace has been corrected by the group index evolution all along the stretched fiber in order to have the right position of the OBR data [4]. The scattering profile of the OBR trace (blue curve) can be interpreted as follows [4]. These typical scattering profiles are due to the characteristics of the different scattering sources, corresponding to the fiber core at the beginning of the transition where light is still guided into the core, the silica cladding when the core is no longer guiding, and the fiber surface when the diameter is very small and the surface scattering takes over the cladding scattering [4]. Concerning the RS trace (red curve in Fig. 3(a)), it is in a very good agreement with the OBR trace which confirms our hypothesis that these scattering signals are proportional. Moreover, RS signal have similar signal-to-noise ratio (SNR) as the OBR, meaning that our technique is sensitive enough to detect a small part of the Rayleigh scattering. A difference is nevertheless visible at the beginning of transition 1 or at the end of transition 2. RS trace presents a more pronounced intensity dip of the signal at about 7 mm of propagation, where the diameter is equal to about 50 $\mu$m (upper frame in 3(a)). This intensity decrease is also observable in the OBR trace however with a much lower amplitude and a broader dip. As previously explained, this signal decrease should correspond to the change of the guiding mode for the fundamental mode, leading to the decrease of the core scattering contribution [4]. Figure 3(b) shows the relative effective indices for the fundamental modes of SMFG652D for $\lambda = 1566$ nm and SM450 fiber for $\lambda = 653$ nm in function of the cladding diameter and relative to the cladding refractive index of each fiber (the $n_{clad}$ values are different for the two fibers but are superimposed in the Fig. 3(b)). It is clear from this figure that the fundamental mode presents an effective index equal to the cladding index of the fiber at around 50 $\mu$m for the both cases, where the guiding mode change from core-cladding to cladding-air guiding process. This confirms that the signal intensity dip observed in the RS trace is due to this change. This behavior in also confirmed by the 2D-scan-XY of the SM450-ONF shown in Fig. 3(c). It is clear that the fundamental mode of the core is spreading in the cladding for a cladding diameter of about 50 $\mu$m in agreement with numerical predictions shown in Fig. 3(b). Moreover, no scattering signal is observed at the surface of the transition which should correspond to no longer guided modes. It is thus important to note that the 1D and 2D RS traces do not show characteristics of higher orders modes as observed in the following multi-mode case (mode beating, other intensity dips than for the fundamental mode), meaning that the transitions are clearly adiabatic for SM450-ONF, in agreement with the losses measured just after the manufacturing. Moreover, as with an OBR or OTDR system, the measures of the signal level before and after the waveguide allow the estimation of the whole waveguide losses. From the RS trace in Fig. 3(a), the losses of the ONF are estimated to $-0.15~dB$, in agreement with the losses measured just after the manufacturing.

 

Fig. 3. (a) OBR trace (blue curve) at $\lambda = 1566$ nm for the SMFG652D-ONF, and scattering trace from our technique (red curve) at $\lambda = 653$ nm for the SM450-ONF. Upper frame: Evolution of the fiber diameter with the longitudinal dimension $X$. (b) Relative effective index compared to the cladding refractive index for the fundamental mode of the SMFG652D fiber (blue curve) and SM450 fiber (red curve). (c) 2D-XY RS trace from our technique at $\lambda = 653$ nm for the SM450-ONF; The white dashed curves correspond to the shape of the fiber; the dotted straight lines correspond to the diameter of 50 $\mu$m.

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It is noteworthy that in Fig. 3(a), a peak of intensity is clearly observed at around 45 mm for the RS trace (red curve) due to a scattering point (dust or local defect). This observation confirms the high sensitivity of our system for defects, dust grain, etc.

To conclude this first part, our technique clearly allows the longitudinal optical characterization of a single-mode ONF as with the OBR system, with the same Signal-to-Noise ratio (SNR), however in the visible range unlike the OBR system. It allows the estimation of the losses induced by the ONF, and the local defects. It also confirms the adiabaticity of the transitions of the tapered fiber. These first results are now extended to the case of a multi-mode fiber in visible range.

4.2 OBR / RS comparison for SMFG652D nanofiber

Figure 4(a) compares the backscattered trace from the OBR system (blue curve) at $1566~nm$ of wavelength with the RS trace from our experimental setup (red curve) obtained with the cw laser source at 653 nm of wavelength for the same SMFG652D-ONF. The fiber is thus in single-mode regime for the OBR trace and multi-mode regime for the RS trace. The scattering profiles are similar to the general expected. However, large differences are noticeable between these two curves.

 

Fig. 4. (a) OBR trace (blue curve) at $\lambda = 1566$ nm and RS trace from our technique (red curve) at $\lambda = 653$ nm for the SMG652D-OMF (see text for the arrow signification). (b) 2D scan for the input transition of the SMFG652D-ONF. White dashed line corresponds to the fiber shape.

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First-of-all, the RS trace at the beginning of the transition 1 (arrow A -Fig. 4(a)) shows two successive decreases of the scattering intensity. These intensity dips originate from the expansion of different optical modes from the fiber core region into the cladding region. On the output side, the same dips of intensity are observed due to the back coupling of the different optical modes from the cladding region into the core at the same fiber diameters as observed at the level of the first transition (arrow A -Fig. 4(a)). These complementary observation lead to the conclusion that these modes are necessarily guided in the ONFs and does not correspond to radiation modes in the transition 1.

On the other hand, for low diameters in first transition, the RS signal is strongly blurred by a large increase of the scattering level (arrow B -Fig. 4(a)), contrary to the RS signal in second transition, and to the OBR measurement in the first transition. Therefore, this RS feature should come from the scattering of different radiation modes which are no more guided in the transition 1 when reaching the cut-off diameter. The fact that this scattering in not visible in transition 2 confirms this observation. To go further into details, Fig. 4(b) represents a 2D map of the RS intensity along the transition 1, which highlights the scattering from the surface of the fiber along the input transition level. We clearly observe the scattering process at the surface of the transition 1 by the radiation HOMs, which are no more guided by the clad of the fiber, in contradiction with the 2D scan of the transition 1 of the SM450-ONF (Fig. 3(c)). This observation could be explained by the fact that the transition are not adiabatic for the SMFG652D-ONF at the visible wavelength contrary to the SM450-ONF. It is noteworthy that the scattering signal in transition 1 depends on the input injection conditions of the laser beam into the fiber. Indeed, the input power is distributed in different modes whose relative power ratio differs according to the input coupling. Consequently, the intensity level of the scattered light in the transition 1 can change significantly.

Concerning the two dips in the beginning of the transition 1, we made a 1D scan with a higher number of points. Mode beating is clearly observable in Fig. 5(a). The contrast of this beating strongly depends on the injection in the standard fiber. Additionally, the mode beating stops at a propagation distance $X=6$ mm (indicated by a vertical dashed line). The spectrogram of the RS signal drawn in Fig. 5(b) shows that the beat frequency of that oscillations evolves from about 2 mm$^{-1}$ to 4.5 mm$^{-1}$, in good agreement this the numerical prediction of Fig. 8(c) for a beating of modes LP01-LP11, reported in this Fig. 5(b) (white straight line). This beating frequency disappears near 6 cm of propagation distance, where the diameter of the taper is approximately equal to 55-60 $\mu$m, i.e., where the mode LP11 is no more guided by the core (Fig. 8(c)). It is also clear that the second numerical frequency curve due to LP11-LP21 (green dashed line) is present before 4 mm of propagation distance, in good agreement with numerical simulations, leading to a broad and intense signal before 4 mm in Fig. 5(b).

 

Fig. 5. (a) Beating of modes in the transition 1. Upper frame: evolution of the diameter of the fiber in the same transition portion. (b) Spectrogram of the experimental data corresponding to the evolution of the beat frequencies in function of the propagation distance X. The solid and dashed color lines correspond to numerical predictions of beat frequencies for different mode pairs.

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Another beating signal is visible in the spectrogram, however with a lower contrast and with a frequency from 4 mm$^{-1}$ to about 6 mm$^{-1}$. The numerical results are in agreement with the beating of modes LP01-LP21 (magenta straight line). This beating is visible up to a distance of about 3 cm, corresponding to a fiber diameter equal to approximately 100 $\mu$m, i.e., where the LP21 mode is no more guided by the core in agreement with the simulations of Fig. 8(c). Moreover, the intensity decreases observed at around 3 cm and 7 cm of propagation distance correspond to fiber diameters equal to about 100 $\mu$m and 50 $\mu$m, respectively, in relative agreement with the change of guiding type for the modes LP21 and LP02 as shown in Fig. 8(c). Therefore, from all these observations, we can state that the modes LP01, LP11 and LP21 are present at the fiber input and output, which confirm that the injection in the SMFG652D is multi-modal.

To conclude this second part, our technique allows us to study the behavior of a multi-mode ONF by observing high order mode characteristics. More precisely, we can discriminate between modes which are no longer guided in the fiber after their cutoff diameter, and high order modes that remain guided throughout the ONF. Indeed, the former radiate outside the fiber and are thus characterized by a scattering signal only in the input transition of the tapered fiber, while the latter show the decrease of the scattering signal in the input and output transition part of the ONF, confirmed by the mode beat observation in both transitions.

4.3 Nonlinear Raman cascading process along nanofiber

The coupling of the spectral and spatial analysis have been tested by measuring the Raman cascading process along the nanofiber. All the difficulty here corresponds to the detection of a weak signal. Indeed, the system have to detect the Raman spectral components, which have lower intensities than for the pump, scattered by the Rayleigh process in the nanofiber. Second, as the laser source is pulsed in order to reach the nonlinear regime, the number of scattered photons decreases once again. Therefore, the intensity of the scattered signal is extremely weak. However, the detection was made possible by the low level of detection of the cooled CCD detector of the spectrometer, and by a long exposure time, typically 60 s per point. Moreover, the 1D-scan-X has been performed only for the nanofiber part, i.e., with the homogeneous diameter, where the scattering process is maximal and constant (see for example Fig. 4(a)). This ensures a scattered recorded signal with a high intensity and a stable SNR in order to observe the real dynamics of Raman amplification. Concerning the experimental setup, the laser source corresponds to a pulse doubled-frequency microchip Nd:YAG laser, with a repetition rate of 21 kHz and 500 ps of pulse duration. The laser beam is first injected in a small core fiber (SM600), connected to the nanofiber corresponding to a SMFG652D-ONF. This system allows the control of the injection in the SMF fiber preferentially in the fundamental mode. The spectrum at the fiber output is measured by an OSA (optical spectrum analyzer) every 7 seconds simultaneously with the Rayleigh scattered signal measurements in order to be sure that the Raman cascading process is stable during all the 1D-scan-X over a duration of approximately one hour. Concerning the Raman spectrometer, three Notch filters were used in order to remove the pump Rayleigh component. The entrance slit was equal to 1 mm, and 50 points were performed, i.e., one point every 400 $\mu$m.

Figure 6(a) shows the evolution of the output spectrum of the fiber recorded by the OSA. The spectrum is clearly stable over one hour without significant variations. Figure 6(b) shows that the first (blue curve) and the last (red curve) spectrum are almost similar. In addition, the spectrum in dashed green curve represents the spectrum at the output of the SM600 fiber. This corresponds to the beam spectrum injected in the standard fiber SMFG652D which includes the nanofiber. As the core diameter of the standard fiber SMFG652D is greater than for the SM600, then the field spectrum should not strongly evolve because of the lower intensity than in the SM600 fiber, up to the beginning of the first transition of the nanofiber.

 

Fig. 6. (a) Evolution of the spectrum at the output of the fiber recorded by an OSA. (b) The first (blue curve) and the last (red curve) OSA spectrum. The dashed green spectrum corresponds to the spectrum at the output of the SM600 fiber.

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Figure 7(a) shows the evolution of the spectrum along the homogeneous part of the nanofiber SMFG652D-ONF. Due to the Notch filters, the pump power is lower than the first Raman Stokes order. It is the first time, to our knowledge, that the spectral longitudinal evolution in nonlinear regime along a nanofiber is observed. The Raman cascading process is indeed clearly visible with the generation of several Raman Stokes and anti-Stokes orders. Even if the distance is short (less than 2 cm), the Raman amplification is efficient due to the large reduction of the effective area of the guided modes, leading to the increase of the pump intensity. Furthermore, Fig. 7(b) compares the first and the last spectrum, and shows that the dynamics is approximately equal to 40 dB, allowing the clear observation of the spectral evolution.

 

Fig. 7. (a) Evolution of the scattered spectrum along the SMFG652D-ONF. (b) the input (blue curve) and the output (red curve) spectrum for the nanofiber. (c) Evolution of the natural logarithm of peaks of the different spectral components with regards of propagation distance X. The dashed lines have been used to estimate slopes.

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Figure 7(c) represents the evolution of the natural logarithm of the different spectral component peaks along the nanofiber. The exponential Raman amplification is therefore characterized by a linear evolution, clearly visible for all the Raman Stokes orders, from their generation up to the depletion of their respective pump. Assuming that the detector response is almost flat for the spectral range concerned between the pump and the 3rd order Raman Stokes (532 nm up to 573 nm), the linear slope of the intensity evolution for the first, second and third Raman Stokes orders can be roughly estimated. The first Raman Stokes order presents a slope of about 0.18 $mm^{-1}$, i.e., a Raman gain of 180 $m^{-1}$, the second Raman order 0.35 $mm^{-1}$, and the third 0.53 $mm^{-1}$. The theory predicts that the slope is multiplied by a factor of 2 between the first and the second Raman Stokes order, and by a factor of three between the first and the third Raman Stokes orders. The experimental measurements thus show that the evolution of the spectra are quantitatively coherent with the theory.

5. Conclusion

This work has been dedicated to record the right-angle Rayleigh scattering along a centimetric length and sub-micrometric size optical fiber. The technique based on a confocal Raman micro-spectrometer presents the advantage of being non-destructive and non-invasive.

In linear regime, compared to OBR measurements in a telecom fiber, the technique shows a similar sensitivity and resolution to the scattering profile of the nanofiber. It identifies optical losses, mode behavior along a variable diameter fiber, mode beating and local defects with high reproducibility. It also presents an equivalent SNR to OBR, but with a laser source in the visible range and at any input power, which is impossible for the state-of-the-art OBR system. In nonlinear regime, we have observed for the first time to our knowledge the Raman cascading process all along the nanofiber with quantitative evolution of the different spectral components generated during the propagation. The right-angle Rayleigh scattering technique is then a powerful tool to characterize photonic waveguides in nominal operation.

The perspective of the technique are numerous. The technique could be extended to study other waveguides with complex geometry such as ridge, spiral, opto-fluidic guides, optical microfiber couplers for instance. Moreover, the operating wavelength can be chosen in the visible range but also extended to infrared wavelength with appropriate optics and detector. The intensity and spatial sensitivity of Rayleigh scattering can be used to detect local scattering process induced in a Bragg grating fiber, local doping or UV exposure [30], or local temperature evolution. Additionally, the spectral dispersion through the monochromator offers an opportunity to study in-situ and locally laser power dependence. For instance, it would be worth to study local temperature heating through inhomogeneous broadening of the Rayleigh scattering, but also other nonlinear processes along a waveguide such as second harmonic generation along ridge waveguides, generation of the hyper-Rayleigh scattering process, or the development of the Raman scattering process into a supercontinuum generation along nonlinear waveguides [19,31].

Appendix

All the geometric and index properties of the fibers used are gathered in the table 1. Note that the refractive index profile has been measured by Fiber Index Profiler (IFA-100) based on mach-zehnder for the standard SM450 fiber.

Table 1. Geometric characteristics and refractive indices for SMFG652D and SM450 standard fibers.

View Table

Figures 8(a,b) represent the effective indices $n_{eff}$ of the modes in a SMFG652D fiber and for a wavelength of 653 nm. Figure 8(a) refers to an ONF surrounded by air with a diameter between $0.1~\mu$m and 1 $\mu$m, whereas Fig. 8(b) represents the effective index of only the core modes for cladding diameters $d_{clad}$ from 20 $\mu$m to 125 $\mu$m. It is clear from these two figures that neither the standard SMFG652D fiber nor the SMFG652D-ONF are single-mode at $\lambda =653~\mu$m and can then support several modes. More precisely, from Fig. 8(b), four core mode families (LP01, LP11, LP11, LP02) can be present at the standard fiber input with a proportion according to the injection of the laser beam. On the other hand, from Fig. 8(a), as the diameter of the studied ONF is equal to about 690 nm, then only 4 modes can be guided in the ONF, i.e., HE11, TE01, TM01 and HE21. These four modes can be present from the beginning of the fiber due to the injection in the four previous core family modes, or coupled along a non-adiabatic transition from the fundamental mode. Concerning the SM450 fiber, it is single mode for diameters higher than about $50~\mu$m (see thereafter 3(b)). Below that diameter, the fundamental mode is guided by the cladding of the fiber and the fiber remains single-mode if the transition 1 is adiabatic. For a diameter equal to 0.69 $\mu$m, SM450-ONF can support four modes as for SMFG652D-ONF because the cladding is supposed to have the same refractive index for the both ONFs at 653 nm.

 

Fig. 8. Evolution of effective indices in function of fiber cladding diameter calculated for a wavelength equal to 653 nm for (a) the modes guided by the SMFG652D-ONF immersed in air for cladding diameters from 0.1 $\mu$m to 1 $\mu$m, (b) the modes guided in the core of SMFG652D fiber for cladding diameters from 20 $\mu$m to 125 $\mu$m, and (c) the corresponding beat frequencies between the guided modes.

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Along an adiabatic transition, when the core diameter is too weak, the core no longer guides and each core mode spreads into a cladding mode of the same geometry. This change from the core-cladding guidance to the cladding-air guidance occurs when the effective index of the mode equals the cladding index, i.e., at around $d_{clad}=20~\mu m$ for LP01, $d_{clad}=60~\mu m$ for LP11, $d_{clad}=98~\mu m$ for LP21, and $d_{clad}=103~\mu m$ for LP02 (Fig. 8(b)). However, for non-adiabatic transitions, some energy can also be coupled from these modes to other higher order cladding modes which can be radiated if they are no more guided by the cladding when the diameter decreases along the transitions below the cutoff diameter.

If all the guided modes are present at the fiber input due to the injection, these modes propagate with different propagation constants leading to intermodal beats in the transitions parts only, i.e., where the diameter varies and thus the difference between effective indices of the modes. Then, the beat frequency between two different modes with effective indices equal to $n_i$ and $n_j$ can be calculated from the equation $f_{b}=\left |n_i-n_j\right |/\lambda$. Figure 8(c) presents the beat frequencies for the core modes before they are transformed to cladding modes. It is important to note that the beating is no more visible when one of the involved mode changes its guidance mode due to the large decrease of the contrast of the oscillations (dashed lines in Fig. 8(c)).

Funding

Conseil régional de Bourgogne-Franche-Comté; Agence Nationale de la Recherche (ANR-15-IDEX-0003, ANR-16-CE24-0010, ANR-17-EURE-0002). This work has been supported by the EIPHI Graduate School (contract ANR-17-EURE-0002) and Bourgogne-Franche-Comté Region.

Acknowledgment

Authors thank Gilles Mélin from Ixblue for refractive index profile measurement. We also thank Thibaut Sylvestre for relevant and helpful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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