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We use high-resolution imaging of Rayleigh scattered light through the side of few-mode optical fibers to measure the local spatial structure of propagating vector fields. We demonstrate the technique by imaging both pure modes and superpositions of modes in the LP01 and LP11 families. Direct imaging not only gives high-resolution beat length measurements, but also records the local propagation dynamics including those due to perturbations. The imaging setup uses polarization discrimination to monitor both the transverse and the longitudinal polarization components simultaneously.
© 2015 Optical Society of America
1. Introduction
The theory of propagation of light in optical fibers and cylindrical waveguides is well understood [1]. In real fibers, however, nondestructive approaches for experimentally analyzing propagating fields along the length of a fiber have been of significant interest for many years [2–9]. Unfortunately, direct measurement of the field profile is only possible at the fiber input and output, so the evolution of the field along the fiber is usually unknown. For single-mode optical fibers, in which only two orthogonally-polarized modes exist, the primary propagation characteristic measured has been the beat length between these two modes due to fiber birefringence [2–9], but beat lengths alone do not provide local information on the field. Local polarization measurements can be made through transverse detection of Rayleigh scattered light [6–9], but these have primarily been used only for single-mode fibers [6–8]. Intermodal beat lengths in high-birefringence fibers have also been measured for few-mode fibers [9] but have not resolved transverse structure in the propagating fields.
Meanwhile, there has been recent interest in using and controlling higher-order spatial modes in optical fibers for several applications, including high-bit-rate communications [10,11], fiber-based temperature sensors [12], evanescent optical traps for atoms [13–15], high-power fiber delivery [16,17], and formation of cylindrical vector beams [18–20]. The near-degeneracy between higher-order modes can lead to significant intermodal coupling and evolution of the local polarization and intensity profiles, and the challenge remains of making in situ local measurements of higher order modes in fibers and waveguides. Probing the local field distribution nondestructively enables measurements of propagation in axially nonuniform fibers (e.g. tapered fibers) and the effects of local perturbations. Local measurement capability could also provide an alternative approach to sensors based on optical time-domain reflectometry, such as fiber-based load sensing [21], when very high spatial resolution is desired and optical access is available.
In this paper, we demonstrate high-spatial-resolution imaging of the Rayleigh scattered (RS) light to observe propagation dynamics in few-mode fibers. We show characteristic RS images for pure modes of fibers supporting the LP01 and LP11 mode families, and directly image the evolution of mode superpositions along the fiber to measure beat lengths. As an example of the ability to probe local perturbations, we image the propagation across a fiber splice joining two different optical fibers. Finally, we also show that differences in the strengths of the longitudinal components of these modes can be detected by simultaneous imaging of the transverse and longitudinally polarized scattered light.
2. Background
Rayleigh scattering, which is one of the primary loss mechanisms in optical fibers, is due to subwavelength fluctuations in the refractive index, and has been used to measure beat lengths in optical fibers [6–9]. The small fluctuations scatter light with a dipolar polarization dependence, providing information on the polarization state of the incident light. Details of RS can be found in many texts [22], but the essential relevant aspect is that only the polarization components of the propagating field orthogonal to the viewing direction are observed. By viewing from two (or more) directions, we obtain information about the total field of the incident light on a scatterer. In a bulk medium, the distribution of Rayleigh scatterers enables spatial variations of intensity and polarization to be sampled and imaged.
The field in the fiber can be decomposed in cylindrical coordinates (r, θ) as follows
where βj are the modal propagation constants, fj are the N guided fiber modes, and z is the propagation distance along the fiber axis,. Two modes interfere with a beat length zb = 2π/(β2 - β1) = λ/(n2 – n1), where n2 and n1 are the effective refractive indices for the two modes. Figure 1(a) shows the first six modes of a cylindrical waveguide that are supported in a few-mode optical fiber. In a single-mode fiber, only two degenerate HE11 modes exist; this degeneracy is lifted for birefringent fiber (e.g. polarization-maintaining fiber). In standard fibers, the vector modes [TE01, TM01, HE21(e), HE21(o)] have spatially-varying polarization profiles and nearly equal propagation constants, with correspondingly large zi. In the weak-guiding approximation [1], superpositions of these higher-order modes form the basis of the LP11 family. Specifically, the vector mode basis functions have the following approximate profiles in the fiber core:In the weak-guiding approximation, the F01 and F11 are Bessel functions; for exact forms, see Ref [1]. In these expressions, we have ignored the longitudinal polarization component, which is negligible in weakly-guiding fibers.
Fig. 1 (a) Vector modes in the LP01 and LP11 families. (b) Example of TM01 in fiber. Only the polarization component orthogonal to the viewing direction is observed, giving a two-lobed profile from each direction. (c) Schematic of setup. CCD camera images fiber-under-test (FUT) over its length. Sample image shown, with TM01 input. Structure of Rayleigh scattering for TM01 is clearly resolved in the core. WP = λ/2 waveplate.
The dipolar behavior of RS maps the spatial variation of the vector mode polarization onto the scattered field. The 2D intensity profile is integrated along the viewing direction to become a 1D line profile for each z position. As an example, Fig. 1(b) shows the expected RS profiles of the transverse components of the TM01 mode, for which RS produces two lobes from any viewing direction; when imaged onto the camera with high magnification, this mode appears as in Fig. 1(c). Because we are viewing the propagating beam from the side, both the longitudinal and transverse polarization components are detected. Light polarized along the z-direction can be imaged separately, but in the weakly-guiding fibers used in this work, this component is small and will be discussed in Sec. 4.4. Images for other pure modes are shown in Sec. 4.1. Mode superpositions, discussed in Sec. 4.2 and Sec. 4.3, produce polarization structures, and hence RS images, that evolve along the length of the fiber according to Eq. (1).
3. Experiment
The basic setup for these experiments is shown in Fig. 1(c). A Gaussian laser beam from a diode laser at 795 nm (Vescent D2-100-DBR) is converted to a cylindrical vector beam (CVB) and coupled into the fiber under test (FUT). RS light emitted orthogonal to the propagation direction is collected by a near-IR Mitutoyo microscope objective (10x or 50x) and imaged onto an EMCCD camera (Andor Luca). By scanning the CCD camera over the fiber length, we build an RS image of the fiber that contains information on the spatial distribution of the field polarization in the fiber core. A typical image with TM01 input is shown for HI1060 fiber. The core shows the structure of the RS light from one direction.
The CVBs are formed using a fiber-based mode converter [18,19]. Briefly, a linearly-polarized Gaussian beam passes through a π-phase-plate mounted on a translation stage. The resulting beam is coupled into HI1060 optical fiber with an inline fiber polarization controller. This controller mixes the modes, and produces either pure CVBs or any superposition of the first 6 modes in the LP01 and LP11 families [23] at its output. This beam is then coupled into the FUT.
Although the imaging is done through the fiber cladding, the main effect of the cladding is to act as a cylindrical lens, which magnifies the core by ~1.6, but it has only a small effect on the image quality. Initially we tried mounting the fiber in an index-matched medium, but we found that the added complexity did not improve the image quality significantly. The 50x objective has a specified imaging resolution of 0.4 μm, but its 0.65 μm depth of field is smaller than the fiber core diameter (~3-5 microns), so the imaging resolution is slightly reduced. Also, because only polarization components orthogonal to the viewing direction are detected in dipole scattering, we also simultaneously image the fiber from a second azimuthal direction using the same objective and camera with a tilted mirror [see Fig. 1(c)]; e.g. for orthogonal views, the mirror is tilted by 45 degrees.
4. Results
4.1. Rayleigh scattering profiles for pure vector modes
Although the intensity distributions of the vector modes in the LP11 family are similar, their azimuthal polarization dependences produce distinct RS behavior that can be resolved when the fiber is viewed from different azimuthal directions. By analogy with the TM01 example shown in Fig. 1(b), each mode in the LP11 family produces profiles with 1 or 2 lobes.
In Fig. 2(a), we show RS images of pure modes within the LP01 (HE11) and LP11 families [TE01, TM01, HE21(e) and HE21(o)] propagating in Corning HI1060 fiber with 795 nm light. To differentiate the modes in the LP11 family, we view the fiber from two azimuthal orientations separated by 45 degrees by tilting the mirror in Fig. 1(c) to 22.5 degrees; for this choice, the two views of the vector modes will have a distinctive combination of 1 or 2 lobes as shown. We have also used the specified fiber parameters to calculate mode profiles [1] to give the expected line profiles underneath the images, along with experimental line profiles.
Fig. 2 (a) Images and line profiles of the core of HI1060 fiber from 0 and 45 degrees for pure modes in the LP01 and LP11 families. Horizontal axis is scaled by the fiber radius. (b) RS images of the TM01 and TE01 modes over the 50 cm fiber length. The images do not change longitudinally, indicating high modal purity.
The images in Fig. 2(a) have not been rescaled; for the four excited modes in the LP11 family, there is equal power radiated along the two directions. For the HE11 mode, however, which has a uniform transverse polarization, the scattered power along a given direction can vary; for the particular case shown in Fig. 2(a), the power in the 45 degree view is less than half that of the 0 degree view. Depending on the polarization, it should be possible to eliminate RS viewed from one direction. However, we have found that it is rarely below 5% of the maximum, likely due to deviations from dipolar scattering [24]. From one direction and longitudinal location, apart from having a slightly smaller width, the shape of the HE11 RS profile is indistinguishable in a practical system from the shape of the TE01 mode RS profile, which also has only one lobe. However, the two can be distinguished by the azimuthal variation in scattered power. In principle, they can also be distinguished by comparing the longitudinal RS power, discussed in Sec. 4.4. For these plots, we ensured >95% mode purity by measuring the modal composition at the input and output [23]. Purity was also confirmed by noting that the RS profiles did not evolve over 50 cm. We show this for the TM01 and TE01 modes in Fig. 2b. RS imaging using a superposition of modes is discussed in Sec. 4.2.
4.2. Superpositions of modes within the LP11 family
When two (or more) modes propagate simultaneously, the local intensity distribution is dependent on the propagation distance according to Eq. (1), leading to RS images that vary over the fiber length. In birefringent single-mode optical fibers, only the two HE11 modes can propagate. These modes have a transverse polarization that is spatially uniform, so the evolution of the polarization leads only to oscillations of the RS power (calculated by summing the transverse pixels of the RS images) detected from one viewing direction, but not of the shape of the intensity profile. When higher-order vector modes are included, however, superpositions can also lead to oscillations of the shape of the intensity profile. In this section, we demonstrate RS imaging of superpositions of the hollow modes within the LP11 family.
Figure 3 shows RS images of the fiber for a superposition of the TM01 and HE21(e) modes [Fig. 3(a)], and for a superposition of the TE01 and HE21(o) modes [Fig. 3(b)]. As shown at the top of Fig. 3(a), the initial TM01-HE21(e) superposition has an intensity profile consisting of two horizontally-separated lobes with horizontal polarization: That this profile matches the TM01-HE21(e) superposition can be deduced from the polarization profiles shown in Fig. 1(a). Such a profile is formed easily from the vector beam generator (e.g. by putting a horizontal polarizer into a pure TM01 beam). The TE01-HE21(o) superposition in Fig. 3(b) is then formed by rotating the polarization to vertical with a half waveplate.
Fig. 3 (a) Top: The superposition of HE21 and TM01 modes produces an oscillating two-lobe pattern. RS images are shown over two beat lengths. Bottom: RS image and profile over 50 cm. Scattered power in each image (blue), calculated by summing the transverse pixels of the RS images, is fit to a sinusoid (black) to determine the beat length. Red: RS intensity along the fiber axis relative to the peak RS intensity in the image. (b) Same plots for a superposition of HE21 and TE01. (c) Cutback measurements of the HE21-TM01 superposition at maximum RS power (top) and minimum RS power (bottom).
The intensity profiles of these superpositions evolve as shown in the schematic diagrams at the top of the figure. We show the corresponding RS images over two beat lengths, imaging from a horizontal (x) direction. For these superpositions, the spatial structure and the power scattered into a particular direction evolve with z, and these variations are observed in the imaging. For the TM01-HE21(e) superposition, at places where the polarization is vertical, the two intensity lobes are also displaced vertically and are easily resolved. When the polarization is horizontal, the two lobes are displaced horizontally and no light is scattered toward the camera. For the TE01-HE21(o) superposition shown in Fig. 3(b), two lobes are never observed from the horizontal direction.
To verify that the local intensity distribution evolves as described, we make cutback measurements of the fiber at two locations and image the beam profile through a polarizer. When we cut the fiber at the z-position of maximum RS having two lobes, the beam is almost entirely y-polarized with the expected two-lobed profile. Alternatively, when the fiber is cut at the minimum of the RS signal, we observe only the x-polarized, two-lobed distribution. Thus, from the evolution of the RS images, or by taking images from two different azimuthal directions as done in Sec. 4.1, we can make estimates of the propagating intensity distribution.
We have also plotted the images and profiles over 50 cm to extract the beat lengths. Because of the large zb, the evolution is slow and captured by taking a single image every 0.5-1 cm. We fit sinusoids to the profiles to extract zb = 10.54 ± 0.01 cm for the HE/TE superposition, and zb = 4.53 ± 0.01 cm for the HE/TM superposition. Superpositions of TE01 and TM01 modes [or of HE21(e) and HE21(o)] do not show beating in the Rayleigh scattering: In these cases the polarizations are always orthogonal so interference does not occur and the sum of the two individual RS distributions would be observed.
Based on the manufacturer specifications for Corning HI1060 fiber, we can estimate the expected beat lengths [1]. For the HE/TE superposition, we estimate zb = 8.1 cm, and for the HE/TM superposition, zb = 15.9 cm. The large differences between these estimated values and the experimental values, particularly for the HE/TM beat length, are not surprising for nearly degenerate modes because zb, which is large, depends inversely on the small difference between the βi. The relatively large uncertainties in the manufacturer specifications can strongly affect the difference between the propagation constants.
The sinusoidal oscillations should reach 0 for dipole scattering. As seen in Figs. 3(a) and 3(b), however, the minimum RS power is at least 20-30% of the maximum. This most likely arises from deviations in dipole scattering from irregularities at the core-cladding interface and fluctuations of the core radius [24]. The reduced contrast is most noticeable for the HE/TM superposition in Fig. 3(a), in which we have also plotted the RS intensity along the fiber axis (relative to the peak image intensity) where no scattering should be detected. Along this line, the RS is constant through the fiber length, even at z positions where no scattering should occur, indicating a process that is independent of polarization and proportional solely to the propagating power. We have found this level to be strongly dependent on the fiber type used, and even for the same fiber but different manufacturing batch. Reducing the microscope objective NA with an iris produced no appreciable change. Imaging is also done with bandpass filters to check for broadband fluorescence, with no observable effect.
4.3. Superpositions of the fundamental and higher order modes
As described in Sec. 4.2, superpositions of modes within the LP11 family give rise to polarization (and intensity) profiles that evolve longitudinally. Superpositions of the HE11 mode (LP01 family), and a vector mode (LP11 family) will also have a z-dependent field profile, but because the HE11 mode has a uniform polarization, the RS power along a given viewing direction is independent of z. For these cases, observation of the beating between the mode families requires transverse spatial resolution of the light in the core. In [9], the transverse magnification was not high enough to visually observe the transverse oscillations, but was enough that a row-by-row Fourier transform of the images could record the beat frequencies.
In our setup, we have used high magnification to clearly resolve these transverse oscillations. Figure 4 shows typical images for 5 different superpositions of the HE11 and TM01 modes. The mode amplitudes are controlled by translating a π-phase-plate across the input Gaussian beam as shown: When the phase plate is centered, the light coupled into the fiber excites only the LP11 family, while when it is removed only the LP01 family is excited. The relative phase between the fundamental and the higher-order modes varies by π when the phase step is on opposite sides of the Gaussian beam center. This π shift is reflected in the 2nd and 4th images shown in Fig. 4(a), in which the oscillations are out of phase. Two line profiles at z locations separated by zb/2 are shown in Fig. 4(b); the peak in the intensity profile shifts transversely by ~2.2 μm for Corning HI1060 fiber. To measure zb, we find the peak center as a function of distance, and fit the resulting sinusoidal curve to extract the period and phase.
Fig. 4 (top) Superpositions of the LP01 and LP11 families are made by translating a pi-phase-plate across a Gaussian beam. (middle) Beam profile entering the fiber. (bottom) Rayleigh scattering image over a 1mm fiber length. (b) Cross sectional profile of the LP01-LP11 superposition at two locations in the image.
For the case of HI1060 at 795 nm, we measure zb = 293 ± 2 μm, again indicating a high level of precision in the measurement. The higher-order-mode beat frequencies measured in Sec. 4.3 also had fractional uncertainties of less than 1%. We estimate zb based on fiber specifications, as in Sec. 4.2, to be 240 μm. This relative agreement is better than that of the higher orders due to the large difference in the βi so that uncertainties in the specifications are less important. In non-polarization-maintaining fibers, intermodal beat lengths are typically 10-50x smaller than the intramodal beat length for the LP11 family, and the two measurements cannot be made in a single image in our setup. We note, however, that one could use anamorphic magnification with cylindrical lenses to achieve both high transverse magnification and large longitudinal distances simultaneously.
4.4. Imaging of the longitudinal field component
With the exception of the TE01 modes, each mode has a field component in the longitudinal direction [1]. It is expected that this this component may be observed by viewing the RS light through a polarizer oriented along the fiber axis. Imaging the longitudinal field of tightly focused, radially-polarized vector modes has been demonstrated using scattering from oriented single molecules [25]. We implement this imaging geometry by placing a calcite displacement prism near the CCD image plane, which separates the transverse and longitudinal RS polarizations on the same image. In standard single-mode fibers with low NAs and core diameters of ~5 μm, the longitudinal component is calculated to be <1% of the transverse component in a typical fiber. Here, we used Fibercore SM1500 optical fiber, which has a NA ~0.3 and a core diameter of ~3.5 μm. In principle, measuring the longitudinal polarization should be useful for maximizing the purity of the TE01 component, which is distinguished by having no longitudinal field. In practice, as will be seen below, the data suggest that other scattering mechanisms obscure quantitative observation of the small longitudinal component.
Figure 5 shows the images of the transverse and longitudinal components for SM1500 fiber. In Fig. 5(a), we show a RS image using polarization discrimination on the TM01 mode; the transverse polarization has the characteristic two-lobed profile, and the longitudinal polarization has a single lobe as predicted. In Fig. 5(b), we plot only the integrated line profiles for the HE11, HE21, TE01, and TM01 modes for the transverse and longitudinal polarizations. Since we are comparing the relative RS power between the longitudinal and transverse polarizations, for visual comparison the line profiles are plotted with amplitudes that give equal total scattered power from the transverse polarization component. This makes the HE11, with a slightly narrower profile, have higher amplitude than the LP11 modes. Because the recorded longitudinal profiles are similar, the differences in the total scattered power of the longitudinal component are clear.
Fig. 5 (a) Simultaneous image (using 50x microscope objective) of the longitudinal and transverse components in the fiber when TM01 is propagating, along the with the line profiles for each. (b) Line profiles for the HE11 (fundamental) mode and the HE21, TE01, and TM01 modes. Profiles have been displaced horizontally from one another for visual comparison.
Quantitatively, the measured longitudinal powers relative to the transverse powers are 0.05, 0.096, 0.144, and 0.11 for the HE11, TE01, TM01, and HE21 modes, respectively. We note that these values are significantly higher than the predicted relative powers of 0.007, 0.0, 0.035, and 0.015, respectively. Unfortunately, although the measured values agree qualitatively with the predicted powers for the higher order modes, the contrast between the different modes is weak, and even for the TE01 mode, we see a strong longitudinal component. This is likely due to similar surface effects at the core-cladding interface and deviations from dipole scattering [24] observed in Sec. 4.2. One indication of this is that, for the HE11 fundamental mode, we observe the smallest z-component and it also has the lowest intensity at the core-cladding interface; the modes in the LP11 family all have significant power at the core-cladding interface, and correspondingly higher detected longitudinal polarization. This qualitative behavior was observed in all fibers we tested.
The objectives have a large collection angle. Off-axis scattered light could have a component along the longitudinal direction and corrupt the measurement. To test this, we also performed the imaging with a small iris in front of the microscope objective. We did not observe any change to the measured polarization ratios. Thus, in our test fibers, longitudinal field imaging shows the correct qualitative behavior but cannot be used for quantitative mode analysis. In very tightly confining waveguides, such as tapered nanofibers, the propagating longitudinal component should be much stronger and more quantitatively compared with the transverse polarization component. Furthermore, the ability to image longitudinal and transverse components simultaneously should be useful for examining propagation in bulk media in which excess surface scattering at interfaces does not arise.
4.5. Imaging local perturbations
In the previous sections, we showed how RS imaging can distinguish propagating modes and provide information of the local, transverse intensity distribution. In this section, we apply RS imaging to observe the effects of a local perturbation on propagating light. In our case, the local perturbation is a splice connecting two fibers with different core size.
Figure 6 shows a RS image for light propagating in HI1060 fiber that has been spliced to 980HP fiber. The core diameter of HI1060 is 5.3 μm; that of 980HP is 3.6 μm. The input light is in a superposition of LP01 and LP11 modes. A number of propagation phenomena are indicated, and the interface between the two fibers is clear. First, the image shows the dramatic change in beat lengths (zb = 293 ± 2 μm for HI1060, and zb = 141 ± 2 μm for 980HP), but the superposition was maintained across the splice. Second, the interface itself is marked by a comparatively large amount of scattering. Third, the light in the input HI1060 fiber that overfills the smaller core of the 980HP fiber can be seen diverging into the cladding of the 980HP fiber. The longitudinal dimension of the image in Fig. 6 has been compressed, and the ~45 degree angle of the divergence in the figure corresponds to an angle of ~80 mrad. This compares well to the expected divergence angle, determined by nsinθ = NA, where n ~1.5 is the cladding index, and NA is the numerical aperture (~0.14). RS imaging could also be useful for analyzing other perturbations, such as bends or localized pressure.
Fig. 6 Image of a fiber splice using RS light. The splice location is indicated by the dashed vertical line. Fiber cores are marked by thin white lines and extensions to the sides of the image. Light from the HI1060 core that is not coupled into the 980HP core can be seen diverging into the 980HP cladding. The fibers have 125 micron cladding diameter.
5. Conclusion
We have demonstrated high-resolution Rayleigh scattering imaging of light propagating in few-mode optical fibers. The polarization dependence of the modes is mapped onto the images so that vector modes can be discriminated from one another when views from two or more directions are recorded. The spatial evolution of modal superpositions is also resolved. Imaging Rayleigh scattering is useful for determining the local intensity distribution and visualizing the effects of perturbations, and should be especially interesting to apply to axially-tailored fibers in which propagation conditions vary longitudinally. In this work, we used a fiber splice as the perturber, but the technique could also be used for imaging the effects of bends or localized pressure on higher order mode coupling. This ability to image fiber modes should also be useful for propagation analysis of planar integrated waveguides.
Acknowledgments
The authors acknowledge helpful discussions with Jonathan Hoffman. This work was supported by the Office of Naval Research and by the Defense Advanced Research Projects Agency.
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