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Abstract
Spatially and spectrally resolved imaging (S2 imaging) is a technique that was developed as an alternative to beam quality (M2) measurements to characterize the modal content of large mode area (LMA) fibers. While it is known that the success of S2 imaging is highly dependent on the broadband source and the launch conditions into the fiber, the information resulting from this method is more limited than may appear at first glance. Experiments and numerical simulations are used to show that (a) the accuracy of the reconstructed LP01 mode profile varies greatly depending on launch conditions, and (b) there are always errors in the reconstructed LP11 mode profile. Not only do these findings reveal that the reconstructed mode areas cannot be accurately determined, but it also shows that the relative modal intensities can only be determined accurately under nearly perfect conditions of fundamental mode launch and propagation through the fiber. Despite these findings, the S2 technique can still be used under nearly ideal launch and propagation conditions to identify the modes and their group delays, and accurately reconstruct only the fundamental mode.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spatially and spectrally resolved imaging (S2 imaging) is a measurement technique that is used to characterize the modes propagating in large mode area (LMA) fibers [1,2] as an alternative to standard measurements of the beam quality (M2) factor. While S2 cannot generally be used on an active laser or amplifier system, it has a particular advantage in being able to assess the fiber alone. The concept behind the S2 measurement is based on the fact that each mode propagates down the fiber at a different speed. For a broadband source, the modes at each wavelength will add coherently with a phase delay that is different at each wavelength, resulting in modulation of the observed optical spectrum. The Fourier transform of the spectrum will result in distinct peaks that are due to beating between the modes at the output of the fiber. Further, if the spectrum is spatially sub-sampled, then the spatial profile of the mode can in principle be reconstructed since the intensity of the Fourier peaks represent the strength of the beating in intensity space.
The S2 measurement was originally developed specifically to be an alternative to the standard M2 measurement technique. It was considered to be superior because it generates the spatial mode profiles and their relative powers, and can be used on an unpumped fiber. The latter in particular is advantageous in that it eliminates the time, cost, and labor otherwise required to build, for example, kilowatt-class lasers or amplifiers to test the modal properties of the fiber. There are other methods that can be used to assess the modal content of the fiber, though each has significant limitations [3,4]. S2 imaging has proven to be a useful technique to identify the number and types of modes propagating in the fibers as well as the relative intensity of the modes [5–7]. The method has also been adapted to measure other properties, such as bend loss [8] and splice performance [9]. However, certain conditions must be met for a successful S2 measurement. First, the source must be sufficiently broadband to allow for many beat periods between the fiber modes [1]. Second, the power in the higher order modes must be small in comparison to the power in the fundamental mode [2]. Two new analysis methods [10,11] have been proposed to ease this limitation, but neither has been validated experimentally. S2 imaging can also be complicated by the presence of “spurious modes” [12], which are not true modes but extra peaks in the Fourier transform caused by the interference between higher order modes. Nonetheless, given the modal imaging properties of the S2 technique, it is reasonable to assume that other information, such as mode area, can be extracted from the data, although this has not yet been demonstrated to date.
In comparison with other techniques, such as the correlation filter technique (CFT), S2 imaging is more sensitive to both the spectral bandwidth of the source and the fraction of power in the fundamental mode, and presents greater challenges when identifying modes in highly multimode fibers. However, S2 imaging has advantages over CFT, most significantly that it does not require prior knowledge of certain fiber parameters and that it has the ability to reconstruct the mode profiles [3].
In this paper, we present an experimental and theoretical study of the ability of S2 imaging to characterize further properties of the modes, and identify any limitations when using S2 imaging. In Section 2, the experimental setup and physical model are described. Experimental measurement results are presented in Section 3, and the theoretical analysis is provided in Section 4. A discussion of the results and concluding remarks are presented in Section 5.
2. Methodology
2.1 Experimental setup
The experimental setup is shown in Fig. 1. A broadband laser was launched into the LMA fiber under test. The input end of the LMA fiber was placed on a stage with five axes of control (x, y, z, tip, and tilt) to allow for adjustments to the launch conditions. These launch conditions were then examined by placing a card between the second lens and the polarizer at the output end of the fiber under test. A set of lenses after the LMA fiber was used to magnify the image of the beam onto a sampling fiber, which was single mode in the wavelength band of the laser. A polarizer was used to ensure that the polarization states of each of the modes were aligned. The sampling fiber was placed on a 2-dimensional scanning stage, and light coupled into this single mode fiber (SMF) was launched into an optical spectrum analyzer (OSA). The magnified image size and the size of the core of the sampling fiber dictated that the stage be scanned over a 30 × 30 grid, with the optical spectrum being measured at each point in the grid. The scanning stage (Stage 2 in Fig. 1) and the OSA data acquisition were fully automated, requiring approximately 2 hours to acquire the full 30 × 30 data set. The acquired data was then processed with the analysis presented in [1].
Fig. 1. Experimental setup. A single mode fiber (SMF) attached to an optical spectrum analyzer (OSA) scans over the magnified image of the output of an LMA fiber using a 2-dimensional stage (Stage 2).
2.2 Simulation
A simulation of the S2 measurement was created for comparison to the experimental results. The intensity given by the interfering fields exiting the fiber has the form given by Eq. (1):
For the purposes of this simulation, only the LP01 and LP11 modes of the fiber under test were considered. As the two lowest order modes, these modes are present in all fibers that can be measured using the S2 technique. The LP01 and LP11 modes of the fiber under test were calculated, along with their effective indices. This information was used with Eqs. (1) and (2) to calculate the interference intensity between the two modes. The resulting wavelength dependent intensity was then divided into a 30 × 30 grid to match the conditions of the experiment. Like the experimental data, the simulated data was processed with the analysis presented in [1]. To follow this method, the Fourier transform of the spectrum was taken at each point in the grid. A sample Fourier transform is shown in Fig. 2. The Fourier transform includes a peak at zero time delay (the DC component) as well as a smaller peak on each side.
Fig. 2. Fourier transform of the intensity resulting from the interference of two modes in the fiber at one (x,y) point.
In principle, integrating over the DC component at each (x,y) point allows for reconstruction of the intensity profile of the fundamental mode. Correspondingly, the intensity profile of the LP11 mode can be reconstructed by integrating over the second peak. These reconstructed modes were then compared to the original input modes to analyze the efficacy of the S2 method.
3. Experimental measurements
Two different broadband sources were used for the measurement: a diode pump laser thermally detuned from its volume Bragg grating (VBG) stabilizer, and an ultrafast laser. The required bandwidth for the S2 technique depends on the fiber specifications and the length of the fiber under test, but generally varies from 10-50 nm. Most importantly, the source must be slowly varying in frequency so that the width of the Fourier transform of each mode is small compared to the period of the beat frequency between any two modes. The two different broadband lasers were tested with 20 m of standard SMF-28 fiber, which is multimode at the wavelengths used. The laser spectra exiting the test fiber are shown in Fig. 3.
Fig. 3. Measured emission spectra for SMF-28 as the fiber under test using different broadband sources: (a) a pump laser thermally detuned from its VBG, and (b) an ultrafast laser.
The difference is most obvious when comparing the Fourier transforms of the two spectra, which are shown in Fig. 4. The integration region for the LP01 mode is indicated by orange shading, while the integration region for the LP11 mode is indicated by green shading. There appears to be a second peak in the transform of the pump laser spectrum, which would indicate the presence of a second mode. However, when the transform at each point in the 2D grid is stitched together, it is clear that measurement technique does not work. The two modes reconstructed from measurement with the pump laser are shown in Fig. 5. The central peak does not represent a clean LP01 mode. More importantly, the data from the second peak is not recognizable as a mode at all. The measurement fails due to the narrow spectral range of the laser source. While the total width of the spectrum is around 12 nm, nearly all of the intensity is contained in a peak at 976 nm. The usable signal in the 5-nm section around 966-971 nm is small compared to the peak at 976 nm, and is too close to the noise floor of the OSA to produce a good measurement (note that averaging could not be used due to the already long data acquisition times). In contrast, the second peak is more well defined when using the ultrafast laser, which had a spectral width of 20 nm and a central wavelength of 1035 nm. When this data is stitched together, the results are significantly improved, as shown in Fig. 6. Here, the reconstruction of both LP01 and LP11 modes are obvious. However, even these two modes are not ideal modes. The LP01 mode has a longer tail towards the bottom of the plot, and the LP11 mode is clearly not symmetric. The asymmetry is likely due to bend induced mode deformation.
Fig. 5. Reconstructed mode images of SMF-28 fiber imaged with the pump laser: (a) the primary peak shows a near-LP01 mode, (b) the second peak shows no recognizable mode. The mode images capture a span of 15 µm in each direction.
Fig. 6. Reconstructed mode images of SMF-28 fiber imaged with the ultrafast laser: (a) the primary peak shows a good LP01 mode, (b) the second peak shows a recognizable LP11 mode. The mode images capture a span of 15 µm in each direction.
In order for the measurement to be successful, the majority of the power must be contained within the fundamental mode. To test what happens if this is not the case, the launch to the SMF-28 fiber was adjusted so the LP11 mode was visible in the fiber output. The results of this measurement are shown in Fig. 7. Here, both LP01 and LP11 modes are still recognizable, but the reconstructed shapes are visibly incorrect. The fundamental mode is no longer round, but appears elongated. Like the previous measurement, the LP11 mode is asymmetric, but here the intensity difference between the two lobes is greater.
Fig. 7. Reconstructed mode images of SMF-28 fiber imaged with the ultrafast laser: (a) the primary peak shows an elongated LP01 mode, (b) the second peak shows an asymmetric LP11 mode. The mode images capture a span of 15 µm in each direction.
With these understandings established, the ultrafast laser was used to attempt S2 imaging on 10 m of a cladded linear index graded (CLING) fiber. The CLING fiber has a 50-µm core and is designed so the fundamental mode matches that of a standard LMA fiber with a 25-µm core, but the higher order modes are broadened significantly [13]. The CLING fiber was carefully aligned in an attempt to obtain a nearly single mode output, with some success. The CLING fiber has many modes, so it was not possible to obtain purely single mode output in our test configuration. The results of the S2 imaging are shown in Fig. 8. Both reconstructed LP01 and LP11 modes are recognizable in both cases, but neither would be considered to be ideal representations as they should essentially look like conventional LP01 and LP11 modes. Most importantly, they seem to be approximately the same size, which is not expected for the CLING fiber under test. For this CLING fiber, the LP11 mode should be identified as substantially larger than the LP01 mode by observation alone.
Fig. 8. Reconstructed mode images of the CLING fiber under test: (a) the primary peak shows an LP01-like mode, (b) the second peak contains a recognizable LP11 mode amongst other features. The mode images capture a span of 70 µm in each direction.
4. Theoretical analysis
To understand the inconsistencies in mode images presented in Fig. 8, the S2 technique was simulated using the calculated modes of the CLING fiber under test. The refractive index profile of the CLING fiber was measured by Interfiber Analysis [14], and is shown in Fig. 9(a). This measured refractive index profile was used to calculate the fundamental and LP11 modes of the CLING fiber using Lumerical Mode Solutions software [15]. The simulated mode profiles shown in Fig. 9(b) and their calculated effective indices of this custom CLING fiber were then used as the input modes to the S2 simulations. Given that (a) the S2 measurement relies heavily on coupling only into the fundamental mode, and (b) coupling strictly into the fundamental mode of the CLING fiber was challenging due to the 50-µm effective core diameter, the fraction of power contained in the various modes was viewed as a critical parameter to understand. Therefore, the simulations were completed for a number of different modal power distributions, with the LP11 mode containing a percentage of the total power ranging from 0.01% to 99.99%.
Fig. 9. (a) Measured refractive index profile of the CLING fiber, and (b) calculated LP01 and LP11 modes.
Figure 10 shows the two input modes of the CLING fiber along with the simulation results for the reconstructed modes at three selected LP11 content levels. The top row shows the results expected to be the LP01 mode, while the bottom row shows the results expected to be the LP11 mode. The expected modes are identical to the input modes. When nearly all of the power is in the fundamental mode, the reconstructed LP01 mode looks identical to the input fundamental mode. As the LP11 mode becomes relatively more intense, the reconstructed LP01 mode elongates and eventually separates into two lobes and looks identical to the LP11 input mode. The lobes of the reconstructed LP11 mode, however, appear smaller and closer together than the lobes of the input LP11 mode. In contrast to the LP01 mode, the output LP11 mode looks identical for each case, and is independent of the fraction of power in the LP11 mode.
Fig. 10. Reconstructed fundamental modes (top) and LP11 modes (bottom) using the S2 simulation of the CLING fiber at different percentages of power in the LP11 mode. The mode images capture a span of 30 µm in each direction.
The mode areas of the fundamental and LP11 modes reconstructed using the S2 method were calculated as a function of power in the LP11 mode, with the results shown in Fig. 11. For the fundamental mode, the reconstructed mode area begins to deviate from that of the input LP01 mode when the percentage of power in the LP11 mode reaches approximately 1%. At this point, the reconstructed mode area begins to increase. The reconstructed LP01 mode area reaches a maximum near 52% power in the LP11 mode, then begins to decrease until it matches the mode area of the input LP11 mode. The reconstructed LP11 mode area is constant regardless of modal power distribution and is always smaller than the input LP11 mode area.
Fig. 11. Reconstructed fundamental (a) and LP11 (b) mode areas as a function of the percentage of power in the LP11 mode using the S2 simulation with the CLING fiber. The simulated data shows the actual results of the simulation, while the expected data represents the mode areas of the input modes.
The simulation was repeated for the SMF-28 fiber. The reconstructed modes are shown in Fig. 12, along with images of the input modes. Similar to the behavior of the CLING fiber modes, the reconstructed fundamental mode elongates and eventually splits into two lobes as the percentage of power contained in the LP11 mode increases. The reconstructed LP11 mode also shows similar behavior to that of the CLING, and is identical for each power distribution: the lobes of the reconstructed LP11 modes are smaller and closer together than the lobes of the input LP11 mode.
Fig. 12. Reconstructed fundamental mode (top) and LP11 mode (bottom) using the S2 simulation of the SMF-28 fiber at different percentages of power in the LP11 mode. The mode images capture a span of 10 µm in each direction.
The reconstructed fundamental and LP11 mode areas for the SMF-28 S2 simulation are shown in Fig. 13. Similar to the CLING fiber, the reconstructed fundamental mode area of the SMF-28 fiber begins to increase after about 1% of the power is contained in the LP11 mode, and reaches a maximum with 31% of the power contained in the LP11 mode. The reconstructed mode area then decreases until it reaches the value of the mode area of the input LP11 mode. Similar to the CLING fiber results, the reconstructed LP11 mode area of SMF-28 remains constant with modal power distribution. However, in this case the simulated mode area is always larger than the mode area of the input mode.
Fig. 13. Reconstructed fundamental (a) and LP11 (b) mode areas as a function of the percentage of power in the LP11 mode using the S2 simulation with SMF-28 fiber. The simulated data shows the actual results of the simulation, while the expected data represents the mode areas of the input modes.
5. Discussion and conclusions
The data resulting from the S2 technique can be explained by examining Eq. (1) and understanding the underlying physics. Most of the caveats of this method result from the fact that the S2 measurement is essentially a heterodyning technique, as can be observed from Eq. (1). First, note that the DC term (when i = j) contains the intensity distribution of all the modes, not just the fundamental mode, and is specifically given by Eq. (3):
The DC term therefore is a weighted average of the mode intensity shapes of all of the modes of the fiber. The data for the reconstructed fundamental mode of the CLING fiber will contain the fundamental mode intensity plus the intensities of all of the much larger modes. This will not only significantly distort the reconstructed mode shape [observable in Fig. 8(a)], but also erroneously report the mode area. Equation (3) also explains why Figs. 10 and 12 show what looks like an LP11 mode as the reconstructed fundamental mode (i.e., as constructed form the DC peak) when the majority of the power is in the LP11 mode. In this case, the contribution from the fundamental mode is negligible, so the sum of the two mode intensities just looks like the LP11 mode. These results underscore the critical nature of launching and retaining the power in the fundamental mode throughout the measurement. These results also call into question any mode area measurement of the reconstructed fundamental mode, which cannot be determined accurately unless the total power in the higher order modes is less than about 1%.
Second, the mixing terms in Eq. (1) (i ≠ j) at a given phase delay contain the spatial envelope of both modes. Therefore, simply processing a side peak in the spectrum is not sufficient to predict the mode shape since it is a spatial product of both modes. Consider the case of the CLING fiber where the LP11 mode is much larger than the LP01 mode. The product of the two amplitudes will always make the reconstructed LP11 mode appear smaller than anticipated. This phenomenon was observed in the simulation results (the lower plots in Fig. 10), and in the experimental measurements shown in Fig. 8(b). In the case of the SMF-28 fiber, the reconstructed LP11 mode area is always larger than the LP11 input mode. Since the LP01 and LP11 modes are nearly the same size, the spatial product of the two modes results in what looks like an LP11 mode with lobes that are closer together. While this makes the simulated visually appear to be smaller, the opposite is in fact true. In the simulated case, the widths of the lobes are larger in both directions, making the calculated size larger.
When there are more than two modes, any given mixing term (side peak) may also contain the spatial envelope of higher order modes that interfere at similar beat frequencies, particularly because the FFT peaks resulting from the OSA spectra must be integrated over a finite range. This explains the extra features in the reconstructed LP11 mode of the CLING fiber in Fig. 8(b), which are caused by the contributions from the beating of higher order modes.
It should be noted that the reconstructed fundamental mode area reaches a maximum value when a certain percentage of power is in the LP11 mode, a point which differs between the CLING and SMF-28 fibers. This occurs because the reconstructed LP01 mode area is based on the sum of the intensities of both the LP01 and LP11 modes. While the LP11 mode is larger than the LP01 mode, the null in the center makes the calculated area smaller than what one might assume based on visual inspection. When the null is filled in by the LP01 mode, the area of the combines modes is therefore larger. Therefore, the reconstructed mode area is maximized when both modes make significant contributions, and begins to decrease as the relative intensity of the fundamental mode decreases. This occurs at different points for different fibers due to the difference in mode overlap. When the mode overlap is smaller, as is the case for the CLING fiber, the maximum reconstructed mode area will occur when a larger fraction of power is contained in the LP11 mode.
Given these findings, the S2 technique cannot in general correctly characterize the mode shape or quantify the mode area of a given fiber. Specifically, the reconstructed LP01 mode profile is highly dependent on launch conditions, while the reconstructed LP11 mode profile is always incorrect. When using the S2 technique, careful consideration must be made to the broadband laser source and the modal content launched into and retained in the fiber under test in order to obtain any useful results. Under these conditions, particularly where the fundamental mode contains more than 99% of the power throughout the fiber, the S2 technique is still useable for identifying higher order fiber modes and their group delays, as stated in the original papers [1,2] and can accurately reconstruct only the fundamental mode.
Funding
Office of Naval Research (N00014-17-1-2534).
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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