The function s(x,y,t) is defined by normalizing I(x,y,t) by the amplitude and field of the reference mode:

The modal weight trace, S(t), is a one dimensional function describing modal power as a function of delay. The C² term (second term) of this function is a sum of signals in time that inform the relative power of propagating modes in the FUT. By determining the power in the desired mode relative to the power in undesired modes, the purity of mono-mode operation can be measured.

The functional form of I(x,y,t) [Eq. (2)] can accurately describe S² imaging if we block the reference arm (α_r → 0). The C² term thus disappears, and the background term describes the interferometric signals obtained during the S² measurement. We can use this formulation to probe the major differences between S² and C² imaging.

First, the interference signal measured is indecipherable if all the modes in the fiber have similar power. It is necessary for one mode to be dominant so that all other modes can be measured against it. In the limit of this dominant-mode approximation, the DC peak describes the power of the reference mode, and each peak in the time domain is the interference between the reference and other guided modes. In C², by contrast, there is no need for a dominant mode in the FUT because the reference is external to the waveguide.

Secondly, a key step in deriving s(x,y,t) is normalization by the amplitude and field of the reference mode. Mathematically, this is trivial, but in practice this requires that the reference field never go to zero – else S(t) in these regions (which carries no information as there is no reference light there to interfere) will diverge. In S² this requires that the extent of the fundamental mode acting as a reference approximately match that of higher order modes (HOMs) in the fiber – a problem for measuring fibers employing HOM delocalization schemes [20]. If a dominant HOM is desired instead, post-processing must be used to remove sections of the signal. In C², one can independently tailor the reference mode with respect to the FUT modes, and so it is simple to ensure a well behaved reference mode.

Finally, C² allows one to alter the shape and delay of signal peaks in the modal weight trace. Significant GVD will cause broadening of the peaks corresponding to each guided mode in the FUT (see the following section for further details). In C², one can match the GVD times length value for a mode in the FUT by changing the length of the reference fiber, or by post-processing the data electronically – thus eliminating broadening of the peaks. In addition, the free-space delay between the arms can be adjusted to move the delay at which the peaks will manifest. In our experiments we often use this degree of flexibility to ensure that the C² signals do not overlap with the DC peak, any significant background peaks, or any spurious peaks due to back reflections in the system. In S², the reference and other modes co-propagate, and thus one cannot tune the relative delay or compensate for GVD.

3. The cross-correlation function

The cross-correlation function, defined by Eq. (3) above, describes the delay and width of signals in the time domain. In the limit where the GVD (and all higher terms) times length difference between the arms of the interferometer is negligible, the cross-correlation function is simply the inverse Fourier transform of the input spectrum shifted to some relative delay. The following section will describe properties of the cross-correlation function and how they can be tailored through apodization and GVD matching to adjust temporal resolution.

3.1 Apodization and temporal resolution

The spectrum of our source is a narrow line-width laser, swept across a fixed bandwidth (ΔΩ). Outside of this bandwidth, the power is zero – therefore we can describe the behavior of the source as a rect function: $\tilde{A}(\omega )=A_{0}rect(\Delta \omega /\Delta \Omega )$. In the dispersionless case, the rect function intuitively transforms to a sinc function. The sinc function has a narrow peak when the argument is zero; however, it also exhibits side-lobes which may be problematic for measuring small adjacent signals. Therefore, it is often useful to apodize the input spectrum to suppress the side-lobes. Apodization of the input spectrum in fC² is completely analogous to apodization of Bragg gratings and long period gratings in optical fibers [21]. In this work, we consider apodization using super-Gaussian functions, however, more complex apodization functions (also referred to as windowing functions in signal processing literature) may be envisaged to analyze the signal, post-measurement, in the digital domain [22]. Indeed one advantage of this technique is that we can use a multitude of digital signal processing tools developed for image processing and communications without sacrificing the measurement speed since it is post-measurement. The form of the super-Gaussian apodization function is described by the equation below:

In Fig. 1(a), we can see a variety of apodized input spectra including a 12^th (orange), 6^th (yellow) and 4^th (green) order super-Gaussian as well as a normal Gaussian function (N = 2, blue), compared against the un-apodized case (rect, red). In Fig. 1(b), we can see |C(t)|² resultant from each of these apodized input spectra for the case where GVD is neglected. In the un-apodized case (rect, red) there are strong characteristic sinc side-lobes. As the order of the apodization function decreases, the gradient of the spectrum in the frequency domain decreases as well, leading to suppression of the side-lobes in the time domain. In the Gaussian case (blue), the side-lobes are entirely suppressed.

 

Fig. 1 (a) Various apodized input spectra including: Gaussian (blue), 4^th (green), 6^th (yellow), and 12^th (orange) order super-Gaussians, and the un-apodized case (red); (b) |C(t)|² for all cases (offset for clarity); (c) the envelopes of the functions in (b) shown for t>0; the −15 dB point is marked with a dashed line – above the line the un-apodized case is narrowest, below it the Gaussian-apodized function is narrowest; (d) Group delay resolution as a function of MPI between a dominant and parasitic C² signal for each apodization function.

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Traditionally, resolution is defined by the full-width-half-maximum (FWHM) of a trace, and this definition suffices if the signals’ relative powers (modal weights) are similar. Since modal discrimination systems (such as this one) are expected to discern trace amounts of undesired modes as well, a more involved definition of resolution, as a function of MPI, is needed. A worst-case estimation can be obtained by considering the envelope of |C(t)|² [Fig. 1(c)]. Given that a parasitic signal must be above this envelope to be resolved, the envelope gives the minimum detectable MPI level for parasitic signals as a function of relative delay. Therefore, the transpose of the envelope gives the group delay resolution [Fig. 1(d)] as a function of MPI.

In Fig. 1(c), at approximately –15 dB down from the peak, we can see that the envelopes for each function cross each other, regardless of the strength of apodization. Correspondingly, there is a crossing point in group delay resolution [Fig. 1(d)]. Below the crossing point, the envelope of the Gaussian-apodized signal is narrowest and accordingly exhibits improved group delay resolution relative to the un-apodized case. Thus, a trade-off becomes evident – if one is interested in resolving parasitic signals for which the MPI with respect to some desired signal is > –15 dB, the un-apodized case (red, rect function) will have the best resolution. For MPI < –15 dB, Gaussian apodization will have the best resolution. Super-Gaussian apodization of any order is effectively a compromise between these two extremes.

For our experiments, we are interested in MPIs of approximately –14 dB, therefore we do not apodize. It is noteworthy, however, that apodization is a post-processing technique – therefore one could potentially process the same data set with different apodization functions to resolve parasitic modes of various MPI values.

3.2 Effects of group velocity dispersion

In this section, we will consider the effects of dispersion on the cross-correlation function. Here we include only the second-order dispersion (GVD), ignoring all higher terms [k > 2 in Eq. (6)]. In Fig. 2, we show |C(t)|² in the presence of increasing GVD times length difference between the arms of the interferometer for the un-apodized case (a), and the Gaussian-apodized case (b). In the un-apodized case, the form of the cross-correlation function [Eq. (3)] is the integral of a quadratic phase over definite bounds, which can be described analytically using the imaginary error function. In the limit of negligible dispersion, the solution is a simple sinc function. As dispersion increases, however, the side-lobes grow significantly as the peak decreases.

 

Fig. 2 (a) The effect of GVD times length mismatch on |C(t)|² with no apodization; (b) the effect of GVD times length mismatch on |C(t)|² with Gaussian apodization; note that the legends for (a) and (b) describe different GVD times length values.

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For GVD times length differences of 0.3 ps/nm or less, the effects of dispersion are negligible. In our experiments, dispersion is matched such that GVD times length is never greater than 0.22 ps/nm – therefore we do expect dispersive broadening. Alternatively, in the post-processing domain, one can fit C(t) and therefore determine the GVD times length difference for each mode in the fiber. Once the dispersion is known, it is possible to compensate for it as discussed in [16] and [19].

In the Gaussian-apodized case, GVD results in a simple broadening of |C(t)|². It is important to note, however, that significantly larger GVD times length values are required to broaden the signal [note that the legends for Fig. 2(a) and (b) are not the same]. This is because the effects of dispersion are coupled to the bandwidth of the source, and in the apodized case, the bandwidth is effectively reduced by a factor of 2.1 given the definition of ΔΩ_N above.

4. Experimental implementation of fC²

We implement fC² using the setup shown in Fig. 3(a).Our source is a tunable external cavity diode laser (ECL, Thorlabs TLK-L1050M) with ~5 mW output power, and 100 kHz linewidth. The light from the laser is split into the reference and FUT arms of the interferometer using a 3 dB coupler. The second input port of the coupler is angle-cleaved and immersed in index-matching gel to mitigate spurious reflections. The FUT arm (red) incorporates a spatial light modulator (SLM, Hammamatsu X10468-07) which allows excitation of select higher order modes in the FUT. The reference arm (green) comprises a single mode fiber (Corning HI1060, L = 5 m) to nominally match the optical delay of the free-space and fiber paths of the FUT arm. The reference mode is highly magnified by a 150 × objective lens (Nikon BD Plan Apo 150). Outside the fiber, the reference mode transmits through the beam splitter, is retro-reflected by the mirror on the delay stage, and is finally reflected by the beam splitter onto a CMOS camera (Thorlabs DCC1645C, max. frame rate of 140 frames per second). The relative delay of the reference (τ) can be adjusted by moving the delay stage with respect to the beam splitter. The FUT beam is reflected by one beam splitter (so that part of the light can be used for other experiments) and then transmitted through a second beam splitter such that it is collinear with the reference beam. Both beams are near field imaged onto the CMOS camera, where they interfere. Polarization optics in both arms are used to maximize the interference on the camera. We tap off approximately 1% of the light from the laser as a monitor (orange path), which is spatially decorrelated from the other arms, to normalize any fluctuations in laser power as it sweeps (a power meter would be sufficient for this purpose, we however found it convenient to use the camera as it was already interfaced). A sample camera frame is shown in Fig. 3(b).

 

Fig. 3 (a) Experimental setup for fC² imaging facilitated by a tunable external cavity laser split into three arms: monitor (orange), reference (green), and fiber under test (FUT, red), all combined on a CMOS camera; (b) Sample image frame showing the monitor beam (orange) spatially decorrelated from the interfering reference (green) and fiber under test (FUT, red) beams.

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For each measurement, the ECL sweeps from 1046 nm – 1040 nm at a constant velocity while camera frames are captured. Therefore the wavelength of the laser during its sweep is encoded in the time at which the frame is captured. The first nanometer of the laser’s sweep is ignored to remove registration errors due to the acceleration of the ECL’s motor. After this buffer, 99 frames are collected over a bandwidth of 5 nm (1045 – 1040 nm). The laser’s wavelength sweep velocity is 6.3 nm/s, and therefore the measurement (including the buffer) can be completed in 950 ms. Because the wavelength of the source is sweeping during the exposure of each frame (exposure time = 6.7 ms), the effective bandwidth of the source is greatly increased (to 42 pm = 11.6 GHz) with respect to its 100 kHz native bandwidth.

The camera frames are collected by a LabVIEW program which normalizes output power fluctuations of the laser (via the power from the monitor), normalizes the interference signal by the reference field (via a stored image of the reference beam with the FUT beam blocked), and computes S(t) over the region of interference (141 × 141 pixels). Reconstructions of the modal intensity can be generated by selecting a given delay value in the S(t) trace. For each scan, the processing time is roughly 2.5 seconds. Conceivably, this processing time could be greatly reduced, indeed, potentially made negligible compared to the 950 ms measurement time with optimized software and coding algorithms (as frequency-domain OCT techniques can regularly process more complex image stacks at MHz rates).

5. Characterization of resolution, sampling, and signal to noise ratio

To determine the temporal resolution of our system, we must consider the form of the cross-correlation function [Eq. (3)]. We expect the narrowest peaks for the case where the GVD is matched between the arms of the interferometer. In this case, the cross-correlation is in the form of a sinc function. As a result, we find that the minimum distance between two adjacent, resolvable features in the modal weight trace, S(t), is the full-width half-max (FWHM) of the sinc² function. This leads to an inverse relationship between temporal resolution, t_FWHM, and frequency bandwidth ΔΩ (or equivalently Δλ) of the source:

It is also necessary to consider sampling in the time domain in addition to resolution. Using a discrete Fourier transform, the sampling is described by δt = 2π/ΔΩ, yielding a maximum temporal resolution of 720 fs. Thus, in this context, time-domain C² offers better resolution than the frequency-domain version we discuss here, because sampling in the time domain is determined by the increment with which the delay stage is moved rather than fixed by a transform relationship. Over-sampling in this manner, of course, leads to longer measurement time which is contrary to the goals of these experiments.

An interesting consequence of the sampling in the un-apodized case is that the sampling points (described by δt above) are coincident with the nulls of the sinc² function. The DC peak is necessarily aligned with the sampling array (it must be centered at t = 0 by definition), thus the data points, rather than following the envelope of the sinc² function, quickly approach the noise floor. As a result, we find that the DC peak in our measurements can be nearly 7 times narrower at the –25 dB point than the envelope predicts [see Fig. 1(c) in section 3.1]. Aliasing can occur with C² peaks as well, but is less dramatic given that the peak does not necessarily align with the sampling array, and any unmatched GVD between the arms will decrease the visibility of the nulls [see Fig. 2(a)]. In the sensitivity measurements described below, the C² peak is approximately 1.5 times narrower than the worst-case described by the envelope of |C(t)|².

We determine the maximum signal to noise ratio (SNR) of our system by measuring fC² traces while employing a single-mode fiber (L = 1.2 m) as the FUT. Because only a single mode can propagate in the FUT for this case, we expect a DC peak, a single C² peak and any other signal would be considered noise [as per the integration of S(x,y,t) in Eq. (11)]. We measure the tradeoff in the SNR of measured S(t) traces with the measurement speed, since the laser scanning speed could potentially affect its performance. We took care to ensure that the effective bandwidth of the source was the same for each measurement time. We measured five S(t) traces for each measurement time [as an example the average of the five 950 ms measurements is shown in Fig. 4(a)]. We calculate the average value of the noise floor in these traces by ignoring the DC peak (blue shaded region) and the C² peak (red shaded region) and averaging all other points (green shaded region). The extent of the C² and DC peaks is defined by where they cross –40 dB (approximately the bottom of the noise floor). SNR is defined as the ratio of the peak of the C² signal (η) to the average value of the noise floor.

 

Fig. 4 (a) Average of 5 S(t) traces, each measured in 950 ms, used for the calculation of signal to noise ratio (SNR); blue-shaded region corresponds to DC, red-shaded region is anticipated C² peak, green-shaded region is noise; (b) Maximum SNR for our system as a function of measurement time – all subsequent measurements are performed with a speed of 950 ms/measurement.

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SNR as a function of measurement time is shown in Fig. 4(b). There is a general degradation in SNR as the measurements get faster, though the standard deviation in measured SNR (shown with the error bars) does not significantly diverge. This is due to the emergence of spurious peaks in S(t) [occurring at 10 ps and 30 ps for the 950 ms measurements shown in Fig. 4(a)]. We believe that these peaks are from unmitigated back reflections in the system or some artifact of the ECL. We note that at 950 ms per measurement, SNR is increased with respect to neighboring measurement times. We areunsure of the cause of this non-monotonic deviation, but since our objective is to operate with as a fast measurement time as possible, we operate at this local maximum.

For substantially longer measurement times, it becomes difficult to stabilize the phase between the interfering arms. If the relative phase slips due to sporadic changes in the optical path of one of the arms, the resulting S(t) trace will be corrupted. This respresents a potential advantage of the S² modality, since a single-arm, in-fiber interferometer is expected to be less perturbed by environmental fluctuations. The dynamic range of the camera used for these experiments (8 bits) was a general limitation for the SNR. Commercial cameras (which we did not have access to) with increased dynamic range and suitable frame rates could allow for high-speed measurements with SNR approaching that of traditional OCT (> 90 dB).

6. Characterization of multi-mode fiber

In our experiments, we used 1.13 meters of double-clad fiber fabricated by Nufern, Inc as the FUT [Fig. 5(a)]. The fiber comprises three regions: core, inner cladding and outer cladding [marked I, II, and II in Fig. 5(a)]. The core is 18 μm in diameter, and doped with phosphorus, germanium, and fluorine, resulting in a Δn of 1 × 10⁻³ with respect to silica. The inner cladding is a pure silica region 50 μm in diameter. The outer cladding is 96 μm in diameter and down-doped with fluorine to reach a Δn of −0.021. Higher order modes are guided with the bulk of their energy residing in the inner cladding. At the wavelengths for this experiment, the highest LP_0,m mode guided is LP_0,12. Further information on this fiber can be found in [3].

 

Fig. 5 (a) Facet image of the double-clad FUT used for multi-mode experiments with the core (I), inner cladding (II), and outer-cladding (III) marked; (b) S(t) trace for a mixture of modes excited in the test fiber (output of the fiber under this excitation condition inset in trace), images (i) through (vi) are points of interest in the S(t) trace reconstructed with gamma correction for visibility; (c) Comparison of simulated and measured relative delays for the excited modes in 1.13 meters of test fiber.

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6.1 Simultaneous reconstruction of multiple guided modes

We begin by intentionally exciting an ensemble of LP_0,m modes (see the setup diagram in Fig. 1). This was achieved by removing all phase patterns from the SLM such that it acts like a mirror. The light from the ECL (λ = 1046 nm) is coupled into the FUT with a spot size that is roughly the same diameter as the inner-cladding. As a result, light is coupled to a mélange of LP_0,m modes (slight tilts allow for coupling to higher order angular modes as well). The mode image for this ensemble can be seen inset into the S(t) trace in Fig. 5(b). We measure the modal weight trace [see Eq. (11)] at the output of the fiber [Fig. 5(b)] by performing 950-ms fC² measurements, as described in section 3.

Figure 5(b) (i) – (vi) show the reconstruction of each peak in S(t) which correspond to the LP_0,4 through LP_0,9 modes respectively. Figure 5(c) compares the simulated relative delays for each mode, calculated using an in-house mode solver, with the measured values. The excellent match between the two, as well as the qualitative agreement between the simulated and reconstructed mode intensity profiles provides a high degree of confidence in the system’s capability to accurately identify propagating modes in sub-second time scales. In addition, previous C² demonstrations [16] have already shown that mode content may be accurately measured down to the noise floor of the system.

6.2 Mono-mode operation

With an SLM (see setup in Fig. 3), we have previously shown [23] that one can construct a binary phase plate to excite LP_0,m modes using the reverse of the technique described in [24]. In the following experiment, we use such a phase plate to preferentially couple into the LP_0,5 mode of our test fiber. The purity of this excitation is dependent on tailoring the phase-plate, but more importantly, it is crucial to optimize the alignment of the free-space beam entering the fiber.

We make use of near-real-time feedback from fC² measurements to finely tune the lateral alignment of the FUT with respect to the coupling lens [schematic in Fig. 6(a)]. As we translate the fiber, we observe a change in the distribution of modes that are excited at the input of the FUT. Figure 6(b) shows a series of S(t) traces as a function of offset from the optimized coupling position, where the red shaded portions correspond to the temporal delay where the peak of our desired LP_0,5 mode is expected (insets show the raw output images for each of these launch conditions).

 

Fig. 6 (a); (b) S(t) traces as a function of the alignment of the FUT with respect to the coupling lens; the red shaded portion of each trace corresponds to signal in the LP_0,5 mode; (c) Reconstruction of the dominant parasitic mode LP_1,4; (c) Reconstruction of the desired LP_0,5 mode for worst case alignment; significant power is coupled in to the LP_2,4 mode which is near-degenerate in group delay with the LP_0,5 mode – thus the reconstruction at this delay value (shown above the −2 μm trace) is a coherent superposition of the two.

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When coupling is not optimized, we excite the LP_1,4 mode [reconstructed for X = ± 1 μm case in Fig. 6(c)] which manifests as a peak at a relative delay of 1.45 ps from the desired LP_0,5 peak [and again, as in the experiment illustrated in Fig. 5(c), this relative delay value matches very well with the 1.88 ps value obtained from fiber waveguide simulations]. This is expected, since lateral misalignments are well-known to preferentially couple light into anti-symmetric (LP_1,m) fiber modes. When the alignment deviates further (X = ± 2 μm) from the optimum condition, the strength of the LP_1,4 mode does not overtake that of the LP_0,5 mode as one might expect. Instead, the shape of the reconstructed image of the main peak (in the red shaded region) becomes distorted [as shown in Fig. 6(d)]. In this fiber, the group velocity of the LP_2,4 and LP_0,5 mode are nearly degenerate. For a short length of test fiber their peaks will coincide in the S(t) trace. As a result, if there is significant power in both modes, the reconstruction at this delay point will be a coherent superposition of the two. Note that thisillustrates the power of this technique (similar to time domain C² and S²) – the ability of spatially reconstructing the field profile provides a good guess as to whether mode coupling exists or not, even when the group delays are too similar to be distinguished – the reconstructions arising from mode-coupled states would not correspond to reasonable theoretically simulated mode profiles of the fiber [as is the case in Fig. 6(d)].

Coupling is optimized by adjusting the alignment until the parasitic modes are suppressed as much as possible. Here we find that the system is capable of exciting a single mode with > 13 dB (95%) purity (purity is defined as the relative power between the peak of the desired mode and the largest parasitic mode). Since the fC² setup is capable of discerning MPI down to –25 dB (as per the SNR, described in section 4), the 13-dB value here points to the inadequacy of the SLM-based coupling setup rather than a limitation of the measurement setup or the fiber (using fiber gratings, we have previously shown [3] that greater than 30 dB mode purity excitation is possible in similar fibers). Free space, SLM-based means of mode conversion would be a subject for future investigations, aided, in no small part, by the rapid mode-content characterization of this fC² setup.

7. Summary and conclusions

We have demonstrated a measurement methodology for quickly and accurately measuring the modal content of multi-mode fibers. This system represents a marked improvement in speed (10 minutes → 1 second) over previous demonstrations of C² imaging, by moving from the time to frequency domain – currently we are capable of conducting measurements in 950 ms. Full theoretical and experimental analysis of the system shows that we are capable of a maximum group delay resolution of 0.72 ps, and maximum signal to noise ratio of 25 dB. Experiments using a multi-mode test fiber demonstrate the system’s ability to reconstruct and identify propagating modes, and measure mono-modedness of operation as adjustments are made in near-real-time.

We expect that further increase in speed is indeed possible. Certainly the processing time for each measurement (currently 2.5 seconds) could be drastically reduced with improved software and coding algorithms. Through the incorporation of a faster sweep rate laser and a higher frame rate camera, we anticipate that the speed for each measurement could also be increased. With improvements in these areas, it may indeed be possible to demonstrate truly real-time modal purity analysis in multi-mode fibers.