1. Introduction

The mode transfer matrix describes the amplitude and phase of the couplings between all the spatial and polarization modes a waveguide supports. It is analogous to the Jones matrix used to describe polarization but extended to support more than a single spatial mode. It completely describes the linear behavior of the waveguide at a given wavelength and maps any field coupled into one end of the waveguide with the corresponding field produced at the other end. Although it is a very basic property of multi-mode systems, it has only been investigated experimentally in a limited way. In the context of Mode Division Multiplexing (MDM) knowledge of at least part of the mode transfer matrix is required to recover the channels at the receiver either through electrical [1–3] or optical [4,5] means. For MDM the mode transfer matrix is also required to characterize the performance of multi-mode devices and components [5–7]. Thus far these techniques have been limited to 6 spatial modes [1,7] and extending them to higher number of modes is constrained by the fact that either the loss scales with the number of spatial modes [6,7] due to the use of beamsplitters and phase plates [6] and/or the use of fiber splitters [6,7], or is constrained by the digital signal processing (DSP) complexity as the number of analogue-to-digital converters scales with the number of modes [1]. Larger numbers of modes have been previously measured [5] using a technique similar to that which will be outlined in this paper, but where only a portion of the mode transfer matrix was measured and particular phase relationships, such as the phase difference between the two polarizations was not of interest and hence not measured.

Another context in which knowledge of the mode transfer matrix is required is imaging through multimode fiber [8–10]. In this case, the transfer matrix is used to reconstruct an image coupled into one end of the fiber, given the pattern observed at the other end, which has undergone distortion due to mode-mixing. Alternatively, the same principle can be applied in the reverse propagation direction to generate a desired image at one end of the fiber, by coupling the required modal pattern into the other end. For imaging applications, the number of measured modes is typically much higher than in telecommunications, however the complete matrix is not measured, either because less modes are measured than actually exist in the fiber and/or because particular phase relationships within the matrix aren’t measured or aren’t meaningful. Typically in imaging, an offset-spot basis is used rather than the eigenmodes of the waveguide. The offset spot basis has an additional disadvantage in that the mode dependent loss (MDL) of this basis increases with the number of modes [11]

In this paper, the amplitude and phase of the mode coupling is measured between all 110 spatial and polarization modes supported by the fiber using spatial light modulators (SLM) at either end of the fiber. This represents the largest complete mode transfer matrix ever measured. The matrix is complete in that it is measured in its entirety, i.e. every element of the matrix has physical significance relative to every other element, both in terms of phase and amplitude. To verify the validity of the measurement, the mode transfer matrix is then inverted and the modal superposition required to produce a desired mode at the receiver are launched at the transmitter using one of the SLMs. In this case, the mode coupling along the length of the fiber serves to transform the launched field into a desired output mode and polarization.

2. Principle of operation

The system used to both measure the mode transfer matrix and generate arbitrary launch conditions is outlined in Fig. 1. An amplified spontaneous emission (ASE) source is filtered (0.5nm at 1545.54nm) and polarized to approximate a high-bandwidth channel. The same technique can also be used with a CW source, however the use of ASE demonstrates this technique is robust against coherence issues associated with linewidth. A polarization controller is then used to ensure the incoming state of polarization is split approximately evenly between the two sides of the SLM corresponding to the horizontal and vertical polarization axes of the system. The SLM launches each mode of the fiber in each polarization one at a time into the fiber under test, a short 2m length of standard OM4 grade 50 μm core multimode fiber. A short length is used so the system can be evaluated in a ‘back-to-back’ configuration where any deviation from ideal performance is likely to be a consequence of the measurement apparatus rather than the fiber under test. The short length also makes the system effectively time invariant such that the transfer matrix can be measured once and used repeatedly over the course of at least several days without recalibration under laboratory conditions. At the receiving end to the right of Fig. 1, the SLM based mode demultiplexer is of the same design except some of the power entering the system is tapped off with a beamsplitter so it can be directed towards a polarization diverse imaging system. This imaging system allows the mode decomposition performed by the receiver using the SLM [5] to be compared with the beam actually observed on the camera.

 

Fig. 1 The mode decomposition and mode generation setup. The asterisk marks the place at which the beam is sampled by the polarization diverse imaging system.

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The mode decomposition process is similar to that discussed previously in [5]. For each mode in each polarization as launched by the transmitter, the receiver SLM displays phase masks for each basis mode and polarization one at a time to measure the amplitude of each mode. The phase of the modes within a polarization is then found by adding together the phase masks for the different basis modes with varying phase shifts and measuring the variation on a power meter. When the phase mask is conjugated with the incoming beam, the power will be maximized. In previous experiments using a similar system [5], the phase difference between the two polarizations was not a parameter of interest and hence was not measured. In this demonstration, the two polarizations are interfered by adjusting the relative phase between the two polarizations on the SLM and observing the interference between the polarizations using an external fiber polarizing beam splitter (PBS) with its polarization axis aligned to 45 degrees with respect to the polarization axis of the SLM receiver.

Given the measurements thus far, the relative phases between all the spatial and polarization modes at the receiver are defined for each of the basis modes launched at the transmitter to within an absolute phase offset. That is, the relative phases between modes along a column of the mode transfer matrix are defined, but the relative phases between columns is not meaningful as no interference has occurred along that axis. For many purposes locking the columns of the matrix to a common phase reference is not required as this has no effect on the measured characteristics of the fiber such as mode dependent loss and is irrelevant for MDM where there is no meaningful phase relationship between independent channels. However in order to know how different launched modes will interfere it is necessary to define all the phases of the mode transfer matrix relative to the same phase reference. To achieve this, as a final step each mode at the transmitter is excited in superposition with a reference mode and the corresponding phase masks for the mode of interest and the reference mode are interfered on the SLM at the receiver to measure their relative phase. Now the phases of the entire matrix are defined relative to the reference mode. Theoretically, the choice of reference mode is arbitrary and could be any basis mode or superposition of modes. However there is some practical advantage to using the fundamental mode as its simplicity makes it straightforward to excite accurately and it is the mode with the least degeneracy in the fiber. The fundamental mode is only degenerate between its two polarizations, in contrast to the other modes of a graded-index multimode fiber which have more complicated degeneracies which in turn lead to more complicated output patterns which are more sensitive to the environment and hence less stable over time. In contrast to Swept-Wavelength Interferometry (SWI) [6,7] which uses an external phase reference arm of an interferometer to define the measured phases, this approach sends the phase reference along the fiber under test itself as the reference mode. Hence all the light for all the measurements travels along the same fibers and as a consequence, the requirements on the coherence of the light source is greatly reduced. There is no need to approximately match path lengths between a phase reference arm and the fiber under test [6] as they are the same fiber in this case. Another advantage of practical significance is that the phases are defined with reference to the plane of the SLMs at the transmitter and receiver for all modes. The reference plane is a part of the mode coupling system itself rather than being located in external splitters, where each mode is fed in using a different input fiber which is likely to not be path length matched and may or may not have the same polarization axis relative to the fiber under test. Although theoretically all such path lengths and polarization rotations could be calibrated out, in practice, doing so for a large number of spatial modes would be very unwieldy and difficult to keep temporally stable in fiber. This system is also convenient in that the apparatus of Fig. 1 is the same regardless of the number of spatial modes being characterized.

3. Mode transfer matrix

The fiber under-test theoretically supports 110 spatial and polarization modes. This consists of 55 modes in each polarization. As the fiber has an approximately parabolic refractive index profile these modes can be organized into approximately degenerate mode-groups, with 10 groups in total where all LP_l,m modes that share the same value of 2m + l have the same propagation constant and hence will mix heavily. The amplitude of the measured mode transfer matrix is shown in Fig. 2(a). The x and y axes run from mode 1 (LP_0,1 Horizontally polarized) to mode 110 (LP_9,1 vertically polarized) and the white lines demarcate the different degenerate groups. Mode coupling occurs mostly between modes within a degenerate group which corresponds to the square white boxes that lie along the diagonal of the matrix in Fig. 2(a).

 

Fig. 2 (a) Amplitude of the mode transfer matrix for all 110 modes. (b) Singular values of that mode transfer matrix representing mode dependent loss.

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Performing the singular value decomposition (SVD) of the measured matrix yields the eigenvalues and the corresponding eigenvectors for each of the 110 orthogonal channels the fiber supports. The eigenvalues, or singular values, of Fig. 2(b) represent the relative loss, sorted in increasing order, for each of these channels represented by the corresponding eigenvector superposition of modes. It can be seen in Fig. 2(b) that there is a steady increase in mode dependent loss as the number of channels is increased until approximately the 90th channel at which point the loss increases sharply. The large loss of these final channels signifies that they are beyond the cutoff of the fiber. Theoretically, a 50 μm core graded-index multimode fiber could support 10 degenerate modes groups (110 modes total), however the 10th group is very close to cutoff even in theory and in practice appears to be beyond cutoff in this particular fiber. The steady accumulation of mode dependent loss for approximately the first 90 channels is likely more a measurement of the quality of the SLM based measurement system than it is an actual measurement of the mode dependent loss of the fiber. Particularly for the heavily-bound modes in the lower mode-groups and given the short length of the fiber, there should be virtually no MDL for these modes. The mode conversion efficiency of the phase masks for each mode are different [5] and when this is taken into account, the coupling efficiency in and out of the fiber remains within +/− 1dB of the theoretical value for the first 100 modes. Although the loss in terms of total power coupled into the fiber remains consistent through the first 100 modes, it can be seen from Fig. 2(b) that the loss in terms of capacity drops off faster. That is, the phase masks are still coupling light into the fiber, but the orthogonality between different launch conditions becomes increasingly difficult to maintain as the structure of the modes becomes more complicated. Higher-order modes place higher demands on the quality of the optics and alignment of the free-space optics of the SLM system of Fig. 1 which makes accurate excitation more difficult in practice. However there is also some mode dependent loss which is accumulated due to phase inaccuracies associated with superimposing the basis phase masks on the SLM. As discussed in previous work [3], the system of Fig. 1 uses a library of 55 basis mode phase masks calculated to accurately excite each of the spatial modes. These masks are interfered on the SLM to measure the relative phases between the modes. When two masks are interfered, numerically the resulting pattern contains both amplitude and phase, however as the SLM can only display phase information, some amplitude information must be discarded in the process. This can result in inaccuracies in the measured phase values of a few degrees. The inaccuracy is generally worse the more dissimilar the modes being superimposed are. The approach taken, consisting of pre-calculating a basis mode library of masks and superimposing them is done for reasons of simplifying computational complexity. However for more accurate results, masks could be calculated directly on-the-fly to generate the desired mode superpositions without the amplitude information loss associated with superimposing masks. In much the same way as the original basis masks of the library are initially calculated.

4. Matrix inversion

To undo the mode coupling which occurs along the length of the fiber, the mode transfer matrix is inverted and multiplied by a vector representing the desired output mode superposition to yield the corresponding vector of mode coefficients that must be launched at the transmitter to generate the desired output. A phase mask is then calculated to generate the desired modal pattern and displayed on the SLM at the transmitter end of the fiber. This phase mask is calculated directly rather than by superposition of masks in the basis library for increased accuracy. That is, mode superpositions are approximated at the receiver using the basis library but not at the transmitter, where all masks are always calculated directly.

In Fig. 3(a) a horizontally polarized Orbital Angular Momentum (OAM) mode of spin + 1 is launched into the fiber using the system on the left of Fig. 1. At the output of the fiber the corresponding intensity pattern is observed on the CCD camera which is consistent with the distribution reconstructed by the amplitude, phase and polarization values measured by the SLM at the receiver. This corresponds to a single column of the mode transfer matrix in Fig. 2(a). It can be seen that even over this relatively short distance, the OAM state has not be maintained at the output. In Fig. 3(b) the mode transfer matrix has been inverted and multiplied by a vector representing a horizontally polarized OAM mode of spin 1. The phase mask required to generate the desired mode superposition is displayed on the SLM at the transmitter and the resulting distribution is observed on the polarization diverse imaging system at the receiver. The modal decomposition performed by the SLM confirms the presence of the spiral wavefront.

 

Fig. 3 Mode distribution at the receiver due to (a) a horizontally polarized OAM 1 launch and (b) due to a launch designed to create an OAM 1 at the receiver.

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More sophisticated examples are illustrated in Fig. 4, where higher-order modes are generated at the output which extend all the way up to the highest-order modes the fiber supports. Figure 4(a) shows the complicated distribution of amplitude, phase and polarization which is required to generate a vertically polarized LP_0,5 at the receiver in theory and Fig. 4(b) is what was observed at the receiver when the mask for the distribution of Fig. 4(a) was programmed onto the surface of the transmitter SLM. LP_0,5 is in the 9th degenerate mode-group which consists of 18 spatial and polarization modes which form the majority of the superposition in Fig. 4(a). Examples of other superpositions are shown in Figs. 4(c)–4(e). Figure 4(c) is the output of the fiber with an excitation at the transmitter designed to generate a horizontally polarized OAM spin 8 mode. The splotchy appearance of the mode is due to one of the constituent LP_8,1 modes, which make up the OAM + 8, having more loss than the other. Figure 4(d) represents a middle-order mode, LP_2,2 vertically polarized and Fig. 4(e) demonstrates how it is possible make different spatial mode patterns exit different ports of the polarizing beamsplitter at the receiver. Specifically a LP_0,4 exits horizontally polarized whilst an LP_4,2 mode exits in the orthogonal polarization.

 

Fig. 4 (a) Theoretical intensity and phase distribution launch required to generate a vertically polarized LP_0,5 mode (b) Corresponding distribution observed on camera. Example for various mode combinations (c) OAM 8 H (d) LP_2,2 V (e) LP_0,4 H, LP_4,2 V

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5. Conclusion

The largest complete mode transfer matrix of a fiber has been measured including 110 modes (55 per polarization). This matrix is then inverted and the validity of the measured values verified optically for the first time.