110 x 110 optical mode transfer matrix inversion

The largest complete mode transfer matrix of a fiber is measured consisting of 110 spatial and polarization modes. This matrix is then inverted and the pattern required to produce a desired output at the receiver are launched at the transmitter. ©2013 Optical Society of America OCIS codes: (060.2350) Fiber optics imaging; (060.2270) Fiber characterization; (070.6120) Spatial light modulators. References and links 1. R. Ryf, S. Randel, N. K. Fontaine, M. Montoliu, E. Burrows, S. Chandrasekhar, A. H. Gnauck, C. Xie, R. Essiambre, P. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Gruner-Nielsen, R. V. Jensen, and R. Lingle, “32-bit/s/Hz spectral efficiency WDM transmission over 177-km few-mode fiber,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper PDP5A.1. 2. C. Koebele, M. Salsi, L. Milord, R. Ryf, C. A. Bolle, P. Sillard, S. Bigo, and G. Charlet, “40km transmission of five mode division multiplexed data streams at 100Gb/s with low MIMO-DSP complexity,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.C.3. 3. X. Chen, A. Li, J. Ye, A. Al Amin, and W. Shieh, “Reception of dual-LP11-mode CO-OFDM signals through few-mode compatible optical add/drop multiplexer,” in National Fiber Optic Engineers Conference, OSA Technical Digest (Optical Society of America, 2012), paper PDP5B.4. 4. N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. Ryf, P. Winzer, L. Buhl, Y. Sun, X. Jiang, and R. Lingle, “Spacedivision multiplexing and all-optical MIMO demultiplexing using a photonic integrated circuit,” in National Fiber Optic Engineers Conference, OSA Technical Digest (Optical Society of America, 2012), paper PDP5B.1. 5. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). 6. N. K. Fontaine, R. Ryf, M. A. Mestre, B. Guan, X. Palou, S. Randel, S. Yi, L. Gruner-Nielsen, R. V. Jensen, and R. Lingle, “Characterization of space-division multiplexing systems using a swept-wavelength interferometer,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW1K.2. 7. N. K. Fontaine and R. Ryf, “Characterization of mode-dependent loss of laser inscribed photonic lanterns for space division multiplexing systems,” in 2013 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching (Optical Society of America, 2013), paper MR2_2. 8. T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express 19(20), 18871–18884 (2011). 9. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012). 10. R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express 20(13), 14321–14337 (2012). 11. R. Ryf, N. K. Fontaine, R. Essiambre, “Spot-based mode coupler for mode-multiplexed transmission in fewmode fiber,” in Photonics Society Summer Topical Meeting Series, 2012 IEEE, 199–200, 9–11 July 2012.


Introduction
The mode transfer matrix describes the couplings between all the spatial and polarization modes a fibre supports. It completely describes the propagation of light within the fibre for a given wavelength and can be used to map any input field to the corresponding output field. In the context of Mode Division Multiplexing (MDM) the mode transfer matrix is of interest for characterizing the modal properties of components 1,2 or in the transmission system itself where knowledge of the coupling between modes is necessary to recover the channels, either electically 3 or optically 2,4 at the receiver. Aside from telecommunications, measurement of part or all of the mode transfer matrix also has wider application to areas of physics and biology requiring modal control, such as imaging and/or focusing through multimode fibres [5][6][7] .
In this paper, the amplitude and phase of the mode coupling is measured between all 110 spatial and polarization modes supported by the fibre representing the largest complete mode transfer matrix ever measured. The measured matrix is then inverted and the patterns required to produce a desired mode at the receiver are launched at the transmitter using a Spatial Light Modulator (SLM). In this case, the mode coupling along the length of the fibre serves to transform the launched field into a desired output mode and polarization. This also represents the first time the complete mode transfer matrix has been measured, inverted and then verified optically by adjusting the launch conditions into the fibre. Partial mode transfer matrices or similar information has also been measured previously in the context of imaging [5][6][7] . Typically the matrix is measured in the basis of offset spots rather than the eigenmodes of the fibre and there are fewer spots than there are modes meaning the entire matrix is not measured. In contrast to these imaging techniques and Swept-Wavelength interferometry (SWI) 1 , the system presented here has no external reference or laser at the receiver 1,5-6 nor does the light ever propagate backwards in the fibre [5][6][7] . Instead the phase reference is sent through the fibre under test itself.

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 and polarized to approximate a high-bandwidth channel. 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 fibre in each polarization one at a time into a short 2m length of standard OM4 grade 50 m core multimode  A short length is used so that the system is effectively time invariant. At the receiving end, 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 2 to be compared with the beam actually observed on the camera.
The mode decomposition process is similar to that discussed previously 2 . 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 similiar system 2 , 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 externally using a fibre interferometer. To define the mode superpositions at the receiver to a common reference, as a final step, the launch mask is superimposed with a reference beam mask consisting of a mixture of modes and the corresponding phase mask at the receiver for the measured mode decomposition is interfered with the corresponding mask for the reference beam as measured at the receiver.

Mode Transfer Matrix
The fibre under-test theoretically supports 110 spatial and polarization modes. This consists of 55 modes in each polarization. As the fibre has an approximately parabolic refractive index profile these modes can be organized into approximately degenerate mode-groups, with 10 groups in total where all LPl,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 axis run from mode 1 (LP0,1 Horizontally polarized) to mode 110 (LP9,1 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). By performing the Singular Value Decomposition (SVD) of the mode transfer matrix it is possible to measure the Mode Dependent Loss (MDL) of the system as a whole, given by the ratio between the largest and smallest singular values. These singular values are shown in Fig. 2(b) normalized to the lowest loss value and sorted by increasing loss. The eigenvalues i and the corresponding eigenvectors need not correspond with the eigenmode basis in which the mode transfer matrix was measured, although there are an equal number of both (110). The mode conversion efficiency of the phase masks for each mode are different by design 2 and when this is taken into account, the coupling efficiency in and out of the fibre remains within +/-1dB of the theoretical value for the first 100 modes. The 10 th mode-group is theoretically very close to cutoff for a standard parabolic refractive index but in practice the losses are very high and it may not be a true bound mode of the fibre. Although the loss in terms of total power coupled into the fibre remains consistent through the first Th.1.C.1.pdf 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 fibre, but the orthogonality between different launch conditions becomes increasingly difficult to maintain as the structure of the modes becomes more complicated.

Matrix Inversion
To undo the mode coupling which occurs along the length of the fibre, 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. In Fig. 3 (a) a horizontally polarized Orbital Angular Momentum (OAM) mode of spin +1 is launched into the fibre using the system on the left of Fig. 1. At the output of the fibre a corresponding intensity pattern is observed on the 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 be 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.
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 fibre supports.

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