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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 1 — Jan. 13, 2014
  • pp: 96–101
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110x110 optical mode transfer matrix inversion

Joel Carpenter, Benjamin J. Eggleton, and Jochen Schröder  »View Author Affiliations


Optics Express, Vol. 22, Issue 1, pp. 96-101 (2014)
http://dx.doi.org/10.1364/OE.22.000096


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Abstract

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

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

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.

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.

] or optical [4

4. N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. Ryf, P. Winzer, L. Buhl, Y. Sun, X. Jiang, and R. Lingle, “Space-division 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

5. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]

] means. For MDM the mode transfer matrix is also required to characterize the performance of multi-mode devices and components [5

5. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]

7

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.

]. Thus far these techniques have been limited to 6 spatial modes [1

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.

,7

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.

] 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

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. [CrossRef]

,7

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.

] due to the use of beamsplitters and phase plates [6

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. [CrossRef]

] and/or the use of fiber splitters [6

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. [CrossRef]

,7

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.

], 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

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.

]. Larger numbers of modes have been previously measured [5

5. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]

] 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.

2. Principle of operation

The system used to both measure the mode transfer matrix and generate arbitrary launch conditions is outlined in Fig. 1.
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.
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

5. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]

] to be compared with the beam actually observed on the camera.

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

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. [CrossRef]

,7

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.

] 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

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. [CrossRef]

] 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 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).
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.
The x and y axes 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).

4. Matrix inversion

Fig. 4 (a) Theoretical intensity and phase distribution launch required to generate a vertically polarized LP0,5 mode (b) Corresponding distribution observed on camera. Example for various mode combinations (c) OAM 8 H (d) LP2,2 V (e) LP0,4 H, LP4,2 V
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 LP0,5 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. LP0,5 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 LP8,1 modes, which make up the OAM + 8, having more loss than the other. Figure 4(d) represents a middle-order mode, LP2,2 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 LP0,4 exits horizontally polarized whilst an LP4,2 mode exits in the orthogonal polarization.

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.

Acknowledgments

We acknowledge the Linkage (LP120100661), Laureate Fellowship (FL120100029), Centre of Excellence (CUDOS, CE110001018), and DECRA (DE120101329) programs of the Australian Research Council.

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. [CrossRef]

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, “Space-division 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). [CrossRef]

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. [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

11.

R. Ryf, N. K. Fontaine, R. Essiambre, “Spot-based mode coupler for mode-multiplexed transmission in few-mode fiber,” in Photonics Society Summer Topical Meeting Series, 2012 IEEE, 199–200, 9–11 July 2012.

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2350) Fiber optics and optical communications : Fiber optics imaging
(070.6120) Fourier optics and signal processing : Spatial light modulators

ToC Category:
Subsystems for Optical Networks and Datacomms

History
Original Manuscript: October 7, 2013
Revised Manuscript: November 19, 2013
Manuscript Accepted: November 19, 2013
Published: December 23, 2013

Virtual Issues
European Conference and Exhibition on Optical Communication (2013) Optics Express

Citation
Joel Carpenter, Benjamin J. Eggleton, and Jochen Schröder, "110x110 optical mode transfer matrix inversion," Opt. Express 22, 96-101 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-1-96


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References

  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. [CrossRef]
  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, “Space-division 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, T. D. Wilkinson, “Degenerate mode-group division multiplexing,” J. Lightwave Technol. 30(24), 3946–3952 (2012). [CrossRef]
  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. [CrossRef]
  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, 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). [CrossRef] [PubMed]
  9. I. N. Papadopoulos, S. Farahi, C. Moser, D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012). [CrossRef] [PubMed]
  10. R. N. Mahalati, D. Askarov, J. P. Wilde, 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). [CrossRef] [PubMed]
  11. R. Ryf, N. K. Fontaine, R. Essiambre, “Spot-based mode coupler for mode-multiplexed transmission in few-mode fiber,” in Photonics Society Summer Topical Meeting Series, 2012 IEEE, 199–200, 9–11 July 2012.

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