Abstract

Orbital angular momentum (OAM) beams allow for increased channel capacity in free-space optical communication. Conventionally, these OAM beams are multiplexed together at a transmitter and then propagated through the atmosphere to a receiver where, due to their orthogonality properties, they are demultiplexed. We propose a technique to demultiplex these OAM-carrying beams by capturing an image of the unique multiplexing intensity pattern and training a convolutional neural network (CNN) as a classifier. This CNN-based demultiplexing method allows for simplicity of operation as alignment is unnecessary, orthogonality constraints are loosened, and costly optical hardware is not required. We test our CNN-based technique against a traditional demultiplexing method, conjugate mode sorting, with various OAM mode sets and levels of simulated atmospheric turbulence in a laboratory setting. Furthermore, we examine our CNN-based technique with respect to added sensor noise, number of photon detections, number of pixels, unknown levels of turbulence, and training set size. Results show that the CNN-based demultiplexing method is able to demultiplex combinatorially multiplexed OAM modes from a fixed set with >99% accuracy for high levels of turbulence—well exceeding the conjugate mode demultiplexing method. We also show that this new method is robust to added sensor noise, number of photon detections, number of pixels, unknown levels of turbulence, and training set size.

Full Article  |  PDF Article
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References

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2015 (1)

O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei-Fei, “ImageNet large scale visual recognition challenge,” Int. J. Comput. Vis. 115, 211–252 (2015).
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2014 (7)

Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. 39, 2845–2848 (2014).
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B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100  Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

W. Nelson, J. P. Palastro, C. C. Davis, and P. Sprangle, “Propagation of Bessel and Airy beams through atmospheric turbulence,” J. Opt. Soc. Am. A 31, 603–609 (2014).
[Crossref]

B. Guan, R. P. Scott, C. Qin, N. K. Fontaine, T. Su, C. Ferrari, M. Cappuzzo, F. Klemens, B. Keller, M. Earnshaw, and S. J. B. Yoo, “Free-space coherent optical communication with orbital angular, momentum multiplexing/demultiplexing using a hybrid 3D photonic integrated circuit,” Opt. Express 22, 145–156 (2014).
[Crossref]

2013 (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

2012 (3)

T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. B. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20, 9396–9402 (2012).
[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

R. L. Nowack, “A tale of two beams: an elementary overview of Gaussian beams and Bessel beams,” Stud. Geophys. Geod. 56, 355–372 (2012).
[Crossref]

2011 (1)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

2010 (1)

2008 (1)

2006 (1)

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44(11), 46–51 (2006).
[Crossref]

2005 (1)

2004 (2)

2003 (1)

D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theory to practice: an overview of MIMO space-time coded wireless systems,” IEEE J. Sel. Areas Commun. 21, 281–302 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref]

2000 (1)

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1992 (4)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

1973 (1)

Ahmed, N.

Akopyan, F.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Allen, L.

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Alvarez-Icaza, R.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Amir, A.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 52.

Anguelov, D.

C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” arXiv:1409.4842 (2014).

Anguita, J. A.

Appuswamy, R.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Arabaci, M.

Arthur, J. V.

P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L. Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy, B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A million spiking-neuron integrated circuit with a scalable communication network and interface,” Science 345, 668–673 (2014).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Bandres, M. A.

Bao, C.

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Berg, A. C.

O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei-Fei, “ImageNet large scale visual recognition challenge,” Int. J. Comput. Vis. 115, 211–252 (2015).
[Crossref]

Berkhout, G. C. G.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
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Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
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Phys. Rev. A (2)

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Proc. Natl. Acad. Sci. USA (1)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143  km,” Proc. Natl. Acad. Sci. USA 113, 13648–13653 (2016).
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Figures (12)

Fig. 1.
Fig. 1.

Simulated u B G ( 5 ) with β = 350 at z = 0 . Phase information is represented by the hue while the energy is represented by the normalized intensity. Colorbar is in radians.

Fig. 2.
Fig. 2.

Effects of simulated multiplexing different OAM modes together for BGB ( β = 350 ) in numerical simulation. Left column is m = { 2,7 } , middle column is m = { 5,7 } , and right column is m = { 2,5 , 7 } . Colorbar is in radians.

Fig. 3.
Fig. 3.

Conjugate mode sorting for an OAM mode set of [ 2,5 , 7 ] and multiplexed signal with modes m = 2 and m = 7 .

Fig. 4.
Fig. 4.

Alexnet architecture. Note that layers 1–5 are convolutional (and include max-pooling, and ReLU) and layers 6–8 are fully connected (and include dropout and ReLU). The smaller black squares represent the receptive field or convolutional filter.

Fig. 5.
Fig. 5.

Photo of laboratory experiment setup.

Fig. 6.
Fig. 6.

Example of two realizations of simulated turbulence screens created by the described methods; scale is in radians.

Fig. 7.
Fig. 7.

Diagram representing the conjugate mode sorting experiment; P = Pinhole and M = Mirror. First, laser light is collimated. Next, a hologram is created to represent the BGB OAM mode-encoded signal with the created random turbulence realization. This hologram is displayed on the first SLM, and after the plane wave interferes with the hologram the beam propagates in free space until reaching the second SLM. Conjugate mode holograms are displayed in serial on the second SLM and the resulting demultiplexed pattern is recorded by the camera.

Fig. 8.
Fig. 8.

Diagram representing the CNN-based mode sorting experiment. First, laser light is collimated. Next, a hologram is created to represent the BGB OAM mode-encoded signal with the created random turbulence realization. This hologram is displayed on the SLM, and after the plane wave interferes with the hologram the beam propagates in free space until it is recorded by the camera.

Fig. 9.
Fig. 9.

Example of experimental data without turbulence for mode set 1. Title for each sub-image is the bit-string (5 digit binary number) and the modes that are active (set of integers in braces). The images have been cropped and a colormap has been applied for visualization purposes.

Fig. 10.
Fig. 10.

Example of reduced spatial image size for the multiplexed BGB with m = { 7,8 , 13 } and D / r 0 = 5 . From left to right, 256 × 256,128 × 128,64 × 64 , and 42 × 42 . Colormap and crop applied for visualization only.

Fig. 11.
Fig. 11.

Example of different quantization levels for the multiplexed BGB with m = { 7,8 , 13 } and D / r 0 = 5 . From left to right, 8-, 7-, and 6-bit. Colormap and crop applied for visualization only.

Fig. 12.
Fig. 12.

Example of added sensor noise for the multiplexed BGB with m = { 7,8 , 13 } and D / r 0 = 5 . From left to right, additive white Gaussian noise=0, 10, 20, 30, and 40. Colormap and crop applied for visualization only.

Tables (11)

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Table 1. Architecture for CNN-Based Demultiplexing Method

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Table 2. Mode Sets Used in Experiment

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Table 3. Hyperparameters for CNN Training

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Table 4. Demultiplexing Accuracy

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Table 5. Demultiplexing BER

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Table 6. Demultiplexing Accuracy for CNN Trained with Only 2 of the 3 Turbulence Levels

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Table 7. Demultiplexing Accuracy for CNN Trained with All 3 Turbulence Levels

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Table 8. Demultiplexing Accuracy with Reduced Spatial Image Sizes for D / r 0 = 15

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Table 9. Demultiplexing Accuracy with Reduced Training Set Size for D / r 0 = 15

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Table 10. Demultiplexing Accuracy with Increased Levels of Quantization for D / r 0 = 15

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Table 11. Demultiplexing Accuracy with Added Sensor Noise for D / r 0 = 15

Equations (18)

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2 u t 2 = c 2 2 u ,
2 U + k 2 U = 0 ,
1 r r ( r U ¯ r ) + 2 U ¯ z 2 + k 2 U ¯ = 0 .
1 r r ( r V r ) + 2 i k V z = 0 .
u B ( m ) ( r , θ , z ) = C B J m ( β r ) exp ( i k z z ) exp ( i m θ ) ,
u B G ( m ) ( r , θ , z ) = C B G w 0 w ( z ) J m ( β r 1 + i z / z r ) × exp [ i ( k β 2 2 k ) z ζ ( z ) + 1 w 2 ( z ) ] × exp [ i k 2 R ( z ) ( r 2 + β 2 z r k 2 ) ] exp ( i m θ ) ,
u B G ( m ) ( r , θ , z = 0 ) = C B G J m ( β r ) exp [ ( r / w 0 ) 2 ] exp ( i m θ ) ,
f ( x ) = a L b L a L 1 b L 1 a 1 b 1 ( x ) ,
Ψ ( κ ) = 0.033 C n 2 ( κ 2 + 1 / L 0 2 ) 11 / 6 exp ( κ 2 / κ 2 ) × ( 1 + 1.082 ( κ / κ ) 0.254 ( κ / κ ) 7 / 6 ) ,
r 0 = ( 0.423 k 2 sec ( α ) Path C n 2 ( z ) d z ) 3 / 5 ,
r 0 = ( 0.423 k 2 Δ z C n 2 ) 3 / 5 .
P = F 1 { Ψ C } ,
S [ x , y ] = exp ( i P [ x , y ] ) j = 1 t u B G ( m j ) [ x , y ] ,
H [ x , y ] = | exp ( i ang ( S [ x , y ] ) ) + exp ( i x ) | 2 .
H ^ [ x , y ] = { π H [ x , y ] > ( | H max | + | H min | ) / 2 0 otherwise .
I ˜ = U x ( D x ( I ) ) .
I ¯ b = Q b ( I ) ,
I ¯ x = Q 8 ( I + N x ) .

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