Abstract

Optical aspects of space-division multiplexing with orthogonal modes of coherent light were considered in theory and experiments with the coherent optical correlator. We resorted to the mathematical tool of generating functions and technologies of diffractive optical elements to implement complex spatial filters matched to rotationally symmetrical transverse modes. Successful multiplexing and demultiplexing in free-space transmission of low-frequency temporally modulated signals through different spatial modes was demonstrated. Experimental results show low cross talk between different mode channels and feasibility of further applications in multimode fiber optical communication data links.

© 2013 Optical Society of America

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2012 (3)

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 138, 1–9 (2012).

S. Shwartz, M. A. Golub, and S. Ruschin, “Generating function approach for creation of coherent multimode beams by diffractive optics,” J. Mod. Opt. 59, 83–89 (2012).
[CrossRef]

D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012).
[CrossRef]

2011 (5)

2010 (2)

2009 (1)

2008 (3)

2007 (1)

2006 (1)

M. R. Duparre, B. Luedge, and S. Schroeter, “Etalons for pure and composite transversal modes,” Proc. SPIE 6101, 61011C (2006).
[CrossRef]

2005 (1)

2001 (3)

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intracavity phase element,” IEEE J. Quant. Electron. 37, 80–86 (2001).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

2000 (2)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

U. W. Krackhardt, R. Klug, and K.-H. Brenner, “Broadband parallel-fiber optical link for short-distance interconnection with multimode fibers,” Appl. Opt. 39, 690–697 (2000).
[CrossRef]

1994 (1)

1982 (2)

S. Berdagué and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. 21, 1950–1955 (1982).
[CrossRef]

M. A. Golub, A. M. Prokhorov, I. N. Sissakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Quantum Electron. 12, 1208–1209 (1982).
[CrossRef]

1981 (1)

U. Levy, H. Kobrinsky, and A. A. Friesem, “Angular multiplexing for multichannel communication in a single fiber,” IEEE J. Quantum Electron 17, 2215–2224 (1981).
[CrossRef]

1979 (1)

Ahmed, N.

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 138, 1–9 (2012).

Anguita, J. A.

Arabaci, M. T.

Barros, D. J. F.

Berdagué, S.

Biswas, R.

S. H. Murshid, A. Chakravarty, and R. Biswas, “Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers,” Opt. Laser Technol. 43, 430–436 (2011).
[CrossRef]

Boffil, P.

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Bolle, C. A.

Brenner, K.-H.

Chakravarty, A.

S. H. Murshid, A. Chakravarty, and R. Biswas, “Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers,” Opt. Laser Technol. 43, 430–436 (2011).
[CrossRef]

Chiu, Y.-J.

Davidson, N.

M. A. Golub, L. Shimshi, N. Davidson, and A. A. Friesem, “Mode-matched phase diffractive optical element for detecting laser modes with spiral phases,” Appl. Opt. 46, 7823–7828 (2007).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Djordjevic, I. B.

Dolinar, S.

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 138, 1–9 (2012).

Dubois, F.

Duparre, M. R.

M. R. Duparre, B. Luedge, and S. Schroeter, “Etalons for pure and composite transversal modes,” Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Duparré, M.

Emplit, P.

Essiambre, R.-J.

Facq, P.

Fan, S.

Fazal, I. M.

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 138, 1–9 (2012).

Flamm, D.

Forbes, A.

Friesem, A. A.

M. A. Golub, L. Shimshi, N. Davidson, and A. A. Friesem, “Mode-matched phase diffractive optical element for detecting laser modes with spiral phases,” Appl. Opt. 46, 7823–7828 (2007).
[CrossRef]

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

U. Levy, H. Kobrinsky, and A. A. Friesem, “Angular multiplexing for multichannel communication in a single fiber,” IEEE J. Quantum Electron 17, 2215–2224 (1981).
[CrossRef]

Gatto, A.

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Gnauck, A. H.

Golub, M. A.

S. Shwartz, M. A. Golub, and S. Ruschin, “Generating function approach for creation of coherent multimode beams by diffractive optics,” J. Mod. Opt. 59, 83–89 (2012).
[CrossRef]

M. A. Golub, L. Shimshi, N. Davidson, and A. A. Friesem, “Mode-matched phase diffractive optical element for detecting laser modes with spiral phases,” Appl. Opt. 46, 7823–7828 (2007).
[CrossRef]

M. A. Golub, A. M. Prokhorov, I. N. Sissakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Quantum Electron. 12, 1208–1209 (1982).
[CrossRef]

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC, 1994).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1998).

Grossman, B.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40, 1030–1036 (2008).
[CrossRef]

Hasman, E.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Huang, H.

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 138, 1–9 (2012).

Hugon, O.

Ip, E.

Kahn, J. M.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Klug, R.

Kobrinsky, H.

U. Levy, H. Kobrinsky, and A. A. Friesem, “Angular multiplexing for multichannel communication in a single fiber,” IEEE J. Quantum Electron 17, 2215–2224 (1981).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Krackhardt, U. W.

Kurihara, K.

Laakonen, P.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Lau, A. P. T.

Lee, W. H.

Leger, J.

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intracavity phase element,” IEEE J. Quant. Electron. 37, 80–86 (2001).
[CrossRef]

Levy, U.

U. Levy, H. Kobrinsky, and A. A. Friesem, “Angular multiplexing for multichannel communication in a single fiber,” IEEE J. Quantum Electron 17, 2215–2224 (1981).
[CrossRef]

Lingle, R.

Liou, J.-H.

Luedge, B.

M. R. Duparre, B. Luedge, and S. Schroeter, “Etalons for pure and composite transversal modes,” Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Makki, S.

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intracavity phase element,” IEEE J. Quant. Electron. 37, 80–86 (2001).
[CrossRef]

Mao, W.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides(Academic, 1974).

Martelli, P.

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Martinelli, M.

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

McCurdy, A.

Murshid, S.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40, 1030–1036 (2008).
[CrossRef]

Murshid, S. H.

S. H. Murshid, A. Chakravarty, and R. Biswas, “Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers,” Opt. Laser Technol. 43, 430–436 (2011).
[CrossRef]

Naidoo, D.

Narakorn, P.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40, 1030–1036 (2008).
[CrossRef]

Neifeld, M. A.

Oron, R.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Otomo, A.

Panicker, R. A.

Peckham, D. W.

Prokhorov, A. M.

M. A. Golub, A. M. Prokhorov, I. N. Sissakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Quantum Electron. 12, 1208–1209 (1982).
[CrossRef]

Randel, S.

Ren, Y.

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 138, 1–9 (2012).

Ruschin, S.

S. Shwartz, M. A. Golub, and S. Ruschin, “Generating function approach for creation of coherent multimode beams by diffractive optics,” J. Mod. Opt. 59, 83–89 (2012).
[CrossRef]

Ryf, R.

Saito, S.

Schroeter, S.

M. R. Duparre, B. Luedge, and S. Schroeter, “Etalons for pure and composite transversal modes,” Proc. SPIE 6101, 61011C (2006).
[CrossRef]

Schulze, C.

Shemirani, M. B.

Shimshi, L.

Shwartz, S.

S. Shwartz, M. A. Golub, and S. Ruschin, “Generating function approach for creation of coherent multimode beams by diffractive optics,” J. Mod. Opt. 59, 83–89 (2012).
[CrossRef]

Sierra, A.

Sissakian, I. N.

M. A. Golub, A. M. Prokhorov, I. N. Sissakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Quantum Electron. 12, 1208–1209 (1982).
[CrossRef]

Skidanov, R. V.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

M. A. Golub, A. M. Prokhorov, I. N. Sissakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Quantum Electron. 12, 1208–1209 (1982).
[CrossRef]

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC, 1994).

Syouji, A.

Tacca, M.

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Taga, H.

Tur, M.

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 138, 1–9 (2012).

Turunen, J.

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Vasic, B. V.

Wang, J.

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 138, 1–9 (2012).

Wang, Z.

Willner, A. E.

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 138, 1–9 (2012).

Winzer, P. J.

Wolf, E.

R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[CrossRef]

Yan, Y.

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 138, 1–9 (2012).

Yang, J.-Y.

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 138, 1–9 (2012).

Yu, C.-P.

Yuan, X.-C.

Yue, Y.

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 138, 1–9 (2012).

Zhang, N.

Appl. Opt. (6)

IEEE J. Quant. Electron. (1)

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intracavity phase element,” IEEE J. Quant. Electron. 37, 80–86 (2001).
[CrossRef]

IEEE J. Quantum Electron (1)

U. Levy, H. Kobrinsky, and A. A. Friesem, “Angular multiplexing for multichannel communication in a single fiber,” IEEE J. Quantum Electron 17, 2215–2224 (1981).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. Shwartz, M. A. Golub, and S. Ruschin, “Generating function approach for creation of coherent multimode beams by diffractive optics,” J. Mod. Opt. 59, 83–89 (2012).
[CrossRef]

J. Opt. (1)

A. Gatto, M. Tacca, P. Martelli, P. Boffil, and M. Martinelli, “Free-space orbital angular momentum division multiplexing with Bessel beams,” J. Opt. 13, 064018 (2011).
[CrossRef]

Nat. Photonics (1)

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 138, 1–9 (2012).

Opt. Commun. (1)

S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000).
[CrossRef]

Opt. Express (5)

Opt. Laser Technol. (2)

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40, 1030–1036 (2008).
[CrossRef]

S. H. Murshid, A. Chakravarty, and R. Biswas, “Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers,” Opt. Laser Technol. 43, 430–436 (2011).
[CrossRef]

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Proc. SPIE (1)

M. R. Duparre, B. Luedge, and S. Schroeter, “Etalons for pure and composite transversal modes,” Proc. SPIE 6101, 61011C (2006).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Supplementary Material (2)

» Media 1: AVI (2136 KB)     
» Media 2: AVI (2136 KB)     

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Figures (11)

Fig. 1.
Fig. 1.

Single-mode DOEs design for transmitter: (a) amplitude, (b) phase, and (c) binary encoded.

Fig. 2.
Fig. 2.

Experimental optical setup for MM free-space optical communication system. SFS: spatial filter system; BS1 and BS2: beam splitters; S1 and S2: beam samplers; D1, D2, D3: DOEs; C1 and C2: choppers; MO: micro objective; CMOS:CMOS camera.

Fig. 3.
Fig. 3.

Intensity of radial LGp0 modes reconstructed from the multichannel GF DOE D3: (a) computer simulated and (b) experimental.

Fig. 4.
Fig. 4.

Magnified 3D graph of experimental intensity of the LG10 mode reconstructed from a multichannel DOE D3.

Fig. 5.
Fig. 5.

Experimental Fourier plane intensity. Correlations 1, 3, and 5 correspond to the convolution (denoted by ) between the transmitter’s mode LG20 created by DOE D1 and the receiver’s modes LG20, LG10, and LG00 created by DOE D3. Correlations 2, 4, and 6 correspond to the transmitter’s mode LG10 created by DOE D2 and the receiver’s modes LG20, LG10, and LG00 created by DOE D3. The correlations corresponding to DOE D1 were intentionally shifted relative to those of DOE D2.

Fig. 6.
Fig. 6.

Graphs for cross sections of cross-correlation Fourier plane intensity between modes LGp0 of the transmitter and LGp0 of the receiver with different orders p, p. Blue solid line: experiment; red dashed line: computer simulations considering finite aperture of the DOE. (a) p=2, p=1. (b) p=0, p=1. (c) p=0, p=2.

Fig. 7.
Fig. 7.

Autocorrelation patterns between corresponding single modes Lp0 of the transmitter and Lp0 of the receiver; p=p=0,1,2: (a) experimental and (b) computer simulated.

Fig. 8.
Fig. 8.

Cross-correlation patterns between different LGp0 modes p=0,1,2: (a) experimental and (b) computer simulated.

Fig. 9.
Fig. 9.

(Media 1). Experimental intensity distribution at the Fourier plane at three sequential moments in time, when either the correlation peaks of one the modes has maximum power or both modes have nearly equal powers. The left spot corresponds to LG00 and the right spot to LG10.

Fig. 10.
Fig. 10.

Time dependence of received temporal signal (dashed curves) and transmitted signals (solid curves): (a) channel of mode LG10 with 8 Hz and (b) LG00 with 28 Hz.

Fig. 11.
Fig. 11.

Fourier spectra of measured temporal signals, calculated with 1D FFT. Channels of mode LG10 are shown by solid curves, and LG00 by dashed curves: (a) transmitter and (b) receiver.

Tables (3)

Tables Icon

Table 1. Modal Power Distribution Ip0 in the LG GF 2ζ2(ζ2Π0)

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Table 2. Measured Modal Powers in Cases of Single-mode and MM Beams Created by Receiver’s DOE D1 and Transmitter’s DOE D3; Powers Normalized to That of LG00 Autocorrelation Power

Tables Icon

Table 3. Computer-simulated Modal Powers in Cases of Single-mode and MM Beams Created by Receiver’s DOE D1 and Transmitter’s DOE D3; Powers Normalized to That of LG00 Autocorrelation Power

Equations (21)

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Gψpl(x)ψpl*(x)d2x=δppδll,
x=(xy)
T(x)=p,lTplψpl*(x)exp(i2πvpl·x),
PT=G|T(x)|2d2x=p,l|Tpl|2,
Ipl=1PT|Tpl|2,αpl=argTpl.
Π=Π(x,ζ,η)=p,lπplψpl(x)ζpηl,
Π(m,n,a,b)(x,ζ,η)=ηbaζmnbnηbζnηaζmΠ(x,ζ,η),
πpl(m,n,a,b)=(p+m)!(p+mn)!(l+a)!(l+ab)!πpl.
ζ(x)=|ζ|exp[i(2πvζ·x+α0)],η(x)=|η|exp[i(2πvη·x+β0)],
vζ=(vζ,xvζ,y),vη=(vη,xvη,y)
Tpl=πp,l(m,n,a,b)*|ζ|p|η|lexp[i2π(pα0+lβ0)]
vpl=pvζ+lvη
ψpl(x)=cpl(ρ2)lLpl(2ρ2)exp(ρ2+ilα),cpl=1σ2p!π(|l|+p)!,
Πl=(2ρ)lexp(ilα)(1ζ)l+1exp(ρ21+ζ1ζ)
πpl=1cpl.
Π=l=0[Πlηl+(Πlηl)*]=2Reexp(ρ21+ζ1ζ)1ζη2ρexp(iα).
w(x)=p,lwplψpl(x)
wpl=Gw(x)ψpl*(x)d2x.
FT(v)=Gw(x)T(x)exp(i2πv·x)d2x=p,lTplGw(x)ψpl*(x)exp[i2π(vvpl)·x]d2x.
FT(vpl)=TplGw(x)ψpl*(x)d2x=Tplwpl.
wpl=1TplFT(vpl)

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