Research Article Optics Letters 1
Spatial Mode Diversity for Robust Free-Space Optical
Communications
M
ITCHELLA. C
OX1,2,*, C
ARMELOR
OSALES-G
UZMÁN2, L
INGC
HENG1,
ANDA
NDREWF
ORBES21School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg 2050, South Africa 2School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa
*Corresponding author: mitchell.cox@wits.ac.za
Compiled November 20, 2017
Free-space communication links are severely affected by atmospheric turbulence, which causes degrada-tion in the transmitted signal. One of the most common soludegrada-tions to overcome this is to exploit diversity. In this approach, information is sent in parallel using two or more transmitters that are spatially sepa-rated, with each beam therefore experiencing different atmospheric turbulence, lowering the probability of a receive error. In this work we propose and experimentally demonstrate a generalization of diversity
based on spatial modes of light, which we have termedmodal diversity. We remove the need for a
phys-ical separation of the transmitters by exploiting the fact that spatial modes of light experience different perturbations, even when travelling along the same path. For this proof-of-principle we selected modes from the Hermite-Gaussian and Laguerre-Gaussian basis sets and demonstrate an improvement in Bit Error Rate by up to 54%. We outline that modal diversity enables physically compact and longer distance free space optical links without increasing the total transmit power.
OCIS codes: (060.2605) Free-space optical communication; (010.1300) Atmospheric propagation; (030.7060) Turbulence.
http://dx.doi.org/10.1364/ol.XX.XXXXXX
Free Space Optical (FSO) communication has been the subject of significant research in recent years due to the impending
capac-ity crunch [1]. Using FSO communications between buildings
and even for Earth to satellite communications is of primary interest because of ultra high bandwidth capabilities when tech-nologies such as Mode Division Multiplexing (MDM) are
em-ployed [2,3]. Here, several orthogonal modes are used, each
carrying a separate data stream to multiply the overall data rate of a system. Historically only the azimuthal subset of the
Laguerre-Gauss (LGℓp) basis has been utilised, specified by mode
indexℓ, which relates to the orbital angular momentum of each
photon,ℓ¯h. The use of the radial LG index,p, is uncommon but
is required to harness the full capacity of the basis [4,5]. The
Hermite-Gauss (HGmn) basis can also be used for MDM and is
specified by mode indicesnandm, and beyond scalar modes,
Cylindrical Vector Vortex (CVV) modes have also been the sub-ject of attention for optical communications in both free space
and optical fibres [6–8].
Upon propagation through the atmosphere, turbulence dis-torts the wave-front, amplitude and phase of the launched modes, called fading. Fading leads to errors in a communi-cation system with a certain probability, and hence deteriorates the overall performance. In an MDM system this fading may also manifest as crosstalk between channels. Fading is obviously dependent on the path that the beam propagates through and
so when several independent paths are used, typically with a
separation of at leastr0(the Fried Parameter, which is a
mea-sure of the transverse distance over which the refractive index is
correlated [9]), the probability of error is typically reduced [10].
This is the basis for what is known as diversity and the so-called diversity gain can often be predicted by using probabilistic
mod-els applied to a range of mode types [11–17]. Diversity can be
considered the complement to MDM, reducing errors rather than increasing capacity. Yet while MDM has been extensively studied, the specific use of optical modes for diversity has not been studied before.
In this letter, we break the paradigm that diversity requires beams to propagate along differing paths in space. Instead, we demonstrate that modes of different spatial profiles are per-turbed differently by atmospheric turbulence, and hence care-fully chosen modes will result in a diversity gain even if they
propagate on the same path, a concept we refer to asmodal
diver-sity. Inspired by recent demonstrations of the robustness of HG
modes in turbulence [18], we provide an experimental
demon-stration of atmospheric turbulence mitigation using modal diver-sity with orthogonal LG and HG co-propagating modes without
r0aperture separation. With this novel diversity method, the
resilience of free-space optical links can be engineered by a ju-dicious choice of the spatial modes, enabling more reliable and longer distance communications, addressing a major drawback
Research Article Optics Letters 2
of FSO communications when compared to radio.
In a typical diversity model, multiple transmitters (lasers)
transmit identical signals,xi(t), at the same time with intensity
gi. These signals propagate through separate channels
repre-sented by channel impulse responses,hi(t). The signal is then
detected by a single receiver (photodiode) with receiver
sensitiv-ity,r. The resulting received signal,y(t)is then found from
y(t) =r
∑
igixi(t)∗hi(t) +n(t). (1)
where an Additive White Gaussian Noise (AWGN) component,
n(t), may be incorporated. Traditionally, “separate channels”
means separate paths, which in turbulence means a physical
separation of at leastr0. Now we will alter this paradigm to
interpret “separate channels” as distinct modes with differing behaviour in turbulence.
If the signal is strong then the contribution of noise to errors is insignificant and the dominant cause of errors is the channel gain itself. If the channel gains are statistically independent then we can write the overall probability of an error in the diversity case as
Pr[Ediversity] =
∏
iPr[Ei], (2)
where the probability of an error occurring for theithchannel is
defined as Pr[Ei]. This always results in a lower probability of
error than a single channel case (since Pr[Ei]≤1). This standard
diversity scheme is similar to Equal Gain Combining (EGC) and will only show a diversity gain if the individual channels are
statistically independent [19].
To implement the above using modal diversity, identical
or-der yet orthogonal HGmn and LGℓpbeams were chosen, motivated
by the fact that HG modes are robust to tip/tilt which are the
primary aberrations of atmospheric turbulence [18]. The order
of the beams is given byN=n+m=2p+|ℓ|for HG and LG
beams respectively. The completeness property of both bases allow us to express any element of one basis as a linear combina-tion of elements from the other basis using the transformacombina-tion
relations below [20]:
LGn,m(x,y,z) =
TheLGmodes have been written in terms ofnandm, which
are indices typically used for HG modes. Traditionally, LG
modes are given in terms of azimuthal indexℓand radial index
p, which can be recovered asℓ=n−mandp=min(n,m). For
example, the LG12mode may be written as
LG12=1
Notice that the HG22component has a zero weighting, making
the HG22mode orthogonal to the LG12mode. Figure1shows an
experimental test of this orthogonality. Conveniently, both of
these modes have an order ofN=4, however, not all LG modes
contain an orthogonal HG mode with the same mode order, and
vice versa. Similarly, HG44 is orthogonal to LG16withN = 8.
Both of these mode sets were tested in the experiment.
HG2
Fig. 1. Experimental demonstration of the orthogonality of
HG2
2and LG12modes with no turbulence (SR=1.0) and slight
turbulence (SR=0.95). On the left, only an HG2
2is transmitted
and it is clear that there is no LG12component after modal
de-composition. Similarly, on the right there is an LG12component
but no HG22component in the decomposed beam.
The experimental setup consisted of two laser diodes which were individually modulated with identical On-Off Keying (OOK-NRZ) signals. The modulation was intentionally done at low bandwidth to avoid non-ideal variables related to high
bandwidth communication. As shown in Fig.2, each beam was
transformed using a Spatial Light Modulator (SLM) into either an LG or HG mode. The first diffraction order from each SLM was spatially filtered and both beams were then combined using a beam splitter to propagate co-linearly. Kolmogorov turbulence and modal decomposition were performed by another SLM
[21,22]. Finally, the resulting beam was filtered using a 50µm
precision pinhole after which the intensity was measured by a photodiode for demodulation. The experiment was performed with varying turbulence strengths corresponding to a range of
atmospheric coherence lengths (r0) from 0.1 to 17 mm. These
coherence lengths correspond to the Strehl Ratios 0.01 to 0.9
when applied to the beams with mode orderN =4. For each
turbulence strength and each mode, one million random bits were tested against 1024 random Kolmogorov phase screens. The Bit Error Rate (BER) was then determined for each test case. Each mode was first tested individually at a specific intensity in a conventional Single Input Single Output (SISO) configuration to determine baseline BER performance for the experimental setup without diversity. Transmit diversity was then tested by transmitting a random bit-stream over the HG and LG modes simultaneously. It is important to note that in the diversity case half of the SISO intensity was used for each transmitter, resulting in the same total intensity as in the SISO case at the receiver. This is done to ensure a fair Signal to Noise Ratio comparison to the SISO case.
The bit error rate results versus turbulence strength (r0) are
shown in Figs.3and4. It is clear that the BER of the EGC case is
on average 23% better than that of the LG SISO case for almost all turbulence strengths. At the weakest turbulence strength, the improvement in diversity BER over both SISO cases is 54%. Due to the fact that the signal powers were normalised, this can only be due to a diversity gain.
Interestingly, the HG case is noticeably better than the LG case at medium turbulence strengths. Given that tip/tilt is the dom-inant aberration in atmospheric turbulence, this result agrees
with the findings in Ref. [18] which demonstrate that HG modes
are more resilient to tip/tilt aberrations than LG modes. At
weaker turbulence strengths, characterised by smallerD/r0
Research Article Optics Letters 3
Fig. 2.Experimental setup showing two transmitters and a single receiver for transmit diversity. The insets to the right are
holo-grams for (a) generating an LG12beam, (b) generating an HG22beam, (c) superposition of the aforementioned holograms and (d)
example turbulence with a grating with SR=0.7. The grey area on the holograms is due to complex amplitude modulation.
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0
10−2
10−1
100
Turbulencer0[mm]
Bit
Err
or
Rate
(BER)
HG2 2 LG12
Diversity
Fig. 3. Bit Error Rate of modes with orderN = 4 with vary-ing turbulence strength showvary-ing a clear improvement for the diversity case (solid lines). The vertical line indicates the ap-proximate size of the beams.
visible. This is expected because the effect of turbulence has become negligible. The diversity gain is still present because of the independent nature of the effect of turbulence on the HG and LG modes.
These results can be put into context by calculating the
ef-fective propagation distance gain at a specific BER [21]. If
Kol-mogorov turbulence is assumed, then we can writer0as a
func-tion ofC2
n, the refractive index structure parameter, the
wave-length,λ, and the propagation distance,z:
r0=0.185
λ2
C2nz 3/5
(6)
This equation can then be solved forz, resulting in:
z≈0.0600647λ
2
C2 nr5/30
(7)
In this experiment,λ = 660 nm and a typical value forC2nis
10−14m−2/3 in strong turbulence [21]. Three arbitrary BERs
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0
10−2
10−1
100
Turbulencer0[mm]
Bit
Err
or
Rate
(BER)
HG44 LG61
Diversity
Fig. 4. Bit Error Rate of modes with orderN =8 with vary-ing turbulence strength showvary-ing a clear improvement for the diversity case (solid lines).
were selected and in Table 1the correspondingr0values are
provided. From Eq.7, the corresponding theoretical
propaga-tion distances are also shown. It is clear that significant im-provements in propagation distance are possible using modal diversity.
Traditionally, diversity has been theoretically and
experimen-tally demonstrated with ar0 (or greater) separation between
transmit and/or receive apertures. In this work we have out-lined the concept of modal diversity, and shown that a significant diversity gain can in fact be achieved in a far more compact
sys-tem, withoutr0separation between apertures or beams. While
we have used orthogonal HG and LG modes of the same order for the demonstration, we expect modal diversity to be observed in any system with judiciously selected mode sets, whether dif-fering in mode order, size or mode type. We speculate that it should be possible to formalise the concept of “separate” in
terms of a newly defined modal distance,rM, akin to the
previ-ous definition in terms of a physical distance,r0. That is, how far
would two modes have to be separated by in mode space,rM,
Research Article Optics Letters 4
Bit Error Rate 0.04 0.08 0.15
LG12r0(mm) 16.6 10.2 4.5
Diversityr0(mm) 12.8 7.4 2.68
LG12Distance (km) 0.24 0.55 2.13
Diversity Distance (km) 0.37 0.93 5.06
Distance Gain (%) 54 71 137
Table 1. Example relations betweenr0and the equivalent
propagation distance,z, withC2
n= 10−14m−2/3for selected
BERs according to Eq.7for the LG and diversity cases
show-ing a clear improvement for the diversity case.
Importantly, the measured diversity gain is shown to increase the allowable propagation distance (determined by an accept-able error rate) by a significant margin at both strong and weak turbulence strengths. This finding can be used to significantly improve future FSO communication links in terms of range and reliability.
FUNDING INFORMATION
The financial assistance of the National Research Foundation (NRF), the Claude Leon Foundation and CONACyT towards this research is acknowledged.
ACKNOWLEDGEMENTS
The authors would like to acknowledge Daniel J. Versfeld for his input during the course of this work.
FULL REFERENCES
1. D. J. Richardson, “Filling the Light Pipe,” Science330, 327 (2010). 2. G. Gibson II, J. Courtial, M. Vasnetsov, S. Barnett, S. Franke-Arnold,
and M. Padgett, “Increasing the data density of free-space optical com-munications using orbital angular momentum,” (International Society for Optics and Photonics, 2004), p. 367.
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,” Nature Photonics6, 488–496 (2012).
4. N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multi-plexed free-space communication,” Nature Photonics9, 822–826 (2015). 5. A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. Ben Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,”6, 27674 (2016).
6. G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett.40, 1980–1983 (2015). 7. M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, D. J. Versfeld, and
A. Forbes, “On the resilience of scalar and vector vortex modes in turbulence,” Optics Express24, 18105 (2016).
8. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” Journal of Lightwave Technology pp. 1–1 (2017). 9. D. L. Fried, “Statistics of a Geometric Representation of Wavefront
Distortion,” Journal of the Optical Society of America55, 1427 (1965). 10. S. Navidpour, M. Uysal, and M. Kavehrad, “BER Performance of
Free-Space Optical Transmission with Spatial Diversity,” IEEE Transactions on Wireless Communications6, 2813–2819 (2007).
11. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” Journal of the Optical Society of America A4, 1923 (1987).
12. M. A. Kashani, M. Uysal, and M. Kavehrad, “A Novel Statistical Channel Model for Turbulence-Induced Fading in Free-Space Optical Systems,” Journal of Lightwave Technology33, 2303–2312 (2015).
13. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Modeling channel interfer-ence in an orbital angular momentum-multiplexed laser link,” (Interna-tional Society for Optics and Photonics, 2009), p. 74640U.
14. J. Zhou, J. Zong, and D. Liu, “Coupled mode theory for orbital an-gular momentum modes transmission in the presence of atmosphere turbulence,” Optics Express23, 31964 (2015).
15. V. P. Aksenov, V. V. Kolosov, G. A. Filimonov, and C. E. Pogutsa, “Orbital angular momentum of a laser beam in a turbulent medium: preservation of the average value and variance of fluctuations,” Journal of Optics18, 054013 (2016).
16. Y. Ren, Z. Wang, G. Xie, L. Li, Y. Cao, C. Liu, P. Liao, Y. Yan, N. Ahmed, Z. Zhao, A. Willner, N. Ashrafi, S. Ashrafi, R. D. Linquist, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Free-space optical communications us-ing orbital-angular-momentum multiplexus-ing combined with MIMO-based spatial multiplexing,” Opt. Lett.40, 4210 (2015).
17. L. Zou, L. Wang, and S. Zhao, “Turbulence mitigation scheme based on spatial diversity in orbital-angular-momentum multiplexed system,” Optics Communications400, 123–127 (2017).
18. B. Ndagano, N. Mphuthi, G. Milione, and A. Forbes, “Comparing mode-crosstalk and mode-dependent loss of laterally displaced orbital angular momentum and Hermite–Gaussian modes for free-space optical com-munication,” Optics Letters42, 4175 (2017).
19. D. Brennan, “Linear Diversity Combining Techniques,” Proceedings of the IRE47, 1075–1102 (1959).
20. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase mode converter,” Optics Communications181, 35–45 (2000).
21. Andrews,Laser Beam Propagation through Random Media(SPIE Press, 2005), 2nd ed.
