Abstract

A multiple-input-multiple-output (MIMO) visible light communication (VLC) system employing vertical cavity surface emitting laser (VCSEL) and spatial light modulators (SLMs) with 16-quadrature amplitude modulation (QAM)-orthogonal frequency-division multiplexing (OFDM) modulating signal is proposed and experimentally demonstrated. The transmission capacity of system is significantly increased by space-division demultiplexing scheme. With the assistance of low noise amplifier (LNA) and data comparator, good bit error rate (BER) performance, clear constellation map, and clear eye diagram are achieved for each optical channel. Such a MIMO VLC system would be attractive for providing services including data and telecommunication services. Our proposed system is suitably applicable to the lightwave communication system in wireless transmission.

© 2014 Optical Society of America

1. Introduction

Recently, visible light communication (VLC) system has attracted a lot of attention as a potential candidate for the indoor wireless communication because VLC has several advantages over traditional radio-frequency (RF)-based wireless communication. It can provide many advantages such as communication links in specific areas in which RF communication is prohibited, like in a hospital or in an aircraft [13]. VLC system employs modulated light wavelength emitted by a variety of light sources, mainly through the use of light emitting diode (LED) [4,5]. However, the performance of VLC system has been limited mostly due to the finite modulation bandwidth (BW) of LED and irradiance decline with distance. To conquer the limitations, laser pointer laser (LPL) has been applied in VLC system [6,7]. Nevertheless, LPL with selected peak resonance frequency to modulate the data signal is needed because of insufficient modulation BW. It increases the complexity of system to have the LPL with selected peak resonance frequency. In recent years, the vertical cavity surface emitting laser (VCSEL) technology has advanced enough that it can be designed to operate in the 680-nm visible light range. VCSEL, with high modulation BW characteristic, is shown to be a prominent one to present its advantage in VLC applications. It is attractive because it avoids the need of peak resonance frequency selection. Such a VCSEL can be employed to replace the LPL in VLC system. For a practical implementation of VLC link, installation simplicity and convenience are beyond disputed issues needed to be solved. VCSEL with high modulation BW and acceptable divergence beam angle is expected to have good performances in VLC system. In this paper, a multiple-input-multiple-output (MIMO) VLC system employing VCSEL and spatial light modulators (SLMs) with 16-quadrature amplitude modulation (QAM)-orthogonal frequency-division multiplexing (OFDM) modulating signal over a 15-m free-space link is proposed and demonstrated. To adapt the SLMs in MIMO VLC system, the optical channel is divided into various separate subchannels. Such a MIMO VLC system has higher transmission capacity than single-input-single-output (SISO) one. A Fresnel lens function is applied to operate the SLM as a dynamic convex lens. After through the SLM, the space-division demultiplexed light is focused into a point to increase the free-space transmission distance [8,9]. OFDM is a promising technology which has very high spectrum efficiency and robust dispersion tolerance to improve the transmission performance of system [10]. To be the first one of employing VCSEL and SLMs in MIMO VLC system, the transmitting lights are successfully modulated with 16-QAM-OFDM signals. With the help of low noise amplifier (LNA) and data comparator at the receiving site, good bit error rate (BER) performance, clear constellation map, and clear eye diagram are achieved for each optical channel. A system of 4 16-QAM OFDM channels over a 15-m free-space link with a total data rate of 10 Gbps (2.5Gbps/Channel × 4Channels) is successfully demonstrated. This proposed MIMO VLC system is shown to be an outstanding one not only to present its convenience in optical free-space transmission, but also to reveal its potential for the real implementation.

2. Experimental setup

The experimental configuration of our proposed MIMO VLC systems employing VCSEL and SLMs with 16-QAM OFDM modulating signal over a 15-m free-space link is shown in Fig. 1. The beam divergence with acceptable divergent angle and the optical spectrum of VCSEL are shown in Fig. 2 and Fig. 3, respectively. The VCSEL, with 3-dB BW/wavelength range/color of 3GHz/684-685.5nm/red, is directly modulated by a 2.5Gbps/2.5GHz 16-QAM OFDM signal. The 16-QAM OFDM signal is generated offline by MATLAB program and uploaded into a Tektronix arbitrary waveform generator (AWG). Such 16-QAM OFDM signal is represented by 64 subcarriers, 512 FFT size, and 2.5 GHz intermediate frequency (IF), respectively. The divergence beam of VCSEL is passed through 4 SLMs for space-division demultiplexing. The SLM has 800 × 600 translucent liquid crystal pixels with pixel pitch of 32 μm, and an active area of translucent liquid crystal of 26.6 × 20 mm. To operate the SLM like a dynamic convex lens, the Fresnel lens function is supplied to the SLM. By means of SLM, the light is focused into a point to extend the free-space transmission distance. The distance between the SLM and the focal point is the focal length of the Fresnel lens. After focused into a point, the light is diverged, launched into the first stage convex lens, transmitted in the free-space, fed into the second stage convex lens, and focused on the high BW photodiode (PD). The distance between the VCSEL and the SLM, the SLM and the first stage convex lens, and the first stage convex lens and the high BW PD is 0.5 m, 1.5 m, and 13 m, respectively. Thereby, the modulated light is transmitted over a distance of 15 m (0.5 + 1.5 + 13 m), and then reached to the high BW PD. The high BW PD has a detection wavelength range of 320-1000 nm, a 3-dB BW of 3.2 GHz, an active area diameter of 0.4 mm, and a responsivity of 0.46 mA/mW (at 685 nm). The received signal is then amplified by the LNA with a low noise figure of about 4.5 dB, and passed through the data comparator for error correction. Finally the 16-QAM OFDM signal is analyzed by an OFDM analyzer, captured by a communication signal analyzer, and processed off-line with a MATLAB program to evaluate the BER performance and the corresponding constellation map.

 

Fig. 1 Experimental configuration of our proposed MIMO VLC systems employing VCSEL and SLMs with 16-QAM-OFDM signal over a 15-m free-space link.

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Fig. 2 The beam divergence with acceptable divergent angle of VCSEL.

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Fig. 3 The optical spectrum of VCSEL.

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3. Experimental results and discussions

A SLM is a transparent or diffractive optical device which can modulate optical amplitude or phase in each pixel. It can modulate light spatially in amplitude or phase, so it can operate as a dynamic diffractive element controlled by electrical signal. With the Fresnel lens function, the SLM can be operated as a dynamic convex lens. The function of the SLM is to increase the transmission capacity of system and to extend the free-space transmission distance. To employ SLMs to divide the acceptable divergent beam emitted from the VCSEL, the transmission capacity of system can be greatly increased. The system uses four channels for 2.5Gbps/2.5GHz 16-QAM-OFDM signals transmissions, a total data rate of 10 Gbps (2.5Gbps/Channel × 4Channels) is obtained. Space-division multiplexing (SDM) transport systems can fully utilize the free-space and increase the transmission data rate of an optical free-space link. A free-space SDM transport system that uses different optical channels to carry the data signal would be quite useful for providing both data and telecommunication services. Since one VCSEL is divided into four channels, yet a 4-channel of optical free-space SDM transport system is established. Optical free-space SDM takes advantage of advanced optical technology that is able to multiplex and demultiplex many channels in the free-space. The transmission data rate can be easily increased by optical free-space SDM technology. In addition, SLM has focused the light into a point, resulting in the increase of free-space transmission distance. Assume that the transmitted optical power Pt is delivered to the receiving site in a beam divergence (in radians) of Div and a free-space link length of L, then the received optical power, Pr can be stated as:

Pr=Pt(DivL)2
To consider the atmospheric attenuation factorα, the Eq. (1) is modified as:
Pr=Pt(DivL)2e(αL)
Assume now that the receiving site has an effective received area Ar, then the free-space optical link equation is given by [11,12]:
Pr=PtAr(DivL)2e(αL)
Since the SLM has focused the light into a point, yet the beam divergence Div is very small. It can be seen that, from Eq. (3), a small Div is related to a large received optical power. In result, the free-space transmission distance can be extended. The distance between the SLM and the first stage convex lens is 1.5 m, including the focal length of the Fresnel lens (1.4 m) and the focal length of the first stage convex lens (0.1 m). A more 1.5-m free-space transmission distance is obtained due to the use of SLM.

A functional block of the data comparator is illustrated in Fig. 4, in which including an amplitude comparator and a phase comparator. Let data signal d(n) has an amplitude a(n) and a phase θ(n):

d(n)=a(n)ejθ(n)
After transmission through a free-space link, the received data signal der(n) has a distorted amplitude aer(n) and a distorted phase error θer(n):
der(n)=aer(n)ejθer(n)
The data comparator has to estimate d(n) from der(n) by error feedback. For amplitude compensation, the output of the amplitude compensator is compared with a stored copy of a(n) to create an amplitude error. For phase compensation, the output of the phase compensator is compared with a stored copy of θ(n) to create a phase error. Amplitude and phase error compensations are crucial for ensuring maximum distortion suppression, the use of data comparator offers significant amplitude and phase error compensations.

 

Fig. 4 A block diagram of the data comparator, in which including a fast comparator.

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To evaluate the transmitted 16-QAM OFDM signal performance, the measured BER curves and constellation map at a data stream of 2.5Gbps/2.5GHz are present in Fig. 5. At a free-space transmission distance of 15 m; without employing LNA and data comparator, the BER is about 10−2; with employing LNA and data comparator, the BER is reached down to 10−6. As LNA and data comparator are employed simultaneously, low BER value and clear constellation map are obtained. Error free transmission is achieved to demonstrate the possibility of establishing a 4-channel 16-QAM OFDM MIMO VLC system. To show a more direct association with LNA and data comparator, we remove one of the schemes and measure the BER value. It can be seen that the BER performance improvement is limited as only one improvement scheme is employed. Such results indicate that LNA and data comparator play important roles for errors corrections. They can further improve the signal-to-noise ratio (SNR) of system, leading to the improvement of BER performance.

 

Fig. 5 The measured BER curves and constellation map at a data stream of 2.5Gbps/2.5GHz.

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Figs. 6(a) and 6(b) display the eye diagrams of SLM1 channel under 15 m free-space link for the conditions of without employing LNA and data comparator [Fig. 6(a)] and with employing LNA and data comparator [Fig. 6(b)], respectively. In order to analyze the eye diagram, a 2.5Gbps/2.5GHz data stream generated from a signal generator is employed at the transmitting site to modulate the VCSEL directly. Over a 15-m free-space link, the detected data stream is down-converted by mixing with a 2.5 GHz sinusoidal signal and then analyzed by a digital communication analyzer (DCA). The amplitude and phase fluctuations in the signal are obviously observed in the case of without employing LNA and data comparator [Fig. 6(a)]. The signal distortion will give an increase in power penalty. For the case of employing LNA and data comparator [Fig. 6(b)], however, a clear eye diagram is achieved due to amplitude and phase fluctuations suppressions.

 

Fig. 6 The eye diagrams of SLM1 channel under 15 m free-space link: (a) without employing LNA and data comparator, (b) with employing LNA and data comparator.

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To have a more association with free-space link length, data rate and channel number, systems of one 16-QAM OFDM channel over 19.5-m and 6-m free-space links with data rate of 2.5-Gbps and 10-Gbps are established in Fig. 7 and Fig. 8, respectively. The performances of systems are evaluated and compared under the conditions of equal VCSEL optical power and BER value. In Fig. 7, since the optical power emitted from VCSEL has not divided into four parts, yet the free-space link length is longer than that of Fig. 1. In Fig. 8, since the 3-dB BW of VCSEL and PD is higher than that of Fig. 1, yet the data rate is higher than that of Fig. 1. From these results, it can be seen that the free-space link length of systems is limited primarily by the optical power of VCSEL, and the data rate of systems is limited primarily by the 3-dB BW of VCSEL and PD. In addition, it is clear that the function of SLM is to focus the light into a point to extend the free-space link length.

 

Fig. 7 System of one 16-QAM OFDM channel over a 19.5-m free-space link with a data rate of 2.5-Gbps.

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Fig. 8 System of one 16-QAM OFDM channel over a 6-m free-space link with a data rate of 10-Gbps.

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4. Conclusions

We proposed and demonstrated a MIMO VLC system employing VCSEL and SLMs with 16-QAM OFDM modulating signal. With the help of LNA and data comparator at the receiving sites, low BER operation, clear constellation map, and clear eye diagram are obtained for each optical channel. A system of 4 16-QAM OFDM channels over a 15-m free-space link with a total data rate of 10 Gbps (2.5Gbps/Channel × 4Channels) is successfully demonstrated. Employing VCSEL and SLMs in MIMO VLC system is a promising option, an attractive feature that can accelerate the MIMO VLC deployment.

Acknowledgment

The authors would like to thank the financial support from the National Science Council of Taiwan under Grant NSC 100-2221-E-027-067-MY3, NSC 101-2221-E-027-040 -MY3, and NSC 102-2218-E-027-002.

References and links

1. F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and H. T. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” Conf. on Opt. Fiber Commun. (OFC) OTh1G4 (2013).

2. Y. F. Liu, Y. C. Chang, C. W. Chow, and C. H. Yeh, “Equalization and pre-distorted schemes for increasing data rate in-door visible light communication system,” Conf. on Opt. Fiber Commun (OFC) JWA83 (2011).

3. T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004). [CrossRef]  

4. Y. Wang, Y. Wang, N. Chi, J. Yu, and H. Shang, “Demonstration of 575-Mb/s downlink and 225-Mb/s uplink bi-directional SCM-WDM visible light communication using RGB LED and phosphor-based LED,” Opt. Express 21(1), 1203–1208 (2013). [CrossRef]   [PubMed]  

5. C. H. Yeh, Y. F. Liu, C. W. Chow, Y. Liu, P. Y. Huang, and H. K. Tsang, “Investigation of 4-ASK modulation with digital filtering to increase 20 times of direct modulation speed of white-light LED visible light communication system,” Opt. Express 20(15), 16218–16223 (2012). [CrossRef]  

6. C. Y. Chen, P. Y. Wu, H. H. Lu, Y. P. Lin, J. Y. Wen, and F. C. Hu, “Bidirectional 16-QAM OFDM in-building network over SMF and free-space VLC transport,” Opt. Lett. 38(13), 2345–2347 (2013). [CrossRef]   [PubMed]  

7. W. Y. Lin, C. Y. Chen, H. H. Lu, C. H. Chang, Y. P. Lin, H. C. Lin, and H. W. Wu, “10m/500 Mbps WDM visible light communication systems,” Opt. Express 20(9), 9919–9924 (2012). [CrossRef]   [PubMed]  

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

9. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012). [CrossRef]  

10. Y. C. Chi, Y. C. Li, H. Y. Wang, P. C. Peng, H. H. Lu, and G. R. Lin, “Optical 16-QAM-52-OFDM transmission at 4 Gbit/s by directly modulating a coherently injection-locked colorless laser diode,” Opt. Express 20(18), 20071–20077 (2012). [CrossRef]   [PubMed]  

11. H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).

12. S. Bloom, “The physics of free-space optics,” AirFiber Inc 1-22 (2002).

References

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  1. F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and H. T. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” Conf. on Opt. Fiber Commun. (OFC) OTh1G4 (2013).
  2. Y. F. Liu, Y. C. Chang, C. W. Chow, and C. H. Yeh, “Equalization and pre-distorted schemes for increasing data rate in-door visible light communication system,” Conf. on Opt. Fiber Commun (OFC) JWA83 (2011).
  3. T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
    [Crossref]
  4. Y. Wang, Y. Wang, N. Chi, J. Yu, and H. Shang, “Demonstration of 575-Mb/s downlink and 225-Mb/s uplink bi-directional SCM-WDM visible light communication using RGB LED and phosphor-based LED,” Opt. Express 21(1), 1203–1208 (2013).
    [Crossref] [PubMed]
  5. C. H. Yeh, Y. F. Liu, C. W. Chow, Y. Liu, P. Y. Huang, and H. K. Tsang, “Investigation of 4-ASK modulation with digital filtering to increase 20 times of direct modulation speed of white-light LED visible light communication system,” Opt. Express 20(15), 16218–16223 (2012).
    [Crossref]
  6. C. Y. Chen, P. Y. Wu, H. H. Lu, Y. P. Lin, J. Y. Wen, and F. C. Hu, “Bidirectional 16-QAM OFDM in-building network over SMF and free-space VLC transport,” Opt. Lett. 38(13), 2345–2347 (2013).
    [Crossref] [PubMed]
  7. W. Y. Lin, C. Y. Chen, H. H. Lu, C. H. Chang, Y. P. Lin, H. C. Lin, and H. W. Wu, “10m/500 Mbps WDM visible light communication systems,” Opt. Express 20(9), 9919–9924 (2012).
    [Crossref] [PubMed]
  8. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Degenerate mode-group division multiplexing,” IEEE /OSA J. Lightwave Technol. 30(24), 3946–3952 (2012).
    [Crossref]
  9. J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012).
    [Crossref]
  10. Y. C. Chi, Y. C. Li, H. Y. Wang, P. C. Peng, H. H. Lu, and G. R. Lin, “Optical 16-QAM-52-OFDM transmission at 4 Gbit/s by directly modulating a coherently injection-locked colorless laser diode,” Opt. Express 20(18), 20071–20077 (2012).
    [Crossref] [PubMed]
  11. H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).
  12. S. Bloom, “The physics of free-space optics,” AirFiber Inc 1-22 (2002).

2013 (2)

2012 (5)

2010 (1)

H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).

2004 (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Carpenter, J.

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

J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012).
[Crossref]

Chang, C. H.

Chen, C. Y.

Chi, N.

Chi, Y. C.

Chow, C. W.

Henniger, H.

H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).

Hu, F. C.

Huang, P. Y.

Komine, T.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Li, Y. C.

Lin, G. R.

Lin, H. C.

Lin, W. Y.

Lin, Y. P.

Liu, Y.

Liu, Y. F.

Lu, H. H.

Nakagawa, M.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Peng, P. C.

Shang, H.

Thomsen, B. C.

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

J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012).
[Crossref]

Tsang, H. K.

Wang, H. Y.

Wang, Y.

Wen, J. Y.

Wilfert, O.

H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).

Wilkinson, T. D.

J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012).
[Crossref]

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

Wu, H. W.

Wu, P. Y.

Yeh, C. H.

Yu, J.

IEEE /OSA J. Lightwave Technol. (1)

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

IEEE Photon. Technol. Lett. (1)

J. Carpenter, B. C. Thomsen, and T. D. Wilkinson, “Mode division multiplexing of modes with the same Azimuthal index,” IEEE Photon. Technol. Lett. 24(21), 1969–1972 (2012).
[Crossref]

IEEE Trans. Consum. Electron. (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consum. Electron. 50(1), 100–107 (2004).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Radioengineering (1)

H. Henniger and O. Wilfert, “An introduction to free-space optical communications,” Radioengineering 19(2), 203–212 (2010).

Other (3)

S. Bloom, “The physics of free-space optics,” AirFiber Inc 1-22 (2002).

F. M. Wu, C. T. Lin, C. C. Wei, C. W. Chen, Z. Y. Chen, and H. T. Huang, “3.22-Gb/s WDM visible light communication of a single RGB LED employing carrier-less amplitude and phase modulation,” Conf. on Opt. Fiber Commun. (OFC) OTh1G4 (2013).

Y. F. Liu, Y. C. Chang, C. W. Chow, and C. H. Yeh, “Equalization and pre-distorted schemes for increasing data rate in-door visible light communication system,” Conf. on Opt. Fiber Commun (OFC) JWA83 (2011).

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Figures (8)

Fig. 1
Fig. 1

Experimental configuration of our proposed MIMO VLC systems employing VCSEL and SLMs with 16-QAM-OFDM signal over a 15-m free-space link.

Fig. 2
Fig. 2

The beam divergence with acceptable divergent angle of VCSEL.

Fig. 3
Fig. 3

The optical spectrum of VCSEL.

Fig. 4
Fig. 4

A block diagram of the data comparator, in which including a fast comparator.

Fig. 5
Fig. 5

The measured BER curves and constellation map at a data stream of 2.5Gbps/2.5GHz.

Fig. 6
Fig. 6

The eye diagrams of SLM1 channel under 15 m free-space link: (a) without employing LNA and data comparator, (b) with employing LNA and data comparator.

Fig. 7
Fig. 7

System of one 16-QAM OFDM channel over a 19.5-m free-space link with a data rate of 2.5-Gbps.

Fig. 8
Fig. 8

System of one 16-QAM OFDM channel over a 6-m free-space link with a data rate of 10-Gbps.

Equations (5)

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P r = P t (DivL) 2
P r = P t (DivL) 2 e (αL)
P r = P t A r (DivL) 2 e (αL)
d(n)=a(n) e jθ(n)
d er (n)= a er (n) e j θ er (n)

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