Free-space optical relaying system with few-mode all-optical relay

Optics Communications 439 (2019) 164–170

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Free-space optical relaying system with few-mode all-optical relay Shanyong Cai ∗, Zhiguo Zhang, Xue Chen State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing, 100876, China

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Keywords: Free-space optical communication Relay-assisted communication Few mode fiber Photonic lantern

ABSTRACT A scheme of relay-assisted transmission system with few-mode all-optical relay is proposed in this paper to resist turbulence for atmospheric channels. The all-optical relay is composed of few-mode EDFA and photonic lantern which could amplify the distorted light field after atmospheric channels and covert the distorted light field to fundamental Gaussian fields. At the relay, using few-mode all-optical relay leads to the improvement of the turbulence induced fading which brings improved system performance. A dual-hop model for all-optical relaying system with few-mode EDFA and photonic lantern is derived. The communication performance is calculated with the help of turbulence phase screens to model atmospheric turbulence channel. Numerical results show that power budgets are increased by more than 4𝑑𝐵 for fixed gain amplification systems compared with all-optical relaying systems with single-mode all-optical relay and compared with few-mode all-optical relaying systems including few-mode EDFA only. Moreover, the BER performance is the best when mode-dependent gain (MDG) of few-mode EDFA is zero and becomes worse with the increase of MDG.

1. Introduction Free-space optical (FSO) communication systems have wide applications in satellite to ground communications, inter-building communications and so on for its high data rate, unlicenced spectrum and inherent security [1]. However, in FSO systems, atmospheric turbulence causes severe channel fading which results in degraded bit error rate (BER) [2]. So, the compensation of atmospheric turbulence fading is essential for FSO systems. A variety of optical domain techniques have been proposed to mitigate turbulence fading including spatial diversity [3,4], wavefront correction [5] and all-optical relaying system [6–10]. Relaying system relays the data signal from the source to destination through intermediate nodes (relays). It exploits the fact that the turbulence induced fading variance is distance-dependent and yields improved BER by taking advantage of the shorter hop distances. Furthermore, since FSO is a line-of-sight technology, relaying system allows optical connection where source and destination do not have a line of sight. Up to now, all-optical relaying systems with single-mode (SM) erbium-doped fiber amplification (EDFA) based relay have been proposed [9]. SM-EDFA only receives and amplifies the power of spatial fundamental mode. However, the distorted light field after turbulence channel is a superposition of multiple spatial modes, thus, the turbulence induced fading variance at the relay is high. If multiple spatial modes are received and amplified, the turbulence induced fading at the relay will be improved. Recently, few-mode fiber (FMF) components which has been widely used in spatial division multiplexing fiber communication system [11–14] had been paid attention to in FSO

systems to resist turbulence. Few-mode (FM) EDFAs which could receive and amplify multiple spatial modes have been used in preamplified receiver in FSO system [15] and as all-optical relay in FSO relaying system [16]. However, the output of FM-EDFAs are few-mode fields which reduce the receiving power at the next hop in relaying system. Fortunately, photonic lantern (PL) could convert higher order modes into fundamental mode in SMF arrays which had been used to enhance collection efficiency of scattered infrared light [17] and utilized in modes diversity coherent receiver [18]. Using PL to achieve beam shaping is more simple compared with wavefront correction. In this paper, a scheme of FM-EDFA and PL based all-optical relaying system to further resist turbulence is proposed. The channel fading is improved because multiple spatial modes contained in distorted light field after atmospheric turbulence channels are received and amplified at the relay and been converted to fundamental spatial modes before transmitting to the next hop. Through modeling and numerical calculating, we found that more than 4-dB improvement in power budget is obtained for fixed gain amplification compared with SM all-optical relaying systems and compared with FM all-optical relaying system including FM-EDFA only. 2. Scheme The scheme of FM-EDFA and PL based dual-hop transmission system is shown in Fig. 1(a). The relay is composed of receiving lens, FM-EDFA, PL and transmitting lens (Fig. 1(a–b)). At the input of relay, the fewmode signal and background light fields are coupled into FM-EDFA. The

∗ Corresponding author. E-mail address: [email protected] (S. Cai).

https://doi.org/10.1016/j.optcom.2019.01.054 Received 8 November 2018; Received in revised form 30 December 2018; Accepted 19 January 2019 Available online 24 January 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

S. Cai, Z. Zhang and X. Chen

Optics Communications 439 (2019) 164–170

Fig. 1. The schematic of (a) dual-hop all-optical relaying system based on FM-EDFA and PL, (b) all-optical relay with FM-EDFA and PL, and (c) all-optical relay with

SM-EDFA and OC. (d) The schematic and the transfer matrices of PL [12,19]. SMF: single-mode fiber, FMF: few-mode fiber. Table 1 Fiber specifications.

Mode field diameter of 𝐿𝑃01 mode

SMF

FMF

10.4 μm

11 μm

is equal to that of OFS’ FMFs. SMF at Source/Relay/Destination or FMF at Relay is placed on the focal point of transmitting/receiving lens. Transmitting lens focal length is 𝑓𝑡 = 20 cm. At this time, the transmitting beam diameter is 𝐷 = 38 mm. Receiving lens focal length is set to 40 cm for coupling to FMF and 38 cm for coupling to SMF respectively to ensure the optimal and same coupling efficiencies. Moreover, transmitting/receiving lens diameter is set large enough to receive all light fields. Aperture spacing of the transmitters at relay is large than the atmospheric coherence length 𝑟0 to guarantee mutually independent fading in the second hop (see Table 2). Compared with SM all-optical relay (Fig. 1(c)), FM relay characterizes lower relative standard deviation (RSD) of turbulence fading and higher average receiving power [16,20]. The reason can be attributed to mode averaging where multiple spatial modes contained in the turbulence-induced distorted light field are received simultaneously. In addition, after passing through the PL, the few-mode fields are transformed into spatial fundamental mode fields, thus optimizing the channel fading in the second hop.

Table 2 Geometric arrangement. Transmitting lens focal length of the first hop (𝑓𝑡1 ) Transmitting lens focal length of the second hop (𝑓𝑡2 ) Transmitting lens diameter (both the first and second hop) Receiving lens focal length of the first hop (𝑓𝑟1 ) Receiving lens focal length of the second hop (𝑓𝑟2 ) Receiving lens diameter (both the first and second hop) Aperture spacing of the transmitters in the second hop (𝐷0 ) Source–destination link length (𝑑𝑠𝑑 )

20 cm 20 cm 50 mm 40 cm 38 cm 200 mm 𝐷0 > 𝑟0 5 km

noisy few-mode light fields are amplified by FM-EDFA and converted to fundamental modes after PL. In this paper, FM-EDFA and PL are assumed to support 3 spatial modes with three SMFs at the output of PL (Fig. 1(d)). After PL, the noisy light field is distributed to three SMFs. Three noisy single-mode light beams are forwarded to the destination using transmit diversity. A SMF is placed on the destination to collect light field. The optical links between source and relay and between relay and destination are atmospheric turbulence channels. The system operates at 𝜆 = 1550 nm with source–destination length 𝑑𝑠𝑑 = 5.0 km. The relay is placed in the middle of the link (𝑑1 = 𝑑2 ). Two turbulence phase screens placed in the middle of two hops are utilized to model the effect of atmospheric turbulence. The simulation of beam propagation is as follows. At first, the transmitted light field propagates 𝑑𝑖 ∕2 distance in free space. Then, it is modulated by the turbulence screen, and continues to propagate 𝑑𝑖 ∕2 distance until it reaches the relay/receiver. Finally, the received light fields at relay or destination are obtained. Mode field diameter of 𝐿𝑃01 modes in SMF/FMF are given in Table 1. SMF is commercially available SMF-28 fiber. FMF is taken to be graded-index fiber with profile parameter 𝜌 = 2, thus the spatial distribution of core mode in the FMF is given by { ( 2) ( ) 𝑐𝑜𝑠(𝑚𝜃) 𝑟 𝑚 𝑟2 𝑟 ∗ ∗ exp(− ) ∗ , (1) 𝜓𝑚,𝑛 (𝑟, 𝜃) = 𝐶 ∗ 𝐿𝑚 𝑛−1 𝑠𝑖𝑛(𝑚𝜃) 𝜔 𝜔2 2𝜔2

3. Model 3.1. Channel model The temporal random atmospheric channel impairments are denoted by 𝐻𝑖 (𝑡) for the 𝑖th √ hop. The channel impairments 𝐻𝑖 (𝑡) can be formulated as 𝐻𝑖 (𝑡) ≈ 𝐿𝑖 ℎ𝑖𝑝 𝜂𝑖 . 𝐿𝑖 = 𝛼𝑎𝑡𝑡𝑛 ∗ 𝑑𝑖 denotes the atmospheric attenuation caused by absorption and scattering effects where 𝛼𝑎𝑡𝑡𝑛 and 𝑑𝑖 represent the attenuation coefficient and hop distance, respectively. It can be considered constant since the weather changes slowly over a long time. ℎ𝑖𝑝 denotes geometric loss caused by the spatial spread of the optical beam. 𝜂𝑖 denotes turbulence-induced fading factor and is a random coefficient. In this paper, to calculate turbulence-induced fading, atmospheric turbulence is simulated by phase screens with appropriate randomness. Turbulence phase screens are generated via Fourier Transform method. The basic idea is using atmospheric phase spectrum according to atmospheric turbulence theory to filter Gaussian random noises. The phase screen in x, y coordinates in space is calculated by [21]

where 𝐶 is the amplitude constant, 𝐿𝑚 𝑛 are the associated Laguerre polynomials, and 𝜔 is a constant relating modal diameter. Here, 𝜔 = 3.89 μm is set, so the mode field diameter of LP01 mode is 11 μm which

𝜑(𝑥, 𝑦) = 𝐼𝐹 𝐹 𝑇 (𝐶 ⋅ 𝜎(𝑘𝑥 , 𝑘𝑦 )) 2

− 53

𝜎 (𝑘𝑥 , 𝑘𝑦 ) = 0.023𝑟0 165

[𝑘2𝑥

11 + 𝑘2𝑦 ]− 6 ,

(2) (3)

S. Cai, Z. Zhang and X. Chen

Optics Communications 439 (2019) 164–170

Fig. 2. Channel model of the dual-hop relaying system (a) with FM-EDFA and PL based all-optical relay, (b) with SM-EDFA and OC based all-optical relay and (c) with FM-EDFA based all-optical relay.

where 𝜎 2 (𝑘𝑥 , 𝑘𝑦 ) is the variance of phase spectrum, and 𝐶 is a matrix of complex Gaussian random numbers. 𝑟0 is the atmospheric coherence length, which is a function of refractive index structure constant (𝐶𝑛2 ), link distance (𝑑𝑖 ) and optical beam. For Gaussian beam, 𝑟0 can be approximated by the simple expression 𝑟0 = [

8 ]3∕5 (0.423𝐶𝑛2 𝑘2 𝑑𝑖 )−3∕5 , 𝑙0 ≪ 𝑟0 ≪ 𝐿0 , 3(𝑎 + 0.62𝛬11∕6 )

background radiation power of single mode and the optical bandwidth of the system, respectively. Following the approach in [22], the electric field corresponding to the background radiation in (5) is written as a sum of 2𝑀 + 1 (𝑀 = 𝐵𝑜 ∕(2𝛿𝜈)) exponential terms at frequencies 𝜔𝑙 = 𝜔0 + 2𝜋𝑙𝛿𝜈 with center frequency 𝜔0 = 2𝜋𝜈0 (𝜈0 : optical center frequency), where 𝛿𝜈 denotes the spacing of the considered frequencies. 𝜙𝑙,𝑚 denotes a random phase corresponding to the 𝑚th mode. FM-EDFA and PL are followed to amplify the received electric field and convert higher-order FMF modes to SMF modes. The transfer matrices of FM-EDFA and PL are expressed as: √ −1 𝛼𝑃 𝐿 0 0 𝑎0 𝑎0 ⎡ 𝑎0 ⎤ ⎡ ⎤ √ 𝑗2𝜋∕3 −𝑗2𝜋∕3 ⎢ ⎥ ⎢ ⎥ 0 𝛼𝑃 𝐿 0 𝐇 = 𝑎1 𝑎1 𝑒 𝑎1 𝑒 √ ⎢ ⎥ ⎢ ⎥ 𝑎1 𝑒𝑗2𝜋∕3 ⎦ ⎣ ⎣ 𝑎1 𝑎1 𝑒−𝑗2𝜋∕3 0 0 𝛼𝑃 𝐿 ⎦

(4)

where 𝑘 is the wave number, 𝑎 and 𝛬 are the parameters related to the beam waist, divergence angle of transmitting Gaussian beam and transmission distance (𝑑𝑖 ). The detailed derivation of 𝑎 and 𝛬 can be referred to reference [2]. 𝐷∕𝑟0 is always used to characterize the effect of turbulence on FSO beams where 𝐷 is the transmitting beam diameter. In the following, source–destination length 𝑑𝑠𝑑 is used to calculate 𝐷∕𝑟0 for the whole source–destination link.

𝐌∶ 𝐦𝐨𝐝𝐞 𝐜𝐨𝐧𝐯𝐞𝐫𝐬𝐢𝐨𝐧 𝐞𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐜𝐲 𝐨𝐟 𝐏𝐋

⎡ ×⎢ ⎢ ⎣

3.2. Dual-hop model with FM-EDFA and PL based all-optical relay To analyze this scheme, a dual-hop model with FM-EDFA and PL is given (Fig. 2(a)). Three fiber modes including LP0,1 , LP+1,1 and LP−1,1 mode (LP+1,1 = LP11a +iLP11b and LP−1,1 = LP11a -iLP11b are the complex notation of two degenerate LP11 modes) are received by FM all-optical relay. Moreover, the total received electric field at the relay is the sum of signal and background radiation electric fields 𝐸𝑟 (𝑟, 𝜃, 𝑡) =

√

2𝑥𝑃𝑡

3 ∑ 𝑚=1

𝑔 0 0

0 √ 𝑔 0

0 0 √ 𝑔

𝐈∶ 𝐢𝐧𝐬𝐞𝐫𝐭𝐢𝐨𝐧 𝐥𝐨𝐬𝐬 𝐨𝐟 𝐏𝐋

⎤ ⎥. ⎥ ⎦

(6)

𝐆∶ 𝐠𝐚𝐢𝐧 𝐨𝐟 𝐅𝐌−𝐄𝐃𝐅𝐀

𝑀 −1 is the mode √ conversion efficiency of PL as mode demultiplexer where 𝑎0 = 𝑎1 = 3∕3 when the mode dependent loss of PL is zero. 𝑔 and 𝛼𝑃 𝐿 represent the gain coefficient of FM-EDFA and the insertion loss of PL, respectively. The mode dependent gain of FM-EDFA is also set to be zero. For fixed gain amplification relaying system, the fixed amplification gain keeps the average output power of the relay limited at 𝑃𝑟 (Supposing 𝑃𝑟 = 𝑃𝑡 in the remainder of this paper.) and is expressed as:

ℜ{ℎ1,1𝑚 𝑒𝑗𝜙1,1𝑚 𝑒𝑗𝜔0 𝑡 𝜓𝑚 (𝑟, 𝜃)}

3 ∑ 𝑀 ∑ √ + 2𝑁𝑏 𝛿𝜈 ℜ{𝑒𝑗𝜙𝑙,𝑚 𝑒𝑗𝜔𝑙 𝑡 𝜓𝑚 (𝑟, 𝜃)},

√

(5)

𝑚=1 𝑙=−𝑀

𝑔𝑓 𝑖𝑥𝑒𝑑 =

where 𝑃𝑡 and 𝑥 ∈ {0, 2} denote the transmit power of the source and the transmitted OOK symbol. ℎ1,1𝑚 𝑒𝑗𝜙1,1𝑚 represent fading coefficients from transmitting LP01 mode to the 𝑚th fiber mode of FMF-EDFA in the first hop (𝑚 = 1 corresponds to LP01 , 𝑚 = 2 corresponds to LP+1,1 and 𝑚 = 3 corresponds to LP−1,1 , respectively.). 𝜓𝑚 (𝑟, 𝜃) is the spatial complex amplitude of the 𝑚th mode. 𝑁𝑏 = 𝑃𝑏 ∕𝐵𝑜 denotes the background radiation power spectral density, where 𝑃𝑏 and 𝐵𝑜 are the

𝑃𝑟 + 3𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 𝛼𝑃 𝐿 (𝑃𝑡 𝐸[𝛼1 ] + 3𝑃𝑏 + 3𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 )𝛼𝑃 𝐿

,

(7)

where 𝐸[𝛼1 ] is the average channel fading of first hop with 𝛼1 = ∑3 2 𝑚=1 ℎ1,1𝑚 . 3𝑃𝑏 represent the total background radiation power of three modes. ASE noise is introduced after FM-EDFA. 𝑛𝑠𝑝 is the amplifier spontaneous emission factor. ℎ denotes Planck‘s constant. The variable amplification gain keeps the output power of relay stabilized at constant 166

S. Cai, Z. Zhang and X. Chen

Optics Communications 439 (2019) 164–170

which denote the variance of relay, destination and relay–destination background–background beat noise, respectively. 𝐵𝑒 is the electronic bandwidth of the system.

value 𝑃𝑟 at all times and is given by 𝑔𝑣𝑎𝑟 =

𝑃𝑟 + 3𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 𝛼𝑃 𝐿 (𝑃𝑡 𝛼1 + 3𝑃𝑏 + 3𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 )𝛼𝑃 𝐿

.

(8)

ASE–ASE beat noise:

As we known the gain of EDFA can be adjusted by choosing pump power. In real applications, to keep the output power of EDFA at constant, the source–relay channel fading 𝛼1 , required in calculating the variable gain coefficient of FM-EDFA can be estimated at the relay based on pilot symbols emitted by the source. Then, the pump power of FM-EDFA is adjusted according to the known channel information. For the sake of brevity, in the remainder of this paper, 𝑔 represents both 𝑔𝑓 𝑖𝑥𝑒𝑑 and 𝑔𝑣𝑎𝑟 of the relay.

3 3 3 ∑ ∑ ∑ 2 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸 = 𝑅2 𝑁02 𝛼𝑃2 𝐿 ( ℎ42,𝑛1 + 4 ℎ22,𝑖1 ℎ22,𝑗1 )(2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ). 𝑛=1

(14) Signal–background beat noise: 3 ∑ 2 2 𝜎𝑠×𝑏,𝑟 = 4𝑅2 𝑥𝑃𝑡 |𝐻1 𝐻 𝑇 𝐻2𝑇 | 𝑁𝑏 𝐵𝑒 𝑔𝛼𝑃 𝐿 ( ℎ22,𝑛1 ),

The total electric field at the destination

2 𝜎𝑠×𝑏,𝑑

𝐸𝑑 (𝑡) = 𝐸𝑠 (𝑡) + 𝐸𝑏,𝑟 (𝑡) + 𝐸𝐴𝑆𝐸 (𝑡) + 𝐸𝑏,𝑑 (𝑡) √ = 2𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | cos(𝜔0 𝑡 + 𝜙𝑠 ) +

𝑛=1 𝑙=−𝑀

(9)

(16)

(17)

Background–ASE beat noise: 3 3 ∑ ∑ 2 ℎ22,𝑛1 )(2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ), 𝜎𝑏×𝐴𝑆𝐸 = 4𝑅2 𝑁𝑏 𝑁0 𝛼𝑃 𝐿 ( ℎ22,𝑛1 )(1 + 𝑔𝛼𝑃 𝐿 𝑛=1

𝑖𝑝ℎ𝑜𝑡𝑜 (𝑡) = 𝑅𝐸𝑑2 (𝑡)

Shot noise:

+ 2𝑞𝐼𝑠 𝐵𝑒 ,

(21)

for transmit symbol 𝑥 = 0, and 2 2 2 2 2 𝜎𝑜𝑛 = 𝜎𝑜𝑓 𝑓 + 2𝑞𝐼𝑠 𝐵𝑒 + 𝜎𝑠×𝐴𝑆𝐸 + 𝜎𝑠×𝑏,𝑟 + 𝜎𝑠×𝑏,𝑑 ,

(22)

2 2 with thermal noise variance 𝜎𝑡ℎ

for transmit symbol 𝑥 = = 4𝐾𝑇 𝐵𝑒 ∕𝑅𝐿 . 𝐾 is the Boltzmann constant, 𝑇 is the temperature in Kelvin, and 𝑅𝐿 is the photodetector load resistance. In general, the signal current 𝐼𝑠 and the noise variances in Eqs. (21)–(22) constitute the model for dual-hop transmission system with FM-EDFA and PL based all-optical relay. 3.3. Dual-hop model with SM-EDFA and OC based all-optical relay

2 𝜎𝑠×𝑏

ASE–ASE beat noise signal–background beat noise = 2 2 𝜎𝑠×𝑏,𝑟 +𝜎𝑠×𝑏,𝑑 , shot noise and .et al. also can be obtained and are expressed in the following.

To evaluate the advantages of the proposed all-optical relaying system based on FM-EDFA and PL, the system performance is compared with that of an all-optical relaying system based on SM-EDFA and optical coupler (OC) (Fig. 1(c)) and that of all-optical relaying system including FM-EDFA only. The dual-hop model only with FM-EDFA is given in Fig. 2(c) and have been analyzed in our recent work. In this section, the dual-hop model with SM-EDFA and OC is given (Fig. 2(b)). Different from FM-EDFA based relay, at this time, only the signal and radiation LP0,1 mode fields will be received and amplified by SM-EDFA. After OC, the amplified signal and radiation light power is equally distributed to three output SMFs and then forwarded to the destination using transmit

Background–background beat noise: (11)

𝑖=1 𝑗=1,𝑗≠𝑖

𝑅2 𝑁𝑏2 𝑔𝛼𝑃 𝐿 ℎ22,𝑛1 (2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ),

(20)

=

2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓

2 2 2 2 2 2 𝜎𝑜𝑓 𝑓 = 𝜎𝑡ℎ + 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓 + 𝜎𝑏×𝑏 + 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸 + 𝜎𝑏×𝐴𝑆𝐸 ,

1.6 × 10−19 denote the efficiency of PD and the charge of an electron, respectively. The useful signal terms and unwanted noise terms can be obtained by inserting Eq. (9) into Eq. (10). Following the approach in reference [9,23], the expressions of signal direct current term 𝐼𝑠 = 𝑅𝑥𝑃𝑡 |𝐇1 𝐻 𝑇 𝐻 𝑇2 |2 , background–background direct current term ∑ at relay 𝐼𝑏×𝑏,𝑟 = 𝑅𝑁𝑏 𝐵𝑜 𝑔𝛼𝑃 𝐿 3𝑛=1 ℎ22,𝑛1 , ASE–ASE direct current term ∑3 𝐼𝐴𝑆𝐸×𝐴𝑆𝐸 = 𝑅𝑁0 𝐵𝑜 𝛼𝑃 𝐿 𝑛=1 ℎ22,𝑛1 and background–background direct current term at destination 𝐼𝑏×𝑏,𝑑 = 𝑅𝑁𝑏 𝐵𝑜 are given. The variance 2 2 2 2 of background–background beat noise 𝜎𝑏×𝑏 = 𝜎𝑏×𝑏,𝑟 + 𝜎𝑏×𝑏,𝑑 + 𝜎𝑏,𝑟×𝑑 ,

2 𝜎𝑏×𝑏,𝑑 = 𝑅2 𝑁𝑏2 (2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ),

(19)

2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑛

symbol 𝑥 = 0 and transmit symbol 𝑥 = 2, respectively. In addition to the noise terms discussed above, the impact of thermal noise of PD at the destination is also important. The total noise variance of relaying system is

denotes the responsivity of the PD, 𝜌 and 𝑞 =

3 3 3 ∑ ∑ ∑ 2 𝜎𝑏×𝑏,𝑟 = 𝑅2 𝑁𝑏2 𝑔 2 𝛼𝑃2 𝐿 ( ℎ42,𝑛1 + 4 ℎ22,𝑖1 ℎ22,𝑗1 )(2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ),

2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓 = 2𝑞(𝐼𝑏×𝑏,𝑟 + 𝐼𝑏×𝑏,𝑑 + 𝐼𝐴𝑆𝐸×𝐴𝑆𝐸 )𝐵𝑒 ,

2 2 and 𝜎𝑠ℎ𝑜𝑡,𝑜𝑛 denote the shot noise variances for transmit where 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓

(10)

2 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸 ,

(18)

𝑛=1

which is composed of beat noises between ASE and relay–background radiation and between ASE and destination–background radiation.

At the destination, after photodetector (PD), the optical signal is converted into electrical signal. The expression of the photocurrent is

2 𝜎𝑏,𝑟×𝑑 =4

(15)

𝑛=1

𝑙=−𝑀

3 ∑

𝑛=1

3 ∑ 2 2 𝜎𝑠×𝐴𝑆𝐸 = 4𝑅2 𝑥𝑃𝑡 |𝐻1 𝐻 𝑇 𝐻2𝑇 | 𝑁0 𝐵𝑒 𝛼𝑃 𝐿 ( ℎ22,𝑛1 ).

is the sum of signal light 𝐸𝑠 , background radiation light at relay 𝐸𝑏,𝑟 , ASE noise 𝐸𝐴𝑆𝐸 and background radiation light at destination 𝐸𝑏,𝑑 . ASE noise is also treated as a sum of 2𝑀 + 1 frequency terms like background radiation noise. 𝑁0 = 𝑛𝑠𝑝 (𝑔 − 1)ℎ𝜈0 is the spectral density of the ASE noise. 𝐇1 = [ℎ1,11 𝑒𝑗𝜙1,11 , ℎ1,12 𝑒𝑗𝜙1,12 , ℎ1,13 𝑒𝑗𝜙1,13 ], 𝐇2 = ∑3 ∑3 [ℎ2,11 𝑒𝑗𝜙2,11 , ℎ2,21 𝑒𝑗𝜙2,21 , ℎ2,31 𝑒𝑗𝜙2,31 ] and 𝐇1 𝐻 𝑇 𝐻 𝑇2 = 𝑛=1 𝑚=1 𝑗𝜙1,1𝑚 𝑇 𝑗𝜙2,𝑛1 ℜ{ℎ1,1𝑚 𝑒 𝐻𝑚,𝑛 ℎ2,𝑛1 𝑒 } represent channel fading (Fig. 2(a)). ℎ2,𝑛1 𝑒𝑗𝜙2,𝑛1 denote the mutual independent fading coefficients from the 𝑛th transmitter at relay to the single receiver at destination. H denotes the transfer matrices of all-optical relay with FM-EDFA and PL. Superscript T represents matrix transposition. 𝜙𝑟,𝑙,𝑛 and 𝜙𝑑,𝑙 are mutual independent random phase.

𝑛=1

2 𝐻2𝑇 | 𝑁𝑏 𝐵𝑒 ,

Signal–ASE beat noise:

𝑀 ∑ √ + 2𝑁𝑏 𝛿𝜈 cos(𝜔𝑙 𝑡 + 𝜙𝑑,𝑙 ),

where 𝑅 =

= 4𝑅 𝑥𝑃𝑡 |𝐻1 𝐻

𝑇

and relay background radiation and that of beat noise between signal and destination background radiation, respectively.

𝑛=1 𝑙=−𝑀

𝜌𝑞 ℎ𝜈0

2

2 2 where 𝜎𝑠×𝑏,𝑟 and 𝜎𝑠×𝑏,𝑑 denote the variance of beat noise between signal

3 ∑ 𝑀 ∑ √ 2𝑁𝑏 𝛿𝜈𝑔𝛼𝑃 𝐿 ℎ2,𝑛1 cos(𝜔𝑙 𝑡 + 𝜙𝑟,𝑙,𝑛 )

3 ∑ 𝑀 ∑ √ + 2𝑁0 𝛿𝜈𝛼𝑃 𝐿 ℎ2,𝑛1 cos(𝜔𝑙 𝑡 + 𝜙𝐴𝑆𝐸,𝑙,𝑛 )

𝑖=1 𝑗=1,𝑗≠𝑖

(12) (13)

𝑛=1

167

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Optics Communications 439 (2019) 164–170 Table 3 System parameters.

diversity. The purpose of adding OC is to compare with PL. The transfer matrices of SM-EDFA and OC are expressed as: √

⎡ 3𝑔𝛼𝑂𝐶 ⎤ ⎢√ 3 ⎥ 𝐇 = ⎢ 3𝑔𝛼𝑂𝐶 ⎥ , ⎢√ 3 ⎥ ⎢ 3𝑔𝛼𝑂𝐶 ⎥ ⎣ 3 ⎦

(23)

where 𝛼𝑂𝐶 denotes the insertion loss of OC. For fixed and variable gain amplifications, the gain coefficients are expressed as 𝑔𝑓 𝑖𝑥𝑒𝑑 = 𝑃𝑟 +𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 𝛼𝑂𝐶

and 𝑔𝑣𝑎𝑟 =

(𝑃𝑡 𝐸[𝛼1 ]+𝑃𝑏 +𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 )𝛼𝑂𝐶 𝛼1 = ℎ21 . The total electric

𝑃𝑟 +𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 𝛼𝑂𝐶

(𝑃𝑡 𝛼1 +𝑃𝑏 +𝑛𝑠𝑝 ℎ𝜈0 𝐵𝑜 )𝛼𝑂𝐶

, respectively with

field at the destination is 𝐸𝑑 (𝑡) = 𝐸𝑠 (𝑡) + 𝐸𝑏,𝑟 (𝑡) + 𝐸𝐴𝑆𝐸 (𝑡) + 𝐸𝑏,𝑑 (𝑡)

Parameter

Value

Wavelength (𝜆) Electrical bandwidth (𝐵𝑒 ) Optical bandwidth (𝐵𝑜 ) Data rate (𝑅𝑏 ) Amplifier spontaneous emission factor (𝑛𝑠𝑝 ) Insertion loss of PL (𝛼𝑃 𝐿 ) Insertion loss of OC (𝛼𝑂𝐶 ) Receiver noise temperature (𝑇 ) Receiver quantum efficiency (𝜌) Photodetector load resistance (𝑅𝐿 ) Background radiation power spectral density (𝑁𝑏 ) Photodetector load resistance (𝑅𝐿 )

1550 nm 2 GHz 125 GHz 2 Gbps 1.4 2 dB 2 dB 300 K 0.75 50 Ω 1.6 × 10−19 W/Hz 50 Ω

𝑀 ∑ √ √ = 2𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | cos(𝜔0 𝑡 + 𝜙𝑠 ) + 2𝑁𝑏 𝛿𝜈|𝐇𝑇 𝐇𝑇2 | cos(𝜔𝑙 𝑡 + 𝜙𝑟,𝑙 ) 𝑙=−𝑀 𝑀 ∑ √ + 2𝑁0 𝛿𝜈∕𝑔|𝐇𝑇 𝐇𝑇2 | cos(𝜔𝑙 𝑡 + 𝜙𝐴𝑆𝐸,𝑙 ) 𝑙=−𝑀 𝑀 ∑ √ + 2𝑁𝑏 𝛿𝜈 cos(𝜔𝑙 𝑡 + 𝜙𝑑,𝑙 ), 𝑙=−𝑀

(24) where 𝐇1 = [ℎ1 𝑒𝑗𝜙1 ], 𝐇2 = [ℎ2,11 𝑒𝑗𝜙2,11 , ℎ2,21 𝑒𝑗𝜙2,21 , ℎ2,31 𝑒𝑗𝜙2,31 ] (Fig. 2(b)). After PD, the optical signal is converted into electrical signal. The expression of the photocurrent is 𝑖𝑝ℎ𝑜𝑡𝑜 (𝑡) = 𝑅𝐸𝑑2 (𝑡). The useful signal terms and unwanted noise terms can be obtained by substituting Eq. (24) and are expressed as follows: 2

𝐼𝑠 = 𝑅𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | , 𝐼𝑏×𝑏,𝑟 =

(25)

2 𝑅𝑁𝑏 𝐵𝑜 |𝐇𝑇 𝐇𝑇2 | ,

(26)

𝐼𝑏×𝑏,𝑑 = 𝑅𝑁𝑏 𝐵𝑜 ,

(27)

2 𝐼𝐴𝑆𝐸×𝐴𝑆𝐸 = 𝑅𝑁0 𝐵𝑜 |𝐇𝑇 𝐇𝑇2 | ∕𝑔, 4 2 𝜎𝑏×𝑏,𝑟 = 𝑅2 𝑁𝑏2 |𝐇𝑇 𝐇𝑇2 | (2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 ),

(28)

2 𝜎𝑏×𝑏,𝑑

(30)

2 𝜎𝑏,𝑟×𝑑

= =

2

𝑅 − 𝐵𝑒2 ), 2 4𝑅2 𝑁𝑏2 |𝐇𝑇 𝐇𝑇2 | (2𝐵𝑒 𝐵𝑜

− 𝐵𝑒2 ),

4 = 𝑅 𝑁02 |𝐇𝑇 𝐇𝑇2 | (2𝐵𝑒 𝐵𝑜 − 𝐵𝑒2 )∕𝑔 2 . 2 2 2 𝜎𝑠×𝑏,𝑟 = 4𝑅2 𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | |𝐇𝑇 𝐇𝑇2 | 𝑁𝑏 𝐵𝑒 , 2 2 𝜎𝑠×𝑏,𝑑 = 4𝑅2 𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | 𝑁𝑏 𝐵𝑒 , 2 2 2 𝜎𝑠×𝐴𝑆𝐸 = 4𝑅2 𝑥𝑃𝑡 |𝐇1 𝐇𝑇 𝐇𝑇2 | |𝐇𝑇 𝐇𝑇2 | 𝑁0 𝐵𝑒 ∕𝑔. 2 2 2 𝜎𝑏×𝐴𝑆𝐸 = 4𝑅2 𝑁𝑏 𝑁0 |𝐇𝑇 𝐇𝑇2 | (1 + |𝐇𝑇 𝐇𝑇2 | )(2𝐵𝑒 𝐵𝑜 2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓 = 2𝑞(𝐼𝑏×𝑏,𝑟 + 𝐼𝑏×𝑏,𝑑 + 𝐼𝐴𝑆𝐸×𝐴𝑆𝐸 )𝐵𝑒 , 2 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸

2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑛

=

(29)

𝑁𝑏2 (2𝐵𝑒 𝐵𝑜

(31)

2

2 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓

(32) (33) (34) (35) − 𝐵𝑒2 )∕𝑔.

+ 2𝑞𝐼𝑠 𝐵𝑒 .

(36) (37) (38)

Fig. 3. The statistical distribution of 10 000 values of Channel fading at (a) relay and (b) the destination under 𝐶𝑛2 = 5 × 10−14 , 𝛼𝑎𝑡𝑡𝑛 = 0 dB∕km for the relaying systems with SM-EDFA and OC, with FM-EDFA and PL and with FM-EDFA only.

Finally, The total noise variance for transmit symbol 𝑥 = 0 2 2 2 2 2 2 𝜎𝑜𝑓 𝑓 = 𝜎𝑡ℎ + 𝜎𝑠ℎ𝑜𝑡,𝑜𝑓 𝑓 + 𝜎𝑏×𝑏 + 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸 + 𝜎𝑏×𝐴𝑆𝐸 ,

(39)

2 = 4𝐾𝑇 𝐵 ∕𝑅 ), shot is the sum of the variance of thermal noise (𝜎𝑡ℎ 𝑒 𝐿 2 = 𝜎2 2 2 noise, background–background beat noise (𝜎𝑏×𝑏 +𝜎𝑏×𝑏,𝑑 +𝜎𝑏,𝑟×𝑑 ), 𝑏×𝑏,𝑟 ASE–ASE beat noise and background-ASE beat noise. The total noise variance for transmit symbol 𝑥 = 2 2 2 2 2 2 2 2 2 , 𝜎𝑜𝑛 = 𝜎𝑡ℎ + 𝜎𝑠ℎ𝑜𝑡,𝑜𝑛 + 𝜎𝑏×𝑏 + 𝜎𝐴𝑆𝐸×𝐴𝑆𝐸 + 𝜎𝑏×𝐴𝑆𝐸 + 𝜎𝑠×𝑏 + 𝜎𝑠×𝐴𝑆𝐸

4. Numerical results To evaluate the advantage of relaying system based on FM-EDFA and PL. System performance is studied at bit-rates (BR) 2 Gbps with on–off keying (OOK) and direct detection. Other system parameters are given in Table 3. In the presence of atmospheric turbulence, fading coefficients ℎ1,1𝑚 𝑒𝑗𝜙1,1𝑚 and ℎ2,𝑛1 𝑒𝑗𝜙2,𝑛1 in H1 and H2 are obtained through calculating the correlation coefficients between the received signal/background light field and fiber mode fields at the relay and receiver. With the aid of 10 000 turbulence screens generated by Fourier Transform method, the 10 000 sets of values for H1 and H2 are calculated over 𝑁 = 10 000 fading states. The statistical distribution of 10 000

(40)

is the sum of the variance of thermal noise, shot noise, background– 2 2 background beat noise, signal–background beat noise (𝜎𝑠×𝑏 = 𝜎𝑠×𝑏,𝑟 + 2 𝜎𝑠×𝑏,𝑑 ), signal-ASE beat noise and .et al. In general, the dual-hop transmission system with SM-EDFA and OC based all-optical relay is composed of signal current in Eq. (25) and the noise variances in Eqs. (39)–(40). 168

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Fig. 5. BER vs. transmitting power for FM-EDFA with different MDGs under weak turbulence and haze (𝐶𝑛2 = 5 × 10−14 , 𝛼𝑎𝑡𝑡𝑛 = 4.2 dB∕km) in fixed gain amplification system with FM-EDFA and PL.

at FM-EDFA based relay is narrower compared with SM-EDFA based relay (Fig. 3(a)). The average receiving power is increased by 2.15 dB and the RSD is restrained by 71%, respectively. In addition, the statistical distribution of channel fading of the whole link for relaying system with FM-EDFA and PL is the narrowest case compared with relaying system with SM all-optical relay and relaying system with FM relay including FM-EDFA only (Fig. 3(b)). The average receiving power is 0.44 dB higher than that of relaying system with SM relay and 0.86 dB higher than that of relaying system with FM-EDFA based relay. The RSD is restrained by 81.5% compared with SM relay and 83.7% compared with FM relay including FM-EDFA only. The FM-EDFA and PL based all-optical relay characterizes higher receiving power and lower fading RSD which is expected to bring improved BER. Here, the BER of relaying system with FM-EDFA and PL is compared with that of relaying system with SM-EDFA and OC and that of relaying system including FM-EDFA only. The BER is averaged over 10 000 different fading states with 2 × 106 bits transmitted per fading state. Blind detection is utilized as the detection approach [24]. Under weak to moderate atmospheric turbulence, the BER vs. transmitting power (𝑃𝑡 ) are shown in Fig. 4. Obviously, compared with relaying system with SM-EDFA and OC and with FM-EDFA only, BER of relaying system based on FM-EDFA and PL is improved for both fixed gain (red line) and variable gain amplifications (pink line). Under 𝐶𝑛2 = 2 × 10−14 (𝐷∕𝑟0 = 1.0934) and clear air 𝛼𝑎𝑡𝑡𝑛 = 0.43 dB∕km, 4-dB improvement in power budget is obtained at 𝐵𝐸𝑅 = 1 × 10−3 for fixed gain amplification (Fig. 4(a)). With the increase of turbulence level, the promotion of power budget is more obvious. 6-dB and 5-dB increase of power budget are realized under 𝐶𝑛2 = 5 × 10−14 (𝐷∕𝑟0 = 1.8948) and 𝐶𝑛2 = 1 × 10−13 (𝐷∕𝑟0 = 2.872) respectively compared with relaying system with FMEDFA only (Fig. 3(b–c)). Mode-dependent gain (MDG) of FM-EDFA is also considered for the relaying system with FM-EDFA and PL. The results are shown in Fig. 5 under refractive index structure constant 𝐶𝑛2 = 5 × 10−14 (𝐷∕𝑟0 = 1.8948) and haze 𝛼𝑎𝑡𝑡𝑛 = 4.2 dB∕km. The BER performance is the best when 𝑀𝐷𝐺 = 0 and becomes worse with the increase of MDG. The BER is worst when SM-EDFA is utilized as relay which corresponds to the case of 𝑀𝐷𝐺 = ∞.

Fig. 4. BER vs. transmitting power under (a) weak turbulence and clear air (𝐶𝑛2 = 2×10−14 , 𝛼𝑎𝑡𝑡𝑛 = 0.43 dB∕km), (b) weak turbulence and haze (𝐶𝑛2 = 5 × 10−14 , 𝛼𝑎𝑡𝑡𝑛 = 4.2 dB∕km) and (c) moderate turbulence and clear air (𝐶𝑛2 = 1 × 10−13 and 𝛼𝑎𝑡𝑡𝑛 = 0.43 dB∕km) for the fixed and variable gain amplification relaying systems with SM-EDF and OC, with FM-EDFA and PL or with FM-EDFA only.

5. Conclusion ∑3

values of channel fading at relay 𝛼1 (𝛼1 = 𝑚=1 ℎ21,1𝑚 ) and channel fading of the whole link (|𝐇1 𝐻 𝑇 𝐻 𝑇2 |2 ) are shown in Fig. 3 under 𝐶𝑛2 = 5×10−14 , 𝛼𝑎𝑡𝑡𝑛 = 0 dB∕km. Obviously, the statistical distribution of channel fading

An all-optical relaying system based on FM-EDFA and PL is proposed in this paper to resist atmospheric turbulence. At the relay, using FMEDFA and PL leads to the improvement of the turbulence induced 169

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Optics Communications 439 (2019) 164–170

fading which brings improved BER. Atmospheric turbulence is modeled with turbulence phase screen. A model for dual-hop FSO system with FM-EDFA and PL is derived and the BER performance is given. Compared with relaying system based on SM relay and FM relaying system including FM-EDFA only, the BER of FM relaying system with FM-EDFA and PL is always better for the fixed gain or variable gain amplification.

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