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The inter-modes mixing effects in Mode Group Diversity Multiplexing
Optics Communications 282 (2009) 3908–3917
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The inter-modes mixing effects in Mode Group Diversity Multiplexing M. Awad, I. Dayoub, A. Okassa M’Foubat, J.M. Rouvaen * Univ Lille Nord de France, F-59000 Lille, France IEMN (UMR 8520 CNRS), Department of OAE, UVHC F- 59313 Valenciennes, France
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 26 June 2009 Accepted 26 June 2009
PACS: 42.79.Sz 42.81.Uv 42.81.i 42.81.Ht
a b s t r a c t Multiplexing technology in indoor optical networks with multimode fibres allows the integration of several services to the end user. The Mode Group Diversity Multiplexing technique (MGDM) in Graded-Index Multi-Mode (GI-MMF) fibre makes the system less expensive with simpler transmitters and receivers, keeping the same information capacity as other multiplexing techniques. The capacity conservation and transmission quality obtained using MGDM technique depends on the transmission conditions and the state of fibre. Bending of the fibre can affect the system by changing modes excited for the different propagation channels in the fibre. In this paper an analytical modelling method for MGDM will be presented. Improvements in system sending and receiving conditions are studied. Modelling of the fibre curvature as well as the effects of coupling patterns on the MGDM are discussed. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Inter-modes mixing Multi Services Interferences (MSI) Multimode fibre links Optical fibre communication Optical MGDM
1. Introduction In optical communication systems, especially for indoor networks, multimode fibre (MMF) links have been selected as the basic infrastructure [1]. They can provide the bandwidth required for short distance communications at much lower expense than single-mode fibres (SMF), mainly due to the ease of installation. However the multiplicity of optical propagation modes in MMF and waveguides has been considered as a huge disadvantage. It produces the intermodal dispersion that limits the distance-bandwidth product of optical communication systems [2]. Several searches were made to weaken this major inconvenient by designing and developing cost-effective methods to increase the bit rate (>1 Gb/s) at large fibre lengths (’1 km) [3]. On the other hand, there is currently a wide variety of wireline networks, each optimised for transporting a particular set of services in indoor networks (Voice telephony, Internet, etc.). But these networks do not allow the integration of new broadband services. Optical fibre is the best candidate for a shared network enabling several services by using multiplexing techniques. In addition, it al-
* Corresponding author. E-mail address: [email protected] (J.M. Rouvaen). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.065
lows to deliver radio-communication signals to the end user. Several multiplexing techniques have been studied for multimode fibre networks. Wavelength Division Multiplexing (WDM) further extends the network capacity and enables the integration of different services in a single-MMF network allowing the optimal use of the optical bandwidth. This technique is limited by the ability to generate the adequate optical carriers and the correct data transmission over acceptable distances. Indeed, WDM requires wavelength-specific sources in addition to wavelength-selective network functions, which are still quite costly [4]. Optical CodeDivision Multiple-Access (OCDMA) is applied for SMF fibre networks. But, for indoor networks, coding services complicates the system and requires high-speed lasers emitters. Moreover, the two techniques (WDM & CDMA) do not take into consideration the intermodal dispersion, and system performance is still limited to low bit rates and short distances. Mode Group Diversity Multiplexing (MGDM) is a multiplexing technique, based on the spatial launching and detection of subgroups of modes to create a number of independent communication channels in a single-MMF. This technique derives from a recently developed wireless transmission scheme known as BLAST (Basic Local Alignment Search Tool) and has been introduced recently to replace the more expensive WDM in LAN networks [5]. This technique exploits the unused capacity of MMF for improving
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the bandwidth times fibre length product. The excitation of sub-groups of modes decreases the intermodal dispersion; an up to fourfold bandwidth increase has been reported by exciting less than 50% of the fibre modes [3]. Each mode group (MG) launched at the input facet of the fibre constitutes a quasi-separate channel associated with a service or a user and, in the end facet of the fibre, this MG may be detected separately [6]. Several applications have demonstrated the feasibility of this technique with liable input and output couplers, that do not complicate transmission services on the MMF fibre [7,8]. To create N propagations channels, N lasers are followed by N virtual SMFs placed at different offsets in input facet of the MMF fibre. The receiver can be achieved by a lens placed on output facet of the fibre, for the projection of the Near Field Pattern (NFP) onto an optoelectronic integrated circuit (OIC) which contains photo-detectors and preamplifiers (PDIC) [9]. However, the MG propagation in each channel is not ideal, subgroups mode mixing and non-orthogonality of transmitters and receivers cause crosstalk between propagation channels producing Multi Services Interferences (MSI). It reduces the capacity of MMF. The improved utilization of the MMF transmission capacity is related to the reduction of the mixing effects between modes and the amelioration of the transmission/reception conditions. The conditions for transmission and reception play a key role in improving the fibre capacity. The way of excitation (offset, spot size, wavelength) determines the excited mode groups associated with each channel. The choice of optimal conditions reduces the common modes (mode coupling) excited between channels. Moreover, improving the spatial reception reduces the MSI. In this paper, the optimal conditions for transmission and reception are studied. Mode mixing is the gradual redistribution of the optical power among the propagating modes. It occurs between modes of the same MG group, which is called intra-mode mixing, or between modes of different MG groups, which is called inter-mode mixing. Inter-mode mixing is the major effect in MMF transmission, apart from external factors acting on the fibre. The modelling method for intra-mixing is presented in [9], by evenly redistributing the optical power among the modes of the MG. It is known that a perfect MMF transmits a guided mode without any power conversion to another guided mode [10]. Indeed, the installation of optical infrastructure in buildings to the end user induces deformations (fibre not straight) and the guided mode propagation is not perfect. However, any perturbation to the index of refraction, which may include a bend or curvature of the optical waveguide, leads to an imperfect light trapping in the fibre. In this case, a conversion of the energy of a mode to other higher order modes occurs and the light loses energy in the cladding and radiates off the guide. This conversion (inter-mode mixing) changes the propagation delay and increases the transmission losses. In this paper we take into account the curvature of the fibre only. In simple terms, installers know that the signal loss is moderate for a large radius of curvature, and becomes increasingly important when this radius decreases. This drawback becomes obvious as optical networks are closer to end users. A small radius of curvature of the fibre affects the MGDM and inter-mode mixing becomes more important than intra-mode mixing. The MSI increases with the number of channels in the MGDM systems. The fibre curvature acts more on the channels excited to offsets that are closest to the cladding. Several models of the fibre curvature were proposed, but these models are mainly concerned in the power loss and attenuation. In this paper, we model the curvature of the fibre in terms of coupling between modes and we present the effects of this curvature on the MGDM system, according to the radii of curvature and the number of channels defined in the fibre.
MGDM with multiple inputs and multiple outputs is similar to MIMO (Multiple Input Multiple Output) systems in radio-communications. The digital signal processing (DSP) used in radio networks to mitigate the effects of multiple channels can be used to mitigate the MSI in optical networks MGDM system at the receiver. Yet, efficient algorithms and hardware implementations facilitates the separation of services after the optoelectronic conversion. For MGDM, the simplest receiver architecture is matrix inversion, a Zero-Forcing (Z-F) equalizer in line with the requirement of service transparency [11]. The paper is organized as follows: In Section 1, MGDM technique description is given, followed by an analytical system modelling. In Section 2, we review the conditions of emission and reception and we present optimum conditions for the system. In Section 3, we study the modelling of the fibre curvature in terms of mode mixing, and we detail the effects of inter-mode mixing. In Section 4, we present an example (3 3) MGDM system with simulation results for performance. Finally, the discussion in Section 4 provides perspective on the key points. 2. MGDM technique description and analytical modelling The MGDM approach is based on the idea to transmit independent digital data signals using different mode groups on the MMF. Therefore, it is necessary to send a signal for each MG group excited at a given offset. For example, in a (2 2) MGDM system, the first signal is launched into the centre of the core fibre by exciting lower order modes (LM) (in the ideal case the lowest mode will carry most of the energy). The second signal is launched at the core fibre extremity by exciting the higher order modes (HM). During the transmission through the MMF fibre, lower order modes stay mostly in the inner region of the core. The second signal excites higher order modes in the offset launch position, which are mainly travelling in the outer region of the MMF core. At the receiver end, the light energy is concentrated in a circular zone at the middle of the fibre core for the first condition launch. But the light energy to the second launch condition (for HM) is distributed in a ring in the outside region of the core [12]. Fig. 1 gives an example to the MGDM system architecture for 2 transmitters and the light distribution at the end facet. 2.1. Analytical modelling In radio-communication field, MIMO systems have received significant research interest. The relationship between transmitters and receivers is analytically presented in matrix form. This matrix describes the multipath channel (attenuation, phase shift, etc.). From a mathematical point of view, the optical transmission MGDM system, which realizes the multipath propagation in the MMF fibre may have a similar representation that the MIMO radio. The relationship between the N received electrical signals (yi ) and the N emitted electrical signals (si ) is written in the form:
Laser
SMF Core/MMF
End facet
F a
w d
Fig. 1. Mode multiplexing process in the input facet, and detection area at output facet of MMF.
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0
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1
0
h11 By C B B B 2 C B h21 B C¼B B .. C B . @ . A @ ..
h12 .. . s
s ..
hN1
hN2
s
y1
yN
s
1 0 1 n1 s1 CB C B C C s n h2N CB 2 C B 2 C B C þ B C; .. C CB .. C B .. C . A@ . A @ . A
h1N
.
10
hNN
sN
The modal scalar fields are governed by the scalar Helmholtz equation:
ð1Þ
nN
al;m ðzÞ ¼ al;m ð0Þ expðcl;m zÞ;
y ¼ Hs þ n;
ð2Þ
where y, s, n are, respectively, the received signal vector, the emitted signal vector and the additive noise from receivers. The matrix elements hij describe the signal transfer from the transmitter i to the detector j. Each of them has an amplitude (representing the path mitigation (i; j)) and a phase (corresponding to the delay of the specific path (i; j)). The fibre dispersion is negligible in the case of MGDM, so that the matrix elements take on real values, expressing the proportion of power transmitted by the ith source and received by the jth detector [13]. Assuming a linear superposition of N power distributions in the fibre output, hij can be expressed by:
Ij ðSi ; LÞ ; Ij ðS; LÞ
ð3Þ
where Ij is the light flux intensity emitted by the jth transmitter, Si is the area of jth receiver, S is the total area of core fibre and L is the fibre length (Fig. 2). The light flux intensity is given by
Z 1X IðS; LÞ ¼ jal;m ðLÞj2 W2l;m ds 2 l;m S Z X al;m ðLÞal0 ;m0 ðLÞ Wl;m Wl0 ;t0 ds þ l – l0 m – m0
h i cos bl;m bl0 ;m0 L ;
al;m ð0Þ ¼
ZZ
Ein ðx; yÞWl;m ðx; yÞ uz dx dy:
ð9Þ
Ein being the electric field of the SMF. uz the unit vector in a z-directed MMF fibre. We make the assumption that the incident field at the MMF input is a Gaussian field representing the SMF fibre to radial offset (F), angular offset (h) and spot size (w) (Fig. 3). The field components of the input Gaussian beam can be expressed in Cartesian coordinates as follows:
pffiffiffi h i 2 Ein ðx; yÞ ¼ pffiffiffiffiffiffiffiffi exp ðx FÞ2 y2 =w2 exp ðik1 hyÞ; pw
ð10Þ
where k1 ¼ n k0 . k0 ¼ 2p=k, k standing for the wavelength. The Eqs. (8) and (9) allow us to determine the magnitude of the modes excited at offsets and the distribution of the modal power, which is given by: m1 X
jal;ml1 j2 ¼ f ðF; w; w0 ; hÞ:
ð11Þ
l¼0
IðS; LÞ ¼ I0 þ q0 ;
ð5Þ
where al;m and Wl;m are respectively the modal amplitude and the modal function of the (l; m) mode. bl;m being the propagation constant. The intensity I is broken into two parts: I0 is the sum of the intensity distributions due to each mode separately, and q0 expresses the change in the whole intensity distribution due to interfering modal fields. The signal propagation along the z axis of the fibre MMF is the superimposition of several harmonic terms. The plane wave electromagnetic field mode (l; m) is described in Cartesian coordinates (x; y; z) by:
X
ð8Þ
with cl;m the attenuation coefficient of the (l; m) mode, defined in [14]. The excited modes at the fibre input are determined by the modal amplitude coefficients al;m . They are determined by the overlap integral [15]:
Pm ð0Þ ¼
S
ð4Þ
Eðx; y; zÞ ¼
ð7Þ
The modal amplitude al;m ðzÞ in the absence of mode mixing is defined by:
or in matrix notation:
hij ¼
r2t Wl;m þ k20 n2 Wl;m ¼ bl;m Wl;m :
h i al;m ðzÞWl;m ðx; yÞ exp jðbl;m zÞ ;
ð6Þ
l ;m
We define the group index m ¼ l þ m þ 1, which represents all modes that have almost the same propagation constant. The characteristic spot size w0 for the fundamental mode is given by:
w0 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a pffiffiffiffiffiffiffi; k0 n0 2D
ð12Þ
where k0 is the wave number in free space, n0 is the core peak index, D is the refractive-index contrast and a is the fibre core radius. All excited modes change with the offset (F), the spot size (w), the incident angle (h) and the characteristics of MMF fibre. Fig. 4 shows an example of modes excited and power distribution in a GI-MMF (62.5/125) fibre for several F and w values, normal injection (h ¼ 0), at a wavelength k ¼ 850 nm. The calculation of the modal scalar field Wl;m and the modal amplitudes al;m , for excited modes by using Eqs. (7) and (9), allows us to determine the light flux intensity given by (5). From the
l and m being the zeros of electric field distribution mode in the x and y directions, respectively.
y
z
F
s1 s2 s3 2w
Fig. 2. End facet fibre area.
x
Fig. 3. Geometry of excitation of the MMF fibre by a Gaussian field.
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
while the channel receiver to F ¼ 13 lm is an annular area with 7 lm < r < 18$ lm. The limiting values are selected here so that the major part of light intensity emitted by the transmitter i will be received in the corresponding area i. The matrix to match the MGDM (3 3) is given by:
20 F = 10µm, w = 6µm F = 20µm, w = 3µm F = 20µm, w = 6µm F = 10µm, w = 3µm
P (%) m
15
2
3 0:8397 0:2293 0:1042 6 7 H ¼ 4 0:1595 0:7545 0:3427 5:
10
0:0008
5
0:0162 0:5531
2.2. Characterization of MGDM channel
0 0
10 20 Mode groups (m)
30
Fig. 4. Mode power distribution versus geometry of excitation.
determination of the intensity I in each area Si , we can deduce the coefficients of the matrix H, which defines the MGDM channel. Owing to the difficulty of determining the coefficients of the matrix H analytically, we use a numerical averaging to calculate the coefficients hi;j . We are interested in the Graded-Index fibre (GI-MMF), and especially to standards 62.5/125 and 50/125. These two types of fibres use the 850 and 1300 nm transmission windows. In our work, we consider the case of parabolic fibre with quadratic distribution (a ¼ 2). In this case the scalar field Wl;m is obtained based on the Gauss–Hermite (GH) expression. For a (33) MGDM system built on a GI-MMF (62.5/125) and offsets F ¼ 0; 13 and 26 lm, the modes excited in each channel are determined and their modal amplitudes computed, with the spot size w ¼ 4lm and the wavelength k ¼ 850 nm. The light intensity is computed in a determined area for each channel at the fibre output. Fig. 5 shows the intensity distribution of light flux of the three channels in the plane perpendicular to the axis of the fibre propagation with L ¼ 100 m. The determination of the matrix coefficients hi;j is linked to the intensity distribution of light flux in three separate surfaces, each corresponding to a channel determined by its excitation offset. For the fundamental channel (F ¼ 0 lm), the area of reception is circular and centered on the fibre axis with a radius r1 ¼ 7 lm,
0.1
channel #3, F = 26µm channel #2, F = 13µm
0.08
r
channel #1, F = 0µm
2
Normalized Intensity
3911
0.06
r
1
0.04
0.02
0 0
10 20 Distance from fibre axis (µm)
30
Fig. 5. The light intensity distribution for three excitation positions (F = 0, 13, 26 lm, and L = 100 m).
Channel State Information (CSI) is information on the current value of the matrix H. This is a mathematical value that represents the channel signal. It is part of the model signal for MIMO radio systems. Applications authorizing the establishment of a feedback receiver to the transmitter allow the state channel return (CSI). Then it is possible to fully exploit this information to optimize overall transmission by anticipating the effect of channel. It is known that the channel capacity for a channel matrix realization H is given by:
q C ¼ B log2 det INR þ s HH> ðbits=secÞ; NT
ð13Þ
where B is the channel bandwidth (Hz), NT the number of transmitters, NR the number of receivers and the mean SNR. The study of the MGDM channel characteristic is similar to that of MIMO radio channel. The (13) is used to determine the status of the MGDM channel. The temporal and spatial static properties of the matrix elements H affect the capacity of the MGDM link. Although these two statistics have been studied and modelled for MIMO radio channels, they are not well characterized for optical links. For temporal properties, MGDM channel can be regarded as constant over a relatively long period (hours). Studies have demonstrated the modification of the coefficients matrix H for long periods [16]. In this case, a feedback channel to the transmitter is used to redefine the channel matrix H. Regarding the spatial properties, the excitation conditions to transmitter (radial/angular offset, wavelength, spot size, etc.) act on the coefficients of matrix H and on the link capacity. Moreover, the fibre state is the factor which affects the most the quality and capacity of MMF fibre transmission. We will study these two factors in the following sections. Fig. 6 shows the capacity gain provided when using the MGDM multiplexing technique. This figure shows the capacity versus SNR compared for SISO and MGDM systems. The capacity in the case of MGDM (3 3) is significantly increased compared to SISO channels (increased from 9 bits/s/Hz to 27 bits/sec/Hz at SNR = 30 dB). The parameters used in the simulation to generate MGDM (3 3) and (2 2) channel matrix H are as follows: F = 0, 13, 26 lm and F = 0, 26 lm for 3 3 and 2 2, respectively, w = 4lm, k = 850 nm and multimode fibre core diameter of 62.5 lm. Fig. 7 depicts the effect of the change of emission and reception spatial conditions on capacity. As shown in this figure, the capacity in the case of 3 3 channels is significantly reduced in non-orthogonality condition compared to orthogonality condition (reduced from 27 bits/s/Hz to 22 bits/sec/Hz at SNR = 30 dB). Orthogonality and non-orthogonality for 3 3 channels transmission depends on F, w, h and r the mean radius of the reception area. The choice of these parameters may increase or decrease the correlation between channels. It is important to improve these conditions before studying the effect of fibre curvature on MGDM system. In the next section we present the best transmission conditions to increase system performance.
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30
60
SISO channel
62.5/125, F = 0 62.5/125, F = 13 62.5/125, F = 26 50/125, F = 0 50/125, F = 12 50/125, F = 21
MGDM (2× 2) channel
50
MGDM (3× 3) channel
Mode Groups number (M)
Capacity (Bits/s/Hz)
25
20
15
10
10 SNR (dB)
20
30
Capacity (Bits/s/Hz)
0
1
2
3
w / w0 Fig. 8. The number of MG according to the spot size, at different offsets.
modes, the weaker the modal dispersion of the fibre. It is possible to excite a few mode groups. In MMF fibre with parabolic refractive index profile (a ¼ 2), the relationship between the spot size w and the number of excited modes group (M) is given by the following:
Orthogonality Non−orthogonality
25
20
0 0
Fig. 6. The MGDM channel capacity versus SNR, and comparison with a SISO channel.
30
30
10
5
0 −10
40
SISO channel
w 2 0 M¼ þ w
20
(
4Fw=w20
if F > w;
ðF þ wÞ2 =w20
if F < w:
ð14Þ
Fig. 8 shows the change in number of excited modes groups versus w for various offsets to the MMF fibres, (62.5/125) and (50/ 125). The beam waist must be located on the fibre front facet and its width w, associated to the fundamental mode, should be:
15
10
0:5w0 6 w 6 0:8w0 :
5
0 −10
0
10
20
30
SNR (dB)
In the case of GI-MMF (62.5/125) w0 ¼ 6:32 lm and thus w ¼ 0:6w0 , w ¼ 3:79 lm at 850 nm wavelength. This spot size still retains almost 80% of the power in the fundamental mode when F ¼ 0 lm. 3.2. The optimization of the angular launching conditions (h)
Fig. 7. Comparison of the capacity for different excitation and reception conditions.
3. Excitation conditions for MGDM systems Excitation coefficients of the guided modes of optical fibre by narrow input Gaussian beams are calculated using (9). The number of excited modes depends on spatial excitation conditions (F, w, h) and wavelength k. As shown before, these parameters affect the capacity and quality of the MGDM transmission system. We present here the optimum choice for these parameters. In the following analysis the effect of mode mixing is not taken into account. The results presented in this paper are obtained by considering a parabolic refractive index profile since this is one of the few profiles with an analytical solution and constitutes a good approximation for the profile in dispersion-optimized fibres. 3.1. The optimal spot size chosen The number of mode excited at each offset depends of the Gaussian beam spot size. The smaller the number of excited
In order to excite selectively an MMF an offset tilted beam is used. It is possible to reduce the crosstalk between channels by launching the light flux tilted at an angular offset [3]. This is compatible with the launch of helical rays. For every radial offset F there is a unique h0 that results in a helical ray. In the case of parabolic index profile the optimal (h0 , F) pairs that result in helical rays is then calculated analytically [17]. The relationship between F and h is given by:
h0 ðFÞ ¼ arcsin
FNA ; anðFÞ
ð15Þ
where NA is the central numerical aperture of the fibre. Fig. 9 shows the optimal angular offset (h0 ) for different radial offsets and two parabolic fibres (62.5/125) and (50/125). The optimal angular offsets associated to for F = 13lm and 26 lm are h0 ¼ 3:3 and h0 ¼ 6:9 , respectively. Note that the transmitter realization for helical ray launching is complicated. In addition this transmission scenario does not take into account the coupling efficiency at the input fibre (g < 100%).
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10
4. Curvature modelling for MGDM systems
MMF(62.5/125) MMF(50/125)
8
θ0 (degree)
F=13 µm 6
θ0=3.3°
4
F=26 µm 2
0 0
θ0=6.9°
10 20 Radial Offset (µm)
30
Fig. 9. Optimal angular versus radial offset for helical ray launching and two types of parabolic MMF fibres (a ¼ 2).
3.3. Influence of the wavelength (k) on MGDM system For standard 62.5/125 lm MMF, the minimum bandwidths are specified to be 200 and 500 MHz km in the 850 and 1300 nm transmission windows, respectively, under uniform overfilled-excitation condition. The modal dispersion change the bandwidth of the GI-MMF fibre and numbers of excited modes at a given offset depend on the wavelength. Fig. 10 shows an example of the power distribution to different modes at two different wavelengths (850 nm, 1300 nm), for the MMF (62.5/125 lm) in optimal excitation conditions (F, h0 , w). We can notice that the modal power is distributed to the lower order modes at 1300 nm. At 850 nm, higher order modes are excited especially at higher offsets. Higher order modes are sensible to attenuation and to external effect on the fibre. The MGDM channel is more robust at 1300 nm. However, if the number of users increases, the mode coupling between the corresponding channels increase and the channel capacity reduces. In this case, the use of 850 nm for excitation separated MG is best that at 1300 nm.
30
F=13, λ = 850nm F=26, λ = 850nm
Power distribution (%)
25
F=13, λ = 1300nm F=26, λ = 1300nm
20
Until now, we presented an ideal analytical modelling of optical transmission with MGDM multiplexing technique. Excitation conditions and the wavelength effect were presented without consideration on the fibre state modifications and inter-mode mixing. In the interesting applications of the fibre (LAN, indoor networks), occurring external perturbations reduce bandwidth. Most studies regarding the fibre external effects are interested in mode coupling-bandwidth and mode mixing-loss radiation [18,19]. The analytical modelling of external effect is limited to the effects of the fibre curvature. In the mode coupling section, use is made of various approximations (paraxial approximation, weakly-guiding fibre) to derive the well-known coupled-mode equations. In case of MGDM multiplexing, we focus our study on the inter-mode mixing MSI. The modal power redistribution caused by the inter-mode mixing increases the overlap between channels in the MGDM system. The inter-mode mixing arises from several external effects (curvature, core heterogeneities, fibre connector, etc.), but we are interested here only in the fibre curvature effect. Although various aspects of the curvature have been analyzed by researchers [20], the results have often been taken to specific simple geometries. A bent fibre can be thought of as a segment of a ring. The Fig. 11a presents a simple geometry of curvature, a straight section of MMF fibre of length d, leading into a curved section of fixed radius R with angular aperture a. We assumed d sufficiently large for the spatial steady state to be reached. We choose the radius R so that the power attenuation is negligible. The power attenuation depending on fibre radius a, on radius of curvature R and on a is studied in [21]. For parabolic MMF fibre, the attenuation power is negligible (2 dB) for R P 1000a. For MMF (62.5/125) the corresponding radius of curvature for negligible attenuation power is R P 62:5 mm. Then, we determine the inter-mode mixing depending on the radius of curvature by approximating the bent fibre with a series of sections in Fig. 11b. The N sections have a length l ¼ Ra=N. The purpose of this decomposition is the determination of new excited modes at the output of curved fibre for each MGDM channel. We assume that each section of length l is tilted from the subsequent section. Each section excites a new MG at the input of the next section. In the last section we can determine the overall excited MG of the total curvature. The Fig. 12 shows the geometry of two sections. (l; m) modes propagate in the Section 1 with the modal amplitude al;m ðlÞ at output of this section. The modal amplitude al;m ð0Þ of the excited mode at the input of Section 2 is calculated in the corresponding coordinate (x, y, z). The calculation of modal amplitude by the overlap integral allows us to identify new excited modes in the second section. The overlap integral calculated using section I axes is done by:
al0 ;m0 ð0Þ ¼
ZZ
! ! ðEI EII Þuz dA;
ð16Þ
! ! where EI and EII denote the electric field in the sections I and II, respectively.
15
10
a
b
l
5
0
0
10 20 Modes Group
30
Fig. 10. Modal power distribution for two offsets at two wavelengths.
Fig. 11. (a) Bent fibre of radius R where a is the angular curvature. (b) The equivalent sections.
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
y
y’
x’
0 0 Illmm
x
Section 1
Section 2
Fig. 12. Geometry for two sections along the fibre curvature.
al0 ;m0 ð0Þ ¼
X
al;m ðLÞ
ZZ
ð~ Eðl;mÞI ~ Eðl0 ;m0 ÞII Þuz dA
l ;m
¼
X
0 0
al;m ðLÞIllmm :
ð17Þ
l ;m
The coupling coefficient is given by: 0 m0
Illm ¼
ZZ
ð~ Eðl;mÞI ~ Eðl0 ;m0 ÞII Þuz dA:
ð18Þ
Note that in calculating the modal amplitude al;m ðlÞ, the attenuation coefficient is not taken into account since l is very low. The (17) may be repeated N times to determine the modal amplitude at the final output section of the curved part of fibre. The modal amplitude al1 ð0Þ on the last section N (output) is given by:
alN ðlÞ ¼
" X X
lN1
lN2
X ðal1 ð0Þejlbl1 Ill21 Þ ejlbl2 Ill32
!
#
l1
ejlblN1 IllNN1 :
ð19Þ
The modal amplitude is calculated for each offset at the input of straight section of the fibre. The determination of the coupling coefficient and the modal amplitude is made using numerical computing with MatlabTM. For parabolic fibre, this coefficient can be determined analytically. Coupling occurs only between modes of the same polarization and same orientation. The coupling coefficients between Ex -modes and Ey -modes, respectively, are equal [22]. Therefore, only Ex -modes will be considered in the following. For LP modes, we may use a normalization condition different from (18) according to: 0 0
Illmm ¼
ZZ
ðExI ExII Þuz dA;
ð20Þ
where ExI and ExII are the transverse electric fields to sections I and II, respectively. The transverse electric field may then be written as:
Ex ðx;yÞ ¼ ðw0 p2lþm l!m!Þ2 Hl
pffiffiffi ! pffiffiffi ! 2x 2y Hm expððx2 þ y2 Þ=w20 Þ; w0 w0 ð21Þ
( ¼
0
p2lþm l!m!gmm0 Lmmm ð2g2 Þ=2 if m 6 m0 ; 0 0 2 0 w0 eg p2lþm l!m0 !gmm Lmm0 m ð2g2 Þ=2 if m P m0 ;
w0 eg
2
N=40 N=70 without curvature N=30
20
15
10
5
0
0
5
10 15 Modes Group
Fig. 13. Modal power redistribution after curved fibre.
and
ExI ¼ Ex ðl; mÞ expðik1 hyÞ;
ExII ¼ Ex ðl; mÞ;
where H is the Hermite–Gauss function, and h ¼ a=N. According to (20), the coupling coefficient is written as: 0 0 Illmm
¼ Al;m Al0 ;m0
Z
Hm
pffiffiffi ! pffiffiffi ! 2x 2x Hl H l0 expð2x2 =w20 Þ dx w0 w0 pffiffiffi ! pffiffiffi ! 2y 2y Hm0 expð2x2 =w20 Þ expðik1 hyÞ dy: w0 w0
Z
ð22Þ Eq. (22) has been solved by use of the integral 7.374.7 in [23]. The required coupling coefficients are identically zero if l–l0 and, otherwise, determined by:
ð23Þ
pffiffiffi where g ¼ ikhw0 =ð2 2Þ and L is a Laguerre–Gauss function. The more N big is, the more we approach the practical case of uniform curvature (Rc ¼ constant). The analytical determination of coupling coefficient follows by determining the modal amplitude at the curved section output. The calculation of the modal amplitude depends on the number of elements considered in the curved section (N). Fig. 13 shows the change of the coupling versus N for a uniform curvature Rc ¼ 100 mm and a ¼ 50 . For N ¼ 30 elements, there is intermode coupling at the output of the uniformly curved section for the fundamental channel, while for 70 elements, we obtain a quasi-null coupling. This result supports well the theoretical results published in the literature [24–26], which state that no (or negligible) coupling occurs between the modes of propagation in a circularly bent fibre with sufficient radius of curvature, leading therefore to negligible inter-mode mixing. In the case of a 2-D (that is bending in a single plane) non-uniform curvature (Rc variable), we decompose the bent part in a series of M uniformly curved sections, as shown in Fig. 14. We take the example of a curved fiber in the form of a half ellipse. This part is decomposed into a series of M ¼ 12 adjacent circular arcs (consisting each in N ¼ 70 straight sections as above). Each arc is defined by a radius Ri and the angle ai between this arc and the following one. We introduce the constraint 1000a 6 Ri 6 4000a. We simulate a fiber of length 100 m with a knot which presents the curvature after 50 m crossed (see Fig. 14). We determine the
Power distribution (%)
3914
Fig. 14. Description of a non-uniform curvature.
20
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M. Awad et al. / Optics Communications 282 (2009) 3908–3917
40 no curvature
35
Capacity (Bits/s/Hz)
distribution of the modal power after curvature. For a (N N) MGDM, the determination of the channel transfer matrix is similar to steps previously explained in Section 2. Fig. 15 shows the conversion of the modal power from the fundamental mode to the higher order modes after the curvature. The higher order modes are excited at the output of this curvature and increase the MSI with other channels. To study the curvature effects on the MGDM system, we compare the capacity of this system with and without curvature. Fig. 16 shows this comparison for a (4 4) MGDM system, without curvature and another with a curved knot. At SNR = 30 dB, the capacity is significantly decreased (from 34 bits/s/Hz to 24 bits/s/ Hz). In presence of curvature, the capacity is not far from the SISO system one. However, the effect of curvature on the system depends on the number of users. The launch space for a large number of users excites the MG close to each other. Hence, the probability of MSI increases. A MGDM system with a larger number of users will be more affected by a non-uniform curvature and an increase of knot curvature number.
30 1 knot
25
2 knots
20 15 SISO
10 5 2x2
3x3
4x4
5x5
N ×N Fig. 17. The system capacity versus number of users (SNR = 30 dB).
25
after curvature before curvature
Power distribution (%)
20
15
10
Fig. 17 shows the effect of curvature versus the number of users in the system. For the knot presented in Fig. 14, the decrease of the capacity in a (5 5) MGDM system is more noticeable than that in a (2 2) system (reduced from 40 bits/s/Hz to 27 bits/s/Hz at SNR = 30 dB for 5 5 and reduced from 20 to 14 bits/s/Hz for 2 2). For one knot, the capacity is reduced to 24 bits/s/Hz for 4 4. This capacity is twice that of a SISO, and this is much like for a (4 4) CO-MIMO system (correlated channel taps) demonstrated by [27]. For two knots, the capacity for a 4 4 MGDM is lower than that of the (4 4) radio MIMO one. 5. MGDM system application and simulation
5
0
0
5
10 Modes group
15
20
Fig. 15. Modal power redistribution after curved fibre.
35 30
MGDM (4 × 4) no curvature MGDM (4 × 4), curved
Capacity (Bits/s/Hz)
SISO
25 20
We analyze system performance for the MGDM communication link model of Section 2, including influence of curvature. For N N MGDM systems we simulated the average BER for several system configurations differing in modulation format, fibre length, SNR and curvature effect. The simulated model is shown on Fig. 18. Our simulation chain is run using the VPI software. We assume N independent inputs. Each input is OOK modulated by an independent pseudo-random binary data stream (2.5 Gb/s, 5 Gb/s). Each laser, associated with a user, modulates the binary data at the same wavelength (1330 nm) with the same power (0 dB m). By co-simulation between Matlab and VPI, we introduce the MGDM channel model of Section 2 in simulations. The simulation using the model matrix H is followed by the effect of changing of the fibre length on the channels. The coupling between the fibres modes slowly varies relative to the symbol period affected by acoustic, thermal, and laser frequency instabilities. Here, we assume that the matrix H is periodically estimated at a rate faster than its effective rate of change.
15 Photodiode Baseband Z-F signal Laser MMF
10 Channel #1 Channel #2
5 0 −10
PBRS PBRS
Channel #N PBRS
0
10 SNR (dB)
20
MGDM channel model
E Q U A L
30
Fig. 16. The effect of curvature on (4 4) MGDM system capacity.
PRBS = Pseudo Random Binary Sequence Fig. 18. The demonstrator model.
BER Received signal
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M. Awad et al. / Optics Communications 282 (2009) 3908–3917
At the receiver, each photodiode captures a mixture of signals. For MGDM, the simplest receiver architecture is matrix inversion, a Zero-Forcing (Z-F) method in line with the requirement of service transparency. The relation between signals given by (2) will be:
^s ¼ s þ nH1 :
0
10
−5
10
with Z−F, channel #1 with Z−F, channel #2 without Z−F
−10
10
2
4
6
8
BitRate (Gb/s) Fig. 20. BER measurements versus bit rate, for different channels in the (3 3) system at fibre length L = 1 km.
10
10
0
−5
BER
However Z-F has a major disadvantage: when the hij values are small, the element of H1 will be large and can amplify the noise component n (AWGN noise), which will degrade the SNR of the estimated symbols. Consequently, erroneous decisions are produced at the receiver. By choosing a SNR = 30 dB for our simulation, we study the performance of the system by measuring the BER before and after equalizer (Z-F), for many users and for various lengths of fibre. We choose the number of users equal to the number of receivers. The BER penalty changes depending on the number of channels created in the fibre and depending on the fibre length. The Fig. 19 corresponds to the case where data is sent over the fundamental channel (offset = 0 lm) at a bit rate of 5 Gb/s. It shows an increase of BER for large distances. For 3 3 systems, a BER < 1010 is observed at L ¼ 500 m, but this value goes up to 104 at L = 1000 m. Moreover, the BER changes significantly between 3 3 and 5 5 systems. The performance can vary from one channel to another within the same system. The BER measured for data associated with the fundamental channel for the (3 3) MGDM system is not the same as those associated with another channel (offset = 13 lm). Fig. 20 shows the change of BER for different bit rates for the fundamental channel (channel #1) compared to that of channel #2 (offset = 13 lm). For D = 3 Gb/s, the BER changes from 1010 (channel #1) to 108 (channel #2). For this bit rate BER = 103 before Z-F equalizer. The MSI between the channels is at its maximum in the channel at core extremity compared with fundamental channel. Finally, we study the effect of the fibre curvature on the system performance. We simulate a 3 3 transmission by taking into account the curvature of the fibre presented in previous section and we compare its performance with that of a chain without curvature. Fig. 21 shows the change in performance. The BER increases
BER
ð24Þ
10
no curvature
−10
with curvature
0
10
BER
10
10
−5
−15
300
400
500 600 Fibre length (m)
700
800
Fig. 21. BER measurements versus fibre length, for a 3 3 system with and without curvature at D = 10 Gb/s.
10
−10
MGDM 3 × 3
10
from 1014 to 107 for L = 300 m. With the curvature, the MSI is becoming important and the degradation of the system is significant. We can get a BER = 104 for D = 10 Gb/s at 300 m in the presence of the curvature. One may notice that this BER decreases for D < 5 Gb/s.
MGDM 4 × 4
−15
MGDM 5 × 5
6. Summary and discussion
10
−20
400
600
800
1000
Fibre length (m) Fig. 19. BER measurements versus fibre length, for different systems using a constant bit rate of 5 Gb/s.
The conversion of the SISO system to a multi-users one encounters several difficulties in the MMF case (bandwidth, modal dispersion and cost of system). MGDM technology solves many of these difficulties and offers good functionality. But to improve the functioning of this technique, it was important to study the restrictive
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
effects (excitation conditions, fibre curvature). In this paper, we have presented an analytical model for N N MGDM systems, taking into account the conditions of excitation and reception. First we presented the best conditions for emission to improve the system (offset, spot size, and wavelength). Applying our statistical model, we have evaluated the MMF MGDM system capacity compared to SISO systems. The inter-mode mixing effect in this system is studied and the analytical modelling is shown. We investigated the effects of the fibre curvature. We compared the capacity of the MGDM system with curvature and without curvature and we demonstrated the minimum allowable radius of curvature. We evaluated the MMF MGDM system performance for various numbers of users. The results indicate that the BER < 1010 for a 5 5 MGDM system for a 5 Gb/s bit rate over several hundreds of meters of MMF. Finally the deteriorated performance of our system in the presence of the curvature model is presented. A followup research direction, based on the results of this paper, is the physical realization and characterization of a MGDM system. References [1] I. Dayoub et al., Eur. Trans. Telecom. 18 (2007) 811. [2] M. Greenberg et al., IEEE J. Lightwave Technol. 25 (2007) 1503.
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[3] L. Raddatz, I.H. White, D.G. Cunningham, M.C. Nowell, IEEE J. Lightwave. Technol. 16 (1998) 324. [4] X.J. Gu et al., IEEE Photonics Technol. Lett. 18 (2006) 244. [5] G.D. Golden et al., Electron. Lett. 35 (1999) 14. [6] H.R. Stuart, Sci. Mag. 289 (2000) 281. [7] S. Schöllmann, C.Xia, W. Rosenkranz, in: Techn. Digest (CD) of OSA Opt. Fiber Conf., 2006. [8] C.P. Tsekrekos et al., Opt. Lett. 32 (2007) 1041. [9] C.P. Tsekrekos et al., Opt. Exp. 15 (2007) 3656. [10] J. Nathan Kutz, J.A. Cox, D. Smith, IEEE J. Lightwave Technol. 16 (1998) 1195. [11] T. Koonen et al., in: Proc. IEEE LEOS Benelux Symp., vol. 87, 2002. [12] R.W. Smink et al., in: Proc. IEEE LEOS Benelux Symp., vol. 257, 2006. [13] C.P. Tsekrekos et al., IEEE Photonics Technol. Lett. 18 (2006) 2359. [14] G. Yabre, IEEE J. Lightwave. Technol. 18 (2000) 166. [15] J. Saijonmaa et al., Appl. Opt. 19 (1980) 2442. [16] C.P. Tsekrekos, IEEE Photonics Technol. Lett. 18 (2006) 2359. [17] L. Jacomme, Opt. Commun. 14 (1975) 134. [18] E.L. Chinnock et al., Proc. IEEE Lett. 61 (1973) 1499. [19] J. Nathan Kutz, J.A. Cox, D. Smith, IEEE J. Lightwave Technol. 16 (1998) 1195. [20] M.-Y. Loke, J.N. McMullin, IEEE J. Lightwave Technol. 8 (1990) 1250. [21] Allan W. Snyder, john D. Love, Optical Waveguide Theory, Springer, 1983. [22] K. Petermann, IEEE J. Quantum Elect. 16 (1980) 761. [23] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals Series and Products, Academic Press, New York, 1965. [24] D. Donlagic, B. Culshaw, IEEE J. Lightwave Technol. 18 (2000) 334. [25] A.-K. Ghatak, I.-C. Goyal, A. Gupta, J. Mod. Optics 25 (1978) 1. [26] R.-T. Schermer, Opt. Exp. 15 (2007) 15674. [27] A. Tarighat, IEEE Commun. Mag. 45 (2007) 57.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
The inter-modes mixing effects in Mode Group Diversity Multiplexing M. Awad, I. Dayoub, A. Okassa M’Foubat, J.M. Rouvaen * Univ Lille Nord de France, F-59000 Lille, France IEMN (UMR 8520 CNRS), Department of OAE, UVHC F- 59313 Valenciennes, France
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 26 June 2009 Accepted 26 June 2009
PACS: 42.79.Sz 42.81.Uv 42.81.i 42.81.Ht
a b s t r a c t Multiplexing technology in indoor optical networks with multimode fibres allows the integration of several services to the end user. The Mode Group Diversity Multiplexing technique (MGDM) in Graded-Index Multi-Mode (GI-MMF) fibre makes the system less expensive with simpler transmitters and receivers, keeping the same information capacity as other multiplexing techniques. The capacity conservation and transmission quality obtained using MGDM technique depends on the transmission conditions and the state of fibre. Bending of the fibre can affect the system by changing modes excited for the different propagation channels in the fibre. In this paper an analytical modelling method for MGDM will be presented. Improvements in system sending and receiving conditions are studied. Modelling of the fibre curvature as well as the effects of coupling patterns on the MGDM are discussed. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Inter-modes mixing Multi Services Interferences (MSI) Multimode fibre links Optical fibre communication Optical MGDM
1. Introduction In optical communication systems, especially for indoor networks, multimode fibre (MMF) links have been selected as the basic infrastructure [1]. They can provide the bandwidth required for short distance communications at much lower expense than single-mode fibres (SMF), mainly due to the ease of installation. However the multiplicity of optical propagation modes in MMF and waveguides has been considered as a huge disadvantage. It produces the intermodal dispersion that limits the distance-bandwidth product of optical communication systems [2]. Several searches were made to weaken this major inconvenient by designing and developing cost-effective methods to increase the bit rate (>1 Gb/s) at large fibre lengths (’1 km) [3]. On the other hand, there is currently a wide variety of wireline networks, each optimised for transporting a particular set of services in indoor networks (Voice telephony, Internet, etc.). But these networks do not allow the integration of new broadband services. Optical fibre is the best candidate for a shared network enabling several services by using multiplexing techniques. In addition, it al-
* Corresponding author. E-mail address: [email protected] (J.M. Rouvaen). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.065
lows to deliver radio-communication signals to the end user. Several multiplexing techniques have been studied for multimode fibre networks. Wavelength Division Multiplexing (WDM) further extends the network capacity and enables the integration of different services in a single-MMF network allowing the optimal use of the optical bandwidth. This technique is limited by the ability to generate the adequate optical carriers and the correct data transmission over acceptable distances. Indeed, WDM requires wavelength-specific sources in addition to wavelength-selective network functions, which are still quite costly [4]. Optical CodeDivision Multiple-Access (OCDMA) is applied for SMF fibre networks. But, for indoor networks, coding services complicates the system and requires high-speed lasers emitters. Moreover, the two techniques (WDM & CDMA) do not take into consideration the intermodal dispersion, and system performance is still limited to low bit rates and short distances. Mode Group Diversity Multiplexing (MGDM) is a multiplexing technique, based on the spatial launching and detection of subgroups of modes to create a number of independent communication channels in a single-MMF. This technique derives from a recently developed wireless transmission scheme known as BLAST (Basic Local Alignment Search Tool) and has been introduced recently to replace the more expensive WDM in LAN networks [5]. This technique exploits the unused capacity of MMF for improving
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M. Awad et al. / Optics Communications 282 (2009) 3908–3917
the bandwidth times fibre length product. The excitation of sub-groups of modes decreases the intermodal dispersion; an up to fourfold bandwidth increase has been reported by exciting less than 50% of the fibre modes [3]. Each mode group (MG) launched at the input facet of the fibre constitutes a quasi-separate channel associated with a service or a user and, in the end facet of the fibre, this MG may be detected separately [6]. Several applications have demonstrated the feasibility of this technique with liable input and output couplers, that do not complicate transmission services on the MMF fibre [7,8]. To create N propagations channels, N lasers are followed by N virtual SMFs placed at different offsets in input facet of the MMF fibre. The receiver can be achieved by a lens placed on output facet of the fibre, for the projection of the Near Field Pattern (NFP) onto an optoelectronic integrated circuit (OIC) which contains photo-detectors and preamplifiers (PDIC) [9]. However, the MG propagation in each channel is not ideal, subgroups mode mixing and non-orthogonality of transmitters and receivers cause crosstalk between propagation channels producing Multi Services Interferences (MSI). It reduces the capacity of MMF. The improved utilization of the MMF transmission capacity is related to the reduction of the mixing effects between modes and the amelioration of the transmission/reception conditions. The conditions for transmission and reception play a key role in improving the fibre capacity. The way of excitation (offset, spot size, wavelength) determines the excited mode groups associated with each channel. The choice of optimal conditions reduces the common modes (mode coupling) excited between channels. Moreover, improving the spatial reception reduces the MSI. In this paper, the optimal conditions for transmission and reception are studied. Mode mixing is the gradual redistribution of the optical power among the propagating modes. It occurs between modes of the same MG group, which is called intra-mode mixing, or between modes of different MG groups, which is called inter-mode mixing. Inter-mode mixing is the major effect in MMF transmission, apart from external factors acting on the fibre. The modelling method for intra-mixing is presented in [9], by evenly redistributing the optical power among the modes of the MG. It is known that a perfect MMF transmits a guided mode without any power conversion to another guided mode [10]. Indeed, the installation of optical infrastructure in buildings to the end user induces deformations (fibre not straight) and the guided mode propagation is not perfect. However, any perturbation to the index of refraction, which may include a bend or curvature of the optical waveguide, leads to an imperfect light trapping in the fibre. In this case, a conversion of the energy of a mode to other higher order modes occurs and the light loses energy in the cladding and radiates off the guide. This conversion (inter-mode mixing) changes the propagation delay and increases the transmission losses. In this paper we take into account the curvature of the fibre only. In simple terms, installers know that the signal loss is moderate for a large radius of curvature, and becomes increasingly important when this radius decreases. This drawback becomes obvious as optical networks are closer to end users. A small radius of curvature of the fibre affects the MGDM and inter-mode mixing becomes more important than intra-mode mixing. The MSI increases with the number of channels in the MGDM systems. The fibre curvature acts more on the channels excited to offsets that are closest to the cladding. Several models of the fibre curvature were proposed, but these models are mainly concerned in the power loss and attenuation. In this paper, we model the curvature of the fibre in terms of coupling between modes and we present the effects of this curvature on the MGDM system, according to the radii of curvature and the number of channels defined in the fibre.
MGDM with multiple inputs and multiple outputs is similar to MIMO (Multiple Input Multiple Output) systems in radio-communications. The digital signal processing (DSP) used in radio networks to mitigate the effects of multiple channels can be used to mitigate the MSI in optical networks MGDM system at the receiver. Yet, efficient algorithms and hardware implementations facilitates the separation of services after the optoelectronic conversion. For MGDM, the simplest receiver architecture is matrix inversion, a Zero-Forcing (Z-F) equalizer in line with the requirement of service transparency [11]. The paper is organized as follows: In Section 1, MGDM technique description is given, followed by an analytical system modelling. In Section 2, we review the conditions of emission and reception and we present optimum conditions for the system. In Section 3, we study the modelling of the fibre curvature in terms of mode mixing, and we detail the effects of inter-mode mixing. In Section 4, we present an example (3 3) MGDM system with simulation results for performance. Finally, the discussion in Section 4 provides perspective on the key points. 2. MGDM technique description and analytical modelling The MGDM approach is based on the idea to transmit independent digital data signals using different mode groups on the MMF. Therefore, it is necessary to send a signal for each MG group excited at a given offset. For example, in a (2 2) MGDM system, the first signal is launched into the centre of the core fibre by exciting lower order modes (LM) (in the ideal case the lowest mode will carry most of the energy). The second signal is launched at the core fibre extremity by exciting the higher order modes (HM). During the transmission through the MMF fibre, lower order modes stay mostly in the inner region of the core. The second signal excites higher order modes in the offset launch position, which are mainly travelling in the outer region of the MMF core. At the receiver end, the light energy is concentrated in a circular zone at the middle of the fibre core for the first condition launch. But the light energy to the second launch condition (for HM) is distributed in a ring in the outside region of the core [12]. Fig. 1 gives an example to the MGDM system architecture for 2 transmitters and the light distribution at the end facet. 2.1. Analytical modelling In radio-communication field, MIMO systems have received significant research interest. The relationship between transmitters and receivers is analytically presented in matrix form. This matrix describes the multipath channel (attenuation, phase shift, etc.). From a mathematical point of view, the optical transmission MGDM system, which realizes the multipath propagation in the MMF fibre may have a similar representation that the MIMO radio. The relationship between the N received electrical signals (yi ) and the N emitted electrical signals (si ) is written in the form:
Laser
SMF Core/MMF
End facet
F a
w d
Fig. 1. Mode multiplexing process in the input facet, and detection area at output facet of MMF.
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0
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
1
0
h11 By C B B B 2 C B h21 B C¼B B .. C B . @ . A @ ..
h12 .. . s
s ..
hN1
hN2
s
y1
yN
s
1 0 1 n1 s1 CB C B C C s n h2N CB 2 C B 2 C B C þ B C; .. C CB .. C B .. C . A@ . A @ . A
h1N
.
10
hNN
sN
The modal scalar fields are governed by the scalar Helmholtz equation:
ð1Þ
nN
al;m ðzÞ ¼ al;m ð0Þ expðcl;m zÞ;
y ¼ Hs þ n;
ð2Þ
where y, s, n are, respectively, the received signal vector, the emitted signal vector and the additive noise from receivers. The matrix elements hij describe the signal transfer from the transmitter i to the detector j. Each of them has an amplitude (representing the path mitigation (i; j)) and a phase (corresponding to the delay of the specific path (i; j)). The fibre dispersion is negligible in the case of MGDM, so that the matrix elements take on real values, expressing the proportion of power transmitted by the ith source and received by the jth detector [13]. Assuming a linear superposition of N power distributions in the fibre output, hij can be expressed by:
Ij ðSi ; LÞ ; Ij ðS; LÞ
ð3Þ
where Ij is the light flux intensity emitted by the jth transmitter, Si is the area of jth receiver, S is the total area of core fibre and L is the fibre length (Fig. 2). The light flux intensity is given by
Z 1X IðS; LÞ ¼ jal;m ðLÞj2 W2l;m ds 2 l;m S Z X al;m ðLÞal0 ;m0 ðLÞ Wl;m Wl0 ;t0 ds þ l – l0 m – m0
h i cos bl;m bl0 ;m0 L ;
al;m ð0Þ ¼
ZZ
Ein ðx; yÞWl;m ðx; yÞ uz dx dy:
ð9Þ
Ein being the electric field of the SMF. uz the unit vector in a z-directed MMF fibre. We make the assumption that the incident field at the MMF input is a Gaussian field representing the SMF fibre to radial offset (F), angular offset (h) and spot size (w) (Fig. 3). The field components of the input Gaussian beam can be expressed in Cartesian coordinates as follows:
pffiffiffi h i 2 Ein ðx; yÞ ¼ pffiffiffiffiffiffiffiffi exp ðx FÞ2 y2 =w2 exp ðik1 hyÞ; pw
ð10Þ
where k1 ¼ n k0 . k0 ¼ 2p=k, k standing for the wavelength. The Eqs. (8) and (9) allow us to determine the magnitude of the modes excited at offsets and the distribution of the modal power, which is given by: m1 X
jal;ml1 j2 ¼ f ðF; w; w0 ; hÞ:
ð11Þ
l¼0
IðS; LÞ ¼ I0 þ q0 ;
ð5Þ
where al;m and Wl;m are respectively the modal amplitude and the modal function of the (l; m) mode. bl;m being the propagation constant. The intensity I is broken into two parts: I0 is the sum of the intensity distributions due to each mode separately, and q0 expresses the change in the whole intensity distribution due to interfering modal fields. The signal propagation along the z axis of the fibre MMF is the superimposition of several harmonic terms. The plane wave electromagnetic field mode (l; m) is described in Cartesian coordinates (x; y; z) by:
X
ð8Þ
with cl;m the attenuation coefficient of the (l; m) mode, defined in [14]. The excited modes at the fibre input are determined by the modal amplitude coefficients al;m . They are determined by the overlap integral [15]:
Pm ð0Þ ¼
S
ð4Þ
Eðx; y; zÞ ¼
ð7Þ
The modal amplitude al;m ðzÞ in the absence of mode mixing is defined by:
or in matrix notation:
hij ¼
r2t Wl;m þ k20 n2 Wl;m ¼ bl;m Wl;m :
h i al;m ðzÞWl;m ðx; yÞ exp jðbl;m zÞ ;
ð6Þ
l ;m
We define the group index m ¼ l þ m þ 1, which represents all modes that have almost the same propagation constant. The characteristic spot size w0 for the fundamental mode is given by:
w0 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a pffiffiffiffiffiffiffi; k0 n0 2D
ð12Þ
where k0 is the wave number in free space, n0 is the core peak index, D is the refractive-index contrast and a is the fibre core radius. All excited modes change with the offset (F), the spot size (w), the incident angle (h) and the characteristics of MMF fibre. Fig. 4 shows an example of modes excited and power distribution in a GI-MMF (62.5/125) fibre for several F and w values, normal injection (h ¼ 0), at a wavelength k ¼ 850 nm. The calculation of the modal scalar field Wl;m and the modal amplitudes al;m , for excited modes by using Eqs. (7) and (9), allows us to determine the light flux intensity given by (5). From the
l and m being the zeros of electric field distribution mode in the x and y directions, respectively.
y
z
F
s1 s2 s3 2w
Fig. 2. End facet fibre area.
x
Fig. 3. Geometry of excitation of the MMF fibre by a Gaussian field.
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
while the channel receiver to F ¼ 13 lm is an annular area with 7 lm < r < 18$ lm. The limiting values are selected here so that the major part of light intensity emitted by the transmitter i will be received in the corresponding area i. The matrix to match the MGDM (3 3) is given by:
20 F = 10µm, w = 6µm F = 20µm, w = 3µm F = 20µm, w = 6µm F = 10µm, w = 3µm
P (%) m
15
2
3 0:8397 0:2293 0:1042 6 7 H ¼ 4 0:1595 0:7545 0:3427 5:
10
0:0008
5
0:0162 0:5531
2.2. Characterization of MGDM channel
0 0
10 20 Mode groups (m)
30
Fig. 4. Mode power distribution versus geometry of excitation.
determination of the intensity I in each area Si , we can deduce the coefficients of the matrix H, which defines the MGDM channel. Owing to the difficulty of determining the coefficients of the matrix H analytically, we use a numerical averaging to calculate the coefficients hi;j . We are interested in the Graded-Index fibre (GI-MMF), and especially to standards 62.5/125 and 50/125. These two types of fibres use the 850 and 1300 nm transmission windows. In our work, we consider the case of parabolic fibre with quadratic distribution (a ¼ 2). In this case the scalar field Wl;m is obtained based on the Gauss–Hermite (GH) expression. For a (33) MGDM system built on a GI-MMF (62.5/125) and offsets F ¼ 0; 13 and 26 lm, the modes excited in each channel are determined and their modal amplitudes computed, with the spot size w ¼ 4lm and the wavelength k ¼ 850 nm. The light intensity is computed in a determined area for each channel at the fibre output. Fig. 5 shows the intensity distribution of light flux of the three channels in the plane perpendicular to the axis of the fibre propagation with L ¼ 100 m. The determination of the matrix coefficients hi;j is linked to the intensity distribution of light flux in three separate surfaces, each corresponding to a channel determined by its excitation offset. For the fundamental channel (F ¼ 0 lm), the area of reception is circular and centered on the fibre axis with a radius r1 ¼ 7 lm,
0.1
channel #3, F = 26µm channel #2, F = 13µm
0.08
r
channel #1, F = 0µm
2
Normalized Intensity
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0.06
r
1
0.04
0.02
0 0
10 20 Distance from fibre axis (µm)
30
Fig. 5. The light intensity distribution for three excitation positions (F = 0, 13, 26 lm, and L = 100 m).
Channel State Information (CSI) is information on the current value of the matrix H. This is a mathematical value that represents the channel signal. It is part of the model signal for MIMO radio systems. Applications authorizing the establishment of a feedback receiver to the transmitter allow the state channel return (CSI). Then it is possible to fully exploit this information to optimize overall transmission by anticipating the effect of channel. It is known that the channel capacity for a channel matrix realization H is given by:
q C ¼ B log2 det INR þ s HH> ðbits=secÞ; NT
ð13Þ
where B is the channel bandwidth (Hz), NT the number of transmitters, NR the number of receivers and the mean SNR. The study of the MGDM channel characteristic is similar to that of MIMO radio channel. The (13) is used to determine the status of the MGDM channel. The temporal and spatial static properties of the matrix elements H affect the capacity of the MGDM link. Although these two statistics have been studied and modelled for MIMO radio channels, they are not well characterized for optical links. For temporal properties, MGDM channel can be regarded as constant over a relatively long period (hours). Studies have demonstrated the modification of the coefficients matrix H for long periods [16]. In this case, a feedback channel to the transmitter is used to redefine the channel matrix H. Regarding the spatial properties, the excitation conditions to transmitter (radial/angular offset, wavelength, spot size, etc.) act on the coefficients of matrix H and on the link capacity. Moreover, the fibre state is the factor which affects the most the quality and capacity of MMF fibre transmission. We will study these two factors in the following sections. Fig. 6 shows the capacity gain provided when using the MGDM multiplexing technique. This figure shows the capacity versus SNR compared for SISO and MGDM systems. The capacity in the case of MGDM (3 3) is significantly increased compared to SISO channels (increased from 9 bits/s/Hz to 27 bits/sec/Hz at SNR = 30 dB). The parameters used in the simulation to generate MGDM (3 3) and (2 2) channel matrix H are as follows: F = 0, 13, 26 lm and F = 0, 26 lm for 3 3 and 2 2, respectively, w = 4lm, k = 850 nm and multimode fibre core diameter of 62.5 lm. Fig. 7 depicts the effect of the change of emission and reception spatial conditions on capacity. As shown in this figure, the capacity in the case of 3 3 channels is significantly reduced in non-orthogonality condition compared to orthogonality condition (reduced from 27 bits/s/Hz to 22 bits/sec/Hz at SNR = 30 dB). Orthogonality and non-orthogonality for 3 3 channels transmission depends on F, w, h and r the mean radius of the reception area. The choice of these parameters may increase or decrease the correlation between channels. It is important to improve these conditions before studying the effect of fibre curvature on MGDM system. In the next section we present the best transmission conditions to increase system performance.
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30
60
SISO channel
62.5/125, F = 0 62.5/125, F = 13 62.5/125, F = 26 50/125, F = 0 50/125, F = 12 50/125, F = 21
MGDM (2× 2) channel
50
MGDM (3× 3) channel
Mode Groups number (M)
Capacity (Bits/s/Hz)
25
20
15
10
10 SNR (dB)
20
30
Capacity (Bits/s/Hz)
0
1
2
3
w / w0 Fig. 8. The number of MG according to the spot size, at different offsets.
modes, the weaker the modal dispersion of the fibre. It is possible to excite a few mode groups. In MMF fibre with parabolic refractive index profile (a ¼ 2), the relationship between the spot size w and the number of excited modes group (M) is given by the following:
Orthogonality Non−orthogonality
25
20
0 0
Fig. 6. The MGDM channel capacity versus SNR, and comparison with a SISO channel.
30
30
10
5
0 −10
40
SISO channel
w 2 0 M¼ þ w
20
(
4Fw=w20
if F > w;
ðF þ wÞ2 =w20
if F < w:
ð14Þ
Fig. 8 shows the change in number of excited modes groups versus w for various offsets to the MMF fibres, (62.5/125) and (50/ 125). The beam waist must be located on the fibre front facet and its width w, associated to the fundamental mode, should be:
15
10
0:5w0 6 w 6 0:8w0 :
5
0 −10
0
10
20
30
SNR (dB)
In the case of GI-MMF (62.5/125) w0 ¼ 6:32 lm and thus w ¼ 0:6w0 , w ¼ 3:79 lm at 850 nm wavelength. This spot size still retains almost 80% of the power in the fundamental mode when F ¼ 0 lm. 3.2. The optimization of the angular launching conditions (h)
Fig. 7. Comparison of the capacity for different excitation and reception conditions.
3. Excitation conditions for MGDM systems Excitation coefficients of the guided modes of optical fibre by narrow input Gaussian beams are calculated using (9). The number of excited modes depends on spatial excitation conditions (F, w, h) and wavelength k. As shown before, these parameters affect the capacity and quality of the MGDM transmission system. We present here the optimum choice for these parameters. In the following analysis the effect of mode mixing is not taken into account. The results presented in this paper are obtained by considering a parabolic refractive index profile since this is one of the few profiles with an analytical solution and constitutes a good approximation for the profile in dispersion-optimized fibres. 3.1. The optimal spot size chosen The number of mode excited at each offset depends of the Gaussian beam spot size. The smaller the number of excited
In order to excite selectively an MMF an offset tilted beam is used. It is possible to reduce the crosstalk between channels by launching the light flux tilted at an angular offset [3]. This is compatible with the launch of helical rays. For every radial offset F there is a unique h0 that results in a helical ray. In the case of parabolic index profile the optimal (h0 , F) pairs that result in helical rays is then calculated analytically [17]. The relationship between F and h is given by:
h0 ðFÞ ¼ arcsin
FNA ; anðFÞ
ð15Þ
where NA is the central numerical aperture of the fibre. Fig. 9 shows the optimal angular offset (h0 ) for different radial offsets and two parabolic fibres (62.5/125) and (50/125). The optimal angular offsets associated to for F = 13lm and 26 lm are h0 ¼ 3:3 and h0 ¼ 6:9 , respectively. Note that the transmitter realization for helical ray launching is complicated. In addition this transmission scenario does not take into account the coupling efficiency at the input fibre (g < 100%).
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10
4. Curvature modelling for MGDM systems
MMF(62.5/125) MMF(50/125)
8
θ0 (degree)
F=13 µm 6
θ0=3.3°
4
F=26 µm 2
0 0
θ0=6.9°
10 20 Radial Offset (µm)
30
Fig. 9. Optimal angular versus radial offset for helical ray launching and two types of parabolic MMF fibres (a ¼ 2).
3.3. Influence of the wavelength (k) on MGDM system For standard 62.5/125 lm MMF, the minimum bandwidths are specified to be 200 and 500 MHz km in the 850 and 1300 nm transmission windows, respectively, under uniform overfilled-excitation condition. The modal dispersion change the bandwidth of the GI-MMF fibre and numbers of excited modes at a given offset depend on the wavelength. Fig. 10 shows an example of the power distribution to different modes at two different wavelengths (850 nm, 1300 nm), for the MMF (62.5/125 lm) in optimal excitation conditions (F, h0 , w). We can notice that the modal power is distributed to the lower order modes at 1300 nm. At 850 nm, higher order modes are excited especially at higher offsets. Higher order modes are sensible to attenuation and to external effect on the fibre. The MGDM channel is more robust at 1300 nm. However, if the number of users increases, the mode coupling between the corresponding channels increase and the channel capacity reduces. In this case, the use of 850 nm for excitation separated MG is best that at 1300 nm.
30
F=13, λ = 850nm F=26, λ = 850nm
Power distribution (%)
25
F=13, λ = 1300nm F=26, λ = 1300nm
20
Until now, we presented an ideal analytical modelling of optical transmission with MGDM multiplexing technique. Excitation conditions and the wavelength effect were presented without consideration on the fibre state modifications and inter-mode mixing. In the interesting applications of the fibre (LAN, indoor networks), occurring external perturbations reduce bandwidth. Most studies regarding the fibre external effects are interested in mode coupling-bandwidth and mode mixing-loss radiation [18,19]. The analytical modelling of external effect is limited to the effects of the fibre curvature. In the mode coupling section, use is made of various approximations (paraxial approximation, weakly-guiding fibre) to derive the well-known coupled-mode equations. In case of MGDM multiplexing, we focus our study on the inter-mode mixing MSI. The modal power redistribution caused by the inter-mode mixing increases the overlap between channels in the MGDM system. The inter-mode mixing arises from several external effects (curvature, core heterogeneities, fibre connector, etc.), but we are interested here only in the fibre curvature effect. Although various aspects of the curvature have been analyzed by researchers [20], the results have often been taken to specific simple geometries. A bent fibre can be thought of as a segment of a ring. The Fig. 11a presents a simple geometry of curvature, a straight section of MMF fibre of length d, leading into a curved section of fixed radius R with angular aperture a. We assumed d sufficiently large for the spatial steady state to be reached. We choose the radius R so that the power attenuation is negligible. The power attenuation depending on fibre radius a, on radius of curvature R and on a is studied in [21]. For parabolic MMF fibre, the attenuation power is negligible (2 dB) for R P 1000a. For MMF (62.5/125) the corresponding radius of curvature for negligible attenuation power is R P 62:5 mm. Then, we determine the inter-mode mixing depending on the radius of curvature by approximating the bent fibre with a series of sections in Fig. 11b. The N sections have a length l ¼ Ra=N. The purpose of this decomposition is the determination of new excited modes at the output of curved fibre for each MGDM channel. We assume that each section of length l is tilted from the subsequent section. Each section excites a new MG at the input of the next section. In the last section we can determine the overall excited MG of the total curvature. The Fig. 12 shows the geometry of two sections. (l; m) modes propagate in the Section 1 with the modal amplitude al;m ðlÞ at output of this section. The modal amplitude al;m ð0Þ of the excited mode at the input of Section 2 is calculated in the corresponding coordinate (x, y, z). The calculation of modal amplitude by the overlap integral allows us to identify new excited modes in the second section. The overlap integral calculated using section I axes is done by:
al0 ;m0 ð0Þ ¼
ZZ
! ! ðEI EII Þuz dA;
ð16Þ
! ! where EI and EII denote the electric field in the sections I and II, respectively.
15
10
a
b
l
5
0
0
10 20 Modes Group
30
Fig. 10. Modal power distribution for two offsets at two wavelengths.
Fig. 11. (a) Bent fibre of radius R where a is the angular curvature. (b) The equivalent sections.
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
y
y’
x’
0 0 Illmm
x
Section 1
Section 2
Fig. 12. Geometry for two sections along the fibre curvature.
al0 ;m0 ð0Þ ¼
X
al;m ðLÞ
ZZ
ð~ Eðl;mÞI ~ Eðl0 ;m0 ÞII Þuz dA
l ;m
¼
X
0 0
al;m ðLÞIllmm :
ð17Þ
l ;m
The coupling coefficient is given by: 0 m0
Illm ¼
ZZ
ð~ Eðl;mÞI ~ Eðl0 ;m0 ÞII Þuz dA:
ð18Þ
Note that in calculating the modal amplitude al;m ðlÞ, the attenuation coefficient is not taken into account since l is very low. The (17) may be repeated N times to determine the modal amplitude at the final output section of the curved part of fibre. The modal amplitude al1 ð0Þ on the last section N (output) is given by:
alN ðlÞ ¼
" X X
lN1
lN2
X ðal1 ð0Þejlbl1 Ill21 Þ ejlbl2 Ill32
!
#
l1
ejlblN1 IllNN1 :
ð19Þ
The modal amplitude is calculated for each offset at the input of straight section of the fibre. The determination of the coupling coefficient and the modal amplitude is made using numerical computing with MatlabTM. For parabolic fibre, this coefficient can be determined analytically. Coupling occurs only between modes of the same polarization and same orientation. The coupling coefficients between Ex -modes and Ey -modes, respectively, are equal [22]. Therefore, only Ex -modes will be considered in the following. For LP modes, we may use a normalization condition different from (18) according to: 0 0
Illmm ¼
ZZ
ðExI ExII Þuz dA;
ð20Þ
where ExI and ExII are the transverse electric fields to sections I and II, respectively. The transverse electric field may then be written as:
Ex ðx;yÞ ¼ ðw0 p2lþm l!m!Þ2 Hl
pffiffiffi ! pffiffiffi ! 2x 2y Hm expððx2 þ y2 Þ=w20 Þ; w0 w0 ð21Þ
( ¼
0
p2lþm l!m!gmm0 Lmmm ð2g2 Þ=2 if m 6 m0 ; 0 0 2 0 w0 eg p2lþm l!m0 !gmm Lmm0 m ð2g2 Þ=2 if m P m0 ;
w0 eg
2
N=40 N=70 without curvature N=30
20
15
10
5
0
0
5
10 15 Modes Group
Fig. 13. Modal power redistribution after curved fibre.
and
ExI ¼ Ex ðl; mÞ expðik1 hyÞ;
ExII ¼ Ex ðl; mÞ;
where H is the Hermite–Gauss function, and h ¼ a=N. According to (20), the coupling coefficient is written as: 0 0 Illmm
¼ Al;m Al0 ;m0
Z
Hm
pffiffiffi ! pffiffiffi ! 2x 2x Hl H l0 expð2x2 =w20 Þ dx w0 w0 pffiffiffi ! pffiffiffi ! 2y 2y Hm0 expð2x2 =w20 Þ expðik1 hyÞ dy: w0 w0
Z
ð22Þ Eq. (22) has been solved by use of the integral 7.374.7 in [23]. The required coupling coefficients are identically zero if l–l0 and, otherwise, determined by:
ð23Þ
pffiffiffi where g ¼ ikhw0 =ð2 2Þ and L is a Laguerre–Gauss function. The more N big is, the more we approach the practical case of uniform curvature (Rc ¼ constant). The analytical determination of coupling coefficient follows by determining the modal amplitude at the curved section output. The calculation of the modal amplitude depends on the number of elements considered in the curved section (N). Fig. 13 shows the change of the coupling versus N for a uniform curvature Rc ¼ 100 mm and a ¼ 50 . For N ¼ 30 elements, there is intermode coupling at the output of the uniformly curved section for the fundamental channel, while for 70 elements, we obtain a quasi-null coupling. This result supports well the theoretical results published in the literature [24–26], which state that no (or negligible) coupling occurs between the modes of propagation in a circularly bent fibre with sufficient radius of curvature, leading therefore to negligible inter-mode mixing. In the case of a 2-D (that is bending in a single plane) non-uniform curvature (Rc variable), we decompose the bent part in a series of M uniformly curved sections, as shown in Fig. 14. We take the example of a curved fiber in the form of a half ellipse. This part is decomposed into a series of M ¼ 12 adjacent circular arcs (consisting each in N ¼ 70 straight sections as above). Each arc is defined by a radius Ri and the angle ai between this arc and the following one. We introduce the constraint 1000a 6 Ri 6 4000a. We simulate a fiber of length 100 m with a knot which presents the curvature after 50 m crossed (see Fig. 14). We determine the
Power distribution (%)
3914
Fig. 14. Description of a non-uniform curvature.
20
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M. Awad et al. / Optics Communications 282 (2009) 3908–3917
40 no curvature
35
Capacity (Bits/s/Hz)
distribution of the modal power after curvature. For a (N N) MGDM, the determination of the channel transfer matrix is similar to steps previously explained in Section 2. Fig. 15 shows the conversion of the modal power from the fundamental mode to the higher order modes after the curvature. The higher order modes are excited at the output of this curvature and increase the MSI with other channels. To study the curvature effects on the MGDM system, we compare the capacity of this system with and without curvature. Fig. 16 shows this comparison for a (4 4) MGDM system, without curvature and another with a curved knot. At SNR = 30 dB, the capacity is significantly decreased (from 34 bits/s/Hz to 24 bits/s/ Hz). In presence of curvature, the capacity is not far from the SISO system one. However, the effect of curvature on the system depends on the number of users. The launch space for a large number of users excites the MG close to each other. Hence, the probability of MSI increases. A MGDM system with a larger number of users will be more affected by a non-uniform curvature and an increase of knot curvature number.
30 1 knot
25
2 knots
20 15 SISO
10 5 2x2
3x3
4x4
5x5
N ×N Fig. 17. The system capacity versus number of users (SNR = 30 dB).
25
after curvature before curvature
Power distribution (%)
20
15
10
Fig. 17 shows the effect of curvature versus the number of users in the system. For the knot presented in Fig. 14, the decrease of the capacity in a (5 5) MGDM system is more noticeable than that in a (2 2) system (reduced from 40 bits/s/Hz to 27 bits/s/Hz at SNR = 30 dB for 5 5 and reduced from 20 to 14 bits/s/Hz for 2 2). For one knot, the capacity is reduced to 24 bits/s/Hz for 4 4. This capacity is twice that of a SISO, and this is much like for a (4 4) CO-MIMO system (correlated channel taps) demonstrated by [27]. For two knots, the capacity for a 4 4 MGDM is lower than that of the (4 4) radio MIMO one. 5. MGDM system application and simulation
5
0
0
5
10 Modes group
15
20
Fig. 15. Modal power redistribution after curved fibre.
35 30
MGDM (4 × 4) no curvature MGDM (4 × 4), curved
Capacity (Bits/s/Hz)
SISO
25 20
We analyze system performance for the MGDM communication link model of Section 2, including influence of curvature. For N N MGDM systems we simulated the average BER for several system configurations differing in modulation format, fibre length, SNR and curvature effect. The simulated model is shown on Fig. 18. Our simulation chain is run using the VPI software. We assume N independent inputs. Each input is OOK modulated by an independent pseudo-random binary data stream (2.5 Gb/s, 5 Gb/s). Each laser, associated with a user, modulates the binary data at the same wavelength (1330 nm) with the same power (0 dB m). By co-simulation between Matlab and VPI, we introduce the MGDM channel model of Section 2 in simulations. The simulation using the model matrix H is followed by the effect of changing of the fibre length on the channels. The coupling between the fibres modes slowly varies relative to the symbol period affected by acoustic, thermal, and laser frequency instabilities. Here, we assume that the matrix H is periodically estimated at a rate faster than its effective rate of change.
15 Photodiode Baseband Z-F signal Laser MMF
10 Channel #1 Channel #2
5 0 −10
PBRS PBRS
Channel #N PBRS
0
10 SNR (dB)
20
MGDM channel model
E Q U A L
30
Fig. 16. The effect of curvature on (4 4) MGDM system capacity.
PRBS = Pseudo Random Binary Sequence Fig. 18. The demonstrator model.
BER Received signal
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M. Awad et al. / Optics Communications 282 (2009) 3908–3917
At the receiver, each photodiode captures a mixture of signals. For MGDM, the simplest receiver architecture is matrix inversion, a Zero-Forcing (Z-F) method in line with the requirement of service transparency. The relation between signals given by (2) will be:
^s ¼ s þ nH1 :
0
10
−5
10
with Z−F, channel #1 with Z−F, channel #2 without Z−F
−10
10
2
4
6
8
BitRate (Gb/s) Fig. 20. BER measurements versus bit rate, for different channels in the (3 3) system at fibre length L = 1 km.
10
10
0
−5
BER
However Z-F has a major disadvantage: when the hij values are small, the element of H1 will be large and can amplify the noise component n (AWGN noise), which will degrade the SNR of the estimated symbols. Consequently, erroneous decisions are produced at the receiver. By choosing a SNR = 30 dB for our simulation, we study the performance of the system by measuring the BER before and after equalizer (Z-F), for many users and for various lengths of fibre. We choose the number of users equal to the number of receivers. The BER penalty changes depending on the number of channels created in the fibre and depending on the fibre length. The Fig. 19 corresponds to the case where data is sent over the fundamental channel (offset = 0 lm) at a bit rate of 5 Gb/s. It shows an increase of BER for large distances. For 3 3 systems, a BER < 1010 is observed at L ¼ 500 m, but this value goes up to 104 at L = 1000 m. Moreover, the BER changes significantly between 3 3 and 5 5 systems. The performance can vary from one channel to another within the same system. The BER measured for data associated with the fundamental channel for the (3 3) MGDM system is not the same as those associated with another channel (offset = 13 lm). Fig. 20 shows the change of BER for different bit rates for the fundamental channel (channel #1) compared to that of channel #2 (offset = 13 lm). For D = 3 Gb/s, the BER changes from 1010 (channel #1) to 108 (channel #2). For this bit rate BER = 103 before Z-F equalizer. The MSI between the channels is at its maximum in the channel at core extremity compared with fundamental channel. Finally, we study the effect of the fibre curvature on the system performance. We simulate a 3 3 transmission by taking into account the curvature of the fibre presented in previous section and we compare its performance with that of a chain without curvature. Fig. 21 shows the change in performance. The BER increases
BER
ð24Þ
10
no curvature
−10
with curvature
0
10
BER
10
10
−5
−15
300
400
500 600 Fibre length (m)
700
800
Fig. 21. BER measurements versus fibre length, for a 3 3 system with and without curvature at D = 10 Gb/s.
10
−10
MGDM 3 × 3
10
from 1014 to 107 for L = 300 m. With the curvature, the MSI is becoming important and the degradation of the system is significant. We can get a BER = 104 for D = 10 Gb/s at 300 m in the presence of the curvature. One may notice that this BER decreases for D < 5 Gb/s.
MGDM 4 × 4
−15
MGDM 5 × 5
6. Summary and discussion
10
−20
400
600
800
1000
Fibre length (m) Fig. 19. BER measurements versus fibre length, for different systems using a constant bit rate of 5 Gb/s.
The conversion of the SISO system to a multi-users one encounters several difficulties in the MMF case (bandwidth, modal dispersion and cost of system). MGDM technology solves many of these difficulties and offers good functionality. But to improve the functioning of this technique, it was important to study the restrictive
M. Awad et al. / Optics Communications 282 (2009) 3908–3917
effects (excitation conditions, fibre curvature). In this paper, we have presented an analytical model for N N MGDM systems, taking into account the conditions of excitation and reception. First we presented the best conditions for emission to improve the system (offset, spot size, and wavelength). Applying our statistical model, we have evaluated the MMF MGDM system capacity compared to SISO systems. The inter-mode mixing effect in this system is studied and the analytical modelling is shown. We investigated the effects of the fibre curvature. We compared the capacity of the MGDM system with curvature and without curvature and we demonstrated the minimum allowable radius of curvature. We evaluated the MMF MGDM system performance for various numbers of users. The results indicate that the BER < 1010 for a 5 5 MGDM system for a 5 Gb/s bit rate over several hundreds of meters of MMF. Finally the deteriorated performance of our system in the presence of the curvature model is presented. A followup research direction, based on the results of this paper, is the physical realization and characterization of a MGDM system. References [1] I. Dayoub et al., Eur. Trans. Telecom. 18 (2007) 811. [2] M. Greenberg et al., IEEE J. Lightwave Technol. 25 (2007) 1503.
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[3] L. Raddatz, I.H. White, D.G. Cunningham, M.C. Nowell, IEEE J. Lightwave. Technol. 16 (1998) 324. [4] X.J. Gu et al., IEEE Photonics Technol. Lett. 18 (2006) 244. [5] G.D. Golden et al., Electron. Lett. 35 (1999) 14. [6] H.R. Stuart, Sci. Mag. 289 (2000) 281. [7] S. Schöllmann, C.Xia, W. Rosenkranz, in: Techn. Digest (CD) of OSA Opt. Fiber Conf., 2006. [8] C.P. Tsekrekos et al., Opt. Lett. 32 (2007) 1041. [9] C.P. Tsekrekos et al., Opt. Exp. 15 (2007) 3656. [10] J. Nathan Kutz, J.A. Cox, D. Smith, IEEE J. Lightwave Technol. 16 (1998) 1195. [11] T. Koonen et al., in: Proc. IEEE LEOS Benelux Symp., vol. 87, 2002. [12] R.W. Smink et al., in: Proc. IEEE LEOS Benelux Symp., vol. 257, 2006. [13] C.P. Tsekrekos et al., IEEE Photonics Technol. Lett. 18 (2006) 2359. [14] G. Yabre, IEEE J. Lightwave. Technol. 18 (2000) 166. [15] J. Saijonmaa et al., Appl. Opt. 19 (1980) 2442. [16] C.P. Tsekrekos, IEEE Photonics Technol. Lett. 18 (2006) 2359. [17] L. Jacomme, Opt. Commun. 14 (1975) 134. [18] E.L. Chinnock et al., Proc. IEEE Lett. 61 (1973) 1499. [19] J. Nathan Kutz, J.A. Cox, D. Smith, IEEE J. Lightwave Technol. 16 (1998) 1195. [20] M.-Y. Loke, J.N. McMullin, IEEE J. Lightwave Technol. 8 (1990) 1250. [21] Allan W. Snyder, john D. Love, Optical Waveguide Theory, Springer, 1983. [22] K. Petermann, IEEE J. Quantum Elect. 16 (1980) 761. [23] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals Series and Products, Academic Press, New York, 1965. [24] D. Donlagic, B. Culshaw, IEEE J. Lightwave Technol. 18 (2000) 334. [25] A.-K. Ghatak, I.-C. Goyal, A. Gupta, J. Mod. Optics 25 (1978) 1. [26] R.-T. Schermer, Opt. Exp. 15 (2007) 15674. [27] A. Tarighat, IEEE Commun. Mag. 45 (2007) 57.