Abstract

We investigate the concept of principal modes and its application for mode division multiplexing in multimode fibers. We start by generalizing the formalism of the principal modes as to include mode dependent loss and show that principal modes overcome modal dispersion induced by modal coupling in mode division multiplexing operation, even for multi-mode-fibers guiding a large number of modes, if the product of modulation bandwidth, fiber length and differential group delay is equal or less than one in each transmission channel. If this condition is not sustained, modal dispersion and crosstalk at the receiver limit the transmission performance, setting very high constraints towards modal coupling.

© 2012 OSA

1. Introduction

The demand for capacity has been increasing exponentially over the last decade, mainly due to the large growing Internet traffic. The standard single mode fiber (SSMF), which is currently the fiber used for long-haul transmission systems, is reaching its capacity limit and new technologies are needed to confront the challenges of higher capacity transmission [1].

Multi-mode fibers (MMF) offer such a possibility by addressing each propagation mode independently as to increase the capacity by the number of propagating modes. Since the complexity of such systems increases with the number of propagating modes, attention has been placed recently towards few mode fibers, which guide a small number of modes. Most of these investigations [2] use the eigenmodes of the unperturbed few mode fibers as transmission carriers and compensate signal distortion induced by crosstalk and modal dispersion at the receiver using multiple-input-multiple-output signal processing. Instead of using the eigenmodes of the unperturbed MMF as carriers, we use so the called principal modes [3,4] as separate co-propagating transmission channels, to minimize the effect of modal dispersion and in order to avoid complex signal processing at the receiver. Some of the challenges of such transmission system, for instance the amplification throughout the propagation by means of a multimode EDFA, the selective excitation of a specific propagating mode at the input and the spatial filtering at the receiver have been investigated by several authors [58]. Our efforts however, will be focused on the modeling of the transmission channel and the characterization of the carriers used for transmission, the principal modes, which is assessed in section 2. Section 3 then analyzes the transmission performance in terms of achievable transmission capacity for an exemplary three mode fiber by comparing the usage of principal modes (PM) and eigenmodes (EM) as carriers. The key aspect of this analysis is to evaluate the transmission performance of the carrier modes in mode division multiplexing operation by using selective mode excitation at the transmitter and spatial filtering together with direct detection at the receiver without the need of multiple-input-multiple-output (MIMO) signal processing. Finally the scalability of this approach will be assessed in section 4 using MMF guiding a larger number of modes.

2. System modeling

2.1. Principal modes and fiber transmission matrix

The bandwidth and transmission distance of a traditional MMF link is limited by modal dispersion. Modal dispersion arises due to the group velocity difference among the eigenmodes of the MMF. A pulse propagating on several eigenmodes arrives distorted at the output of the fiber. Even if only one eigenmode of the ideal MMF is selectively excited at the input of the MMF, the pulses shape is not guaranteed to be preserved because of modal coupling. For this reason it would be useful to find some propagation state which suffers minimal distortion in the presence of modal coupling. Finding such a propagation state is equivalent of finding a propagation state in the frequency domain which does not change to the first order in frequency. This condition leads to the principal modes [3]. These modes are frequency independent to the first order and offer a possibility of distortion-less transmission in a MMF. In order to understand the principle behind the principal modes (PM) we start to describe the transmission through a MMF by describing the input and output fields. In general one can describe the transversal input field in a MMF for one polarization as a weighted superposition of the eigenmodes of the unperturbed fiber as:

Ein(x,y)=i=1IaiEi(x,y).

Hereaiis a complex weighting coefficient describing the excitation of the ith eigenmode of the unperturbed MMF and I is the total number of modes.

The eigenmodes of the unperturbed MMF are normalized so that they obey the orthonormality condition:

1ξ2Ei(x,y)·Ej*(x,y)·dxdy=δi,j,
whereδi,jis the Kronecker delta symbol. After propagating through the MMF, we can describe the output field also as a superposition of eigenmodes with different complex weighting coefficientsbi:

Eout(x,y)=i=1IbiEi(x,y).

The relation between the input excitation coefficientaiand the output coefficientbican be expressed as follows:

b=ejϕ0(ω)T(ω)·a.

Hereaandbdescribe the column vectora=(a1,a2,,an)Tandb=(b1,b2,,bn)T respectively, T(ω)is the transmission matrix describing the propagation along the MMF and ϕ0(ω)is a common phase to all modes. Here we have usedωas the deviation from the angular center frequencyω0. The common phase ϕ0is arbitrarily chosen to be the phase of the fundamental mode given asϕ0(ω)=β(ω)L(β0+τ0ω)L, where β(ω)is expanded in terms of a Taylor series. The termτ0stands for the group delay per unit lengthβ0is the phase constant and L is the transmission length. The exponential term in (4) is pulled out of the matrixT(ω)in order to normalize propagation inside the matrix with respect to the fundamental mode to make T(ω)a slowly varying matrix, which is advantageous for numerical simulations. The transmission matrix T(ω)is modeled, similar to the model proposed in [4], by assuming that the complete MMF link consists of an assemble of short MMF segments as shown in Fig. 1(b) . We assume ideal propagation along each segment m (no mode coupling) described by the diagonal matrixM(ω). Each of the diagonal elements contains the differential group delay and differential propagation constant, as well as a random phase as follows:

Mm(ω)=(1000ej(Δϕ1+ζ1,m)00000ej(ΔϕI+ζI,m)),
whereΔϕi=Δβ(ω)Lm(Δβi+Δτiω)Lm.HereΔβiandΔτiare both relative to the fundamental mode as mentioned earlier and represent the differential phase constant and differential group delay, respectively, of the ith eigenmode. The random phases ζi,maccounts for variation in the phase constants due to strain or temperature. The variable Lm represents the segment length of the MMF, where m is the index of the segment.

 

Fig. 1 (a) Fiber mismatch in x and y direction, rotated by an angle, (b) Complete fiber is made out of 3 km long segments.

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Mode coupling is modeled by assuming fiber alignment mismatches at splice points as shown in Fig. 1(a). This is due to the assumption that longer terrestrial transmission systems will rely on spliced fibers which will in turn induce mode coupling. We expect mode coupling induced by micro bending along a MMF segment to be relatively small compared to the influence of splices since the segment length is only 3km and we proceed by calculating the induced mode coupling through overlap integrals as:

Kij,m=1ξ2Ei,m(x,y)·Ej,m+1*(x´,y´)dxdy.

Here Ei,m is the ith eigenmode of the unperturbed MMF in the segment m and Ej,m+1 respectively the jth eigenmode of the MMF in segment m + 1. Our transmission matrix is then given by:

T(ω)=m=1MMm(ω)·Km,
where matrix Kmdescribes the coupling between segment m and m + 1; M is the total number of MMF segments. As data signals occupy a certain bandwidth, it is necessary to consider the frequency dependence of the output vector b given a fixed input vectora. For this reason we start by evaluating the derivative of Eq. (4) with respect to the angular frequency denoted asω. This leads to:
ωb=[jT(ω)ωϕ0(ω)+ωT(ω)]ejϕ0(ω)a,
where we have assumed ωa=0 since the input vector ais fixed. By rearranging Eq. (8) and using Eq. (4) we obtain the following expression:

ωb=[jT(ω)ωϕ0(ω)+ωT(ω)]T(ω)1b.

This equation can rearranged such as:

ωb=[jτ0LI+G(ω)]b,
whereIis the identity matrix and  G(ω)=ωT(ω)·T(ω)1. The frequency derivative ofϕ0(ω)can be identified asτ0L, the group delay of the fundamental mode over the whole transmission length L. Equation (10) tells us that an output field pattern, represented by the output vectorb, changes with frequency to the first order due to the matrixG(ω). Nevertheless, it is possible to find a vectorbpthat is frequency independent to the first order if the frequency dependent matrixG(ω)acting onbcomplies with the following equation:

G(ω)·bp=γpbp.

This can be rewritten as an eigenvalue equation as follows:

G(ω)γpI=0,
whereγpare the complex eigenvalues of the matrixG(ω). The eigenvectors computed through Eq. (11) are the PMs at the output of the MMF and we designate them asbp. Inserting Eq. (11) in Eq. (10) we obtain:
ωbp=[j(τ0+τp)LαpL]bp,
where we separatedγpinto a real and imaginary part asγp=(αp+jτp)L. Equation (13) shows thatbpis frequency independent sinceαpandτpare scalar values. The eigenvalues of Eq. (12) are in general complex as mentioned earlier since our matrixT(ω)is not unitary due to losses into radiating modes at splicing points. The imaginary part ofγ,Im[γ]=τpLcan be interpreted as the differential group delayτpof the PM times the transmission length L. The real part ofγ,Re[γ]=-αpLcan be related to a frequency dependent loss over the transmission length for each PM. As a consequence of this the PMs are not orthogonal to each other, but still linearly independent. Figure 2 illustrates the idea of the PMs. The PMs at the input can be computed either by using: Eq. (4) or by following a similar approach as described from Eq. (9) to Eq. (14). This results in the following equation:
ωa=[jωϕ0(ω)IF(ω)]a=0,
where  F(ω)=T(ω)1ωT(ω)Solving the eigenvalue equation:
[F(ω)τpI]ap=0
we obtain the same eigenvaluesτpand are capable of computing the eigenvectors corresponding to the PMs at the input of the MMF, which we designate withap.

 

Fig. 2 Main idea behind the PMs; (a) shows and input vector apwith only three components (Three mode fiber for example) corresponding to a PM at the input of the fiber. As we change the frequency interval fromω1ω2...ωnthe output vectorbpstays unchanged; (b) shows the result if the input vector a is not a PM. As we change the frequency over a small frequency interval ω1ω2...ωn the output vector b(ω)changes

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We emphasize here that Eq. (15) and Eq. (12) are not identical sinceG(ω)F(ω)which is a direct consequence of T(ω)not being unitary. This means that Eq. (15) and Eq. (12) have different eigenvectors and as a consequence, the principal modes at the inputapare not equal to the principal modes at the outputbpand our equations differ at this point from the derivation presented in [3]. We will now proceed with the details of the MMF and the eigenmodes of the MMF.

2.2. Eigenmodes of the MMF

The eigenmodes of the MMF and their propagation constants depend strongly on the fiber geometry (radius and index profile) as well as on the relative index differenceΔ=(n1n2)/n1. Assuming a weakly guiding MMF, that isΔ1, with a parabolic index profile given as [9]:

n(r)=n1(12Δ(r/r0)2),
wheren1is the refractive index profile in the center of the core andr0the core radius, we can formulate the transverse field distribution of the MMF in good approximation in terms of the Laguerre-Gauss modes. We obtain the LP Modes (for one polarization) as given by [9] as:

El,q(r,φ)=Cl,q(rξ)lLql(r2ξ2)er22ξ2{sinlφcoslφ.

Here l describes de circumferential order and q the radial order, Lql the Laguerre polynomial and ξ is given as:

ξ=r0/(k0n12Δ).
Cl,qis a normalization constant, normalizing the field as shown in Eq. (2). The propagation constant and group delay coefficient are given for the Laguerre Gauss modes according to [9] by:
βq,l=n1k0(122Δn1k0r0(2q+2l+1))
and
τq,l=N1c(1+Δ(2q+l+1n1k0r0  )2),
where N1 is the group index in the core. Equation (20) neglects profile dispersion. Using this model, we will now proceed to analyze a simple three-mode system, in order to study the main properties of principal mode transmission in mode division multiplexing operation without the usage of MIMO equalization techniques and compare them to the traditional eigenmode transmission.

3. Performance evaluation in a three mode system

To understand some of the main limitations when using PMs for MDM purposes we investigate a three-mode system numerically. A layout of the simulated three mode transmission system is presented in Fig. 3 . A single coherent light source is used as transceiver and its output power is divided equally into three different modulators, resulting in three individual signals at the same wavelength. The spatial field distribution of each carrier is modified to match a specific principal mode or eigenmode in the optical domain by spatial filtering. The field conversion can be realized as mentioned in [7] with free space optics using a spatial light modulator, by using a mode converter similar to the one proposed in [10,11] or byusing a diffractive optics approach as shown in [12]. If the spatial field distribution is matched to a PM, adaptive techniques would be required due to temporal channel variations. Here we will only consider the time interval where the channel is practically stationary so that PM estimation is required only once per simulation. The modes are then multiplexed into the MMF and de-multiplexed at the output of the MMF. Multiplexing and de-multiplexing can be considered as mirror images and it is therefore possible to apply the same concepts as mentioned for the multiplexing. After de-multiplexing, the signal is amplified to compensate for losses and direct detection is applied.

 

Fig. 3 Exemplary three mode transmission system. optical fields coming out of each modulator (MOD) are encoded and then matched to a desired spatial mode (M-MUX), which can either be a PM or EM. These are then multiplexed into the MMF. At the output of the MMF, the sum of all PMs or EMs is mode-de-multiplexed (M-DE-MUX) and detected at the receiver (Rx). The M-DE-MUX can be realized for instance by a diffractive element or a photodiode array together with a local oscillator to obtain space and phase information.

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The transverse field distributions given in Eq. (17) now simplifies for the LP01 mode as:

E01(r,φ)=1πer2 2ξ2 
and the LP11 modes as:
E11(r,φ)=2/πrer2 2ξ2{sinφcosφ
for the odd and even mode respectively. The coupling matrix for a three mode system can be evaluated for a radial offset b and a rotation angleφ0as shown in Fig. 1 using Eq. (6) as:

Km=(eb22ξ2beb22ξ2(cosφ0+sinφ0)2ξbeb22ξ2(sinφ0cosφ0)2ξbeb22ξ22ξeb22ξ2(sinφ0b2+(b22ξ2)cosφ0)2ξ2eb22ξ2((b22ξ2)sinφ0b2cosφ0)2ξ2beb22ξ22ξeb22ξ2(cosφ0b2+(b22ξ2)sinφ0)2ξ2eb22ξ2(b2sinφ0(b22ξ2)cosφ0)2ξ2)

The simulation parameters are given in Table 1 and some values require explanation; αrepresents the total transmission loss induced by splices and is calculated by exciting all guided modes in the MMF input and measuring the overall loss at the output. ξ is the mode field size of the LP01 mode and can be calculated using Eq. (18), Lm is the fiber segment length in which we assume ideal propagation, b/ξ is the splice mismatch to mode field ratio and induces the splice loss α along the MMF; φ0 is used to randomize the mode coupling between two MMF segments by rotating the axis with respect to the previous. The refractive index n1 and the group index N1 were calculated using the Sellmeyer equation [13] for a fiber with 90% SiO2 concentration andΔτmis the maximal differential group delay (DGD) between the fastest and slowest propagating mode.

Tables Icon

Table 1. Fiber Simulation Parameters Used for MDM Simulation in Three Mode Fiber

Using these parameters we can estimate the bandwidth B of the three mode fiber limited by modal dispersion as [13]:

B1ΔτmL0.7GHz.

Since the PMs are frequency independent to the first order, we expect distortion effects due to modal dispersion and mode coupling to be negligible up to a modulation bandwidth of 0.7GHz. We now proceed to compare the transmission quality of PMs under MDM operation against the well-known eigenmode launch as presented for example in [2]. This is realized however without the usage of MIMO signal processing and using direct detection.

We start by transmitting three random bit sequences with OOK-NRZ modulation format containing 512 symbols over the MMF at0.7Gbit/s. Each bit sequence si(t)is encoded on one spatial mode and multiplexed into the MMF as:

aT(ω)=i=1ISi(ω)ai.

Here Si(ω)=F{si(t)} denotes the Fourier transform of the bit sequencesi(t)and aidenotes the spatial mode, which can either be an EM or PM. At the output of the MMF we spatially decouple each mode as discussed further on in section 3.1, detect the output power and compute the eye opening penalty. Additive Gaussian white noise is not included in this transmission and losses are compensated at the receiver in order to study the eye opening penalty caused by inter-symbolic interference induced by mode coupling. Figure 4 shows an exemplary result for the eye opening computed at the receiver using (a) EMs as carriers and (b) PMs as carriers. We notice that the eye opening in Fig. 4(a) has several quantized amplitude levels inside of what should be a perfect eye.

 

Fig. 4 Eye diagram plotted for a PRBS OOK signal at 1 Gbits/s over 50 km MMF; a) using EM as carriers; b) using PMs as carriers.

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This can be explained considering the discrete nature of the mode coupling induced by splices in our transmission link. PMs on the other hand do not show these discrete amplitude levels since they are computed to overcome this problem, as shown in Fig. 4(b).Nevertheless we observe a slight deformation of the pulse since the signal bandwidth is slightly above the MMF bandwidth. Figure 5 shows the eye opening penalty (EOP) for a transmission link with different number of total splice losses using the (a) EMs and (b) PM as carriers, at a transmission rate of 0.7Gbit/s. The EOP is defined as:

EOP=10log(EOBTBEO),
where EO stands for the eye opening and is defined as the difference between the minimum 1 and the maximum 0 level of the eye and E0BTB stands for back-to-back eye opening. As we can see, there is one eigenmode that has a lower eye opening penalty, which we can identify as the LP01 mode. This result is of course expected since the LP01 mode suffers less from each splice mismatch because its field strength is concentrated in the center of the MMF. The LP11 modes on the other hand have more extended field distributions making them more susceptible to splice mismatches and with it the induced mode coupling, which explains the larger EOP values. Figure 5(b) shows that each PM is practically transparent to modal dispersion. The eye opening penalty is less than one decibel for the simulated range and for this particular bit rate, which makes signal processing at the receiver easier.

 

Fig. 5 EOP for a three mode fiber under MDM operation at 0.7 Gbit/s. a) EM as carriers and b) PM as carriers. MMF simulation parameters are given in Table 1.

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Before we proceed to scale the bit rate in the MMF transmission we define a 2 dB eye opening penalty criterion which allows us to define whether a transmission mode is usable for transmission. In other words a mode is adequate for transmission if its eye opening penalty is less than 2 dB. Using this criterion we obtain the results for the usable modes as presented in Fig. 6 (a) where EMs are used as carriers and (b) where PMs are used as carriers.

 

Fig. 6 EOP of MDM transmission for various bitrates in a three mode fiber; a) Using EM as carriers, b) using PM as carriers.

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First we realize in Fig. 6(a) that the number of usable eigenmodes is not three even for 0 dB loss. This is due to modal coupling, which rises at each splice point due to fiber angular misalignment between two MMF segments as shown in Fig. 1(a). Mode coupling occurs now, without radial offset b, only between the two degenerate LP11 modes, which have the same propagation constants.

The bit pattern contained in each LP11 mode arrives at the same time at the receiver, but closes the eye because of the additional amplitude levels induced through modal coupling as shown in Fig. 4(a). As the splice mismatch b increases, the number of usable modes decreases to the point where no eigenmodes achieve our usability criteria. Additionally we observe that as we increase the bit rate, the curve decreases earlier as well. Figure 6(b) shows the number of usable PMs, which stays constant for the one Gbit per channel transmission. As the transmission rate increases though, a reduced number of usable PMs can be observed as the total splice loss and modal coupling increases. These simulations show that the PMs operate well if the product of frequency deviationω/2πand maximal time delay between modesΔtmis less or equal than one, that is if ω/(2π)Δtm1,whereΔtm=ΔτmL. assuming that mode dependent loss is compensated. Sinceω/2πis roughly proportional to the modulation bandwidth of the signal and the maximal time delay is proportional to one over the MMF bandwidthΔtm1/B, we can say that modal dispersion can be disregarded if the ratio of modulation frequency and MMF bandwidth is less than one. This will be reviewed in section 4 when the number of guided modes is increased and with itΔtm.

Another important factor which needs to be analyzed is crosstalk at the receiver, which will be discussed in more detail in the next section together with the spatial demultiplexer used to demultiplex each EM or PM.

3.1. Crosstalk limitations and spatial filtering

In order to operate the MMF in a MDM operation, it is necessary to demultiplex each transmission mode at the receiver. This has been realized using lenses [2], holograms [14] and diffractive gratings [12]. The goal of this is to modify the output field distribution in such a way, that the orthogonality condition can be applied to discriminate the mode of interest. In our model, where each vector component represents the weighting coefficient for each eigenmode of the unperturbed MMF, it reduces to a scalar multiplication of the output vector b with a detection vectord.This means for instance, that if we would want to discriminate the LP01 mode at the receiver, we would have to scalar multiply the output vectorbwithd=g1(1,0,0)T,whereg1is the gain factor needed to compensate for transmission loses, which are different for each transmission mode. In the case of PM detection, this would extend to the scalar multiplication of the output PM bp,iwith its conjugate complex vectorbp,i*.This is only correct if the PMs form a complete orthogonal basis, which in our case they do not, as mentioned in section 2.1. Computing bp,i·bp,j0 at the output would then lead to residual crosstalk from the remaining PMs at the receiver. Crosstalk at the jth channel can be defined as [15]:

PjC(ω)=i=1,ijI(bi·dj)(bi·dj)*.

Herebiis the ith output field distribution at the output of the MMF anddjis the jth modal filter used at the receiver. In order to avoid crosstalk at the receiver when using PMs as carriers we compute a set of detection vectorsdp,ithat are capable of detecting the ith principal modes at the output selectively for the center frequencyω0/2π=c/λ0, where c is the speed of light. They can be estimated solving following equation:

P·D=I.

HerePis the matrix containing in its rows the principal modes at the outputbp,Iis the identity matrix andDis the detection matrix in which each column represents one detection vector used to detect the ith PM crosstalk free. It is worthwhile noting here that the definition of the detection vectors in Eq. (28) automatically implies the compensation of mode dependent losses. Figure 7 shows the crosstalk at the receiver for three different transmission scenarios; (a) represents the crosstalk at the receiver when eigenmodes are used as carriers, (b) is the crosstalk at the receiver when PMs are used as carriers using their complex conjugate di=gibifor spatial de-multiplexing (gi = gain factor to compensate for their losses) and (c) is the crosstalk at the receiver when PMs are used as carriers and Eq. (28) is used for spatial de-multiplexing.

 

Fig. 7 Left: Crosstalk of: (a) LP11 (b) PM 2 and (c) PM 2 at the output of the MMF for three different splice losses. Right: Crosstalk for each guided (a) EM (b) PM and (c) PM using the detection vectors for constant total splice loss of 0.5 dB; Fig. (b) and (c) differ only at the demultiplexing stage at the receiver; (b) uses the conjugate complex of the PMs as mode filter whereas (c) uses the detection vector defined in Eq. (28). Each EM or PM exited at the input of the MMF contains unit power.

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We can see in Fig. 7(c) that the use of Eq. (28) forces zero crosstalk at the carrier frequency, while the use of the complex conjugates in Fig. 7(b) leaves some residual crosstalk at the receiver at zero frequency deviationω/2π.As the frequency deviation increases, crosstalk increases by about 10 dB per 100 MHz and then fluctuates randomly around 10 dB. The maximal crosstalk value depends mainly on the mode coupling strength which grows as the total loss increases, as shown in Fig. 7(b) and 7(c) on the left hand side. It is also important to notice that each PM has a slight different crosstalk value. If we compare for instance PM1 and PM3 atω/2π=0.5GHzwe observe a crosstalk difference of about 5 dB. This leads to the conclusion that each PM perceives crosstalk differently and not only modal coupling limits the transmission rate but also crosstalk at the receiver when transmitting PMs in MDM operation. This could have a great impact when using a MMF guiding a larger number of eigenmodes in MDM, since the number of modes at the outputbleads to increased crosstalk. In addition, we plot in Fig. 7(a) the crosstalk value when the EMs are used as carriers. Here we notice that the crosstalk value at the receiver stays rather constants over the plotted frequency interval and its variations are not as pronounced as for the PMs. These crosstalk values could be improved, at least around the carrier frequencyω0/2π,by launching into the eigenmodes of the complete MMF. Knowing this we now proceed to scale the number of guided modes in the MMFs.

4. Scaling properties for systems with larger number of modes

We now proceed to scale the number of propagating modes in the fiber by increasing the numerical aperture and core radius r0 such as to keep the mode field radiusξof the LP01 mode constant. This value is chosen here to beξ=5.5μm, which is a value common to the OM4 multimode fiber withNA=0.2and core radius ofr0=25μm.The maximal group delay values Δτm for the different MMF are given in Table 2 . Using these values and our previous suggestion that(ω/2π)ΔτmL1, we can estimate the maximal transmission rate at which modal dispersion is avoided. For a MMF guiding 36 modes the maximal allowable transmission rate is about0.15Gbit/s, assuming a mode delay ofΔtm=6.7ns.This is now numerically verified by simulating a MDM transmission using PMs as carriers for various bit rates.

Tables Icon

Table 2. Maximal Group Delays for Different MMF

The bit sequence is 512 bits long using NRZ-OOK modulation format. Spatial demultiplexing is realized applying Eq. (28) at the receiver, which means zero crosstalk at the carrier frequencyω0/2π. The results are presented in Fig. 8 .

 

Fig. 8 Number of usable principal modes in a 36 mode fiber for various bitrates. The maximal differential group delay has the value ofΔτm=134ps/km and limits the maximal transmission rate down to0.1Gbit/s.

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As we can see the number of usable PMs is constant 36 for the bit rate of0.1Gbit/s. As we increase the bit rate to0.2Gbit/swe observe a slight reduction of one usable PM at 1 dB total loss. This behavior agrees with our previous suggestion of a maximal transmission rate of0.15Gbit/s. The amount of usable PMs reduces as we increase the transmission rate further to the point where only three PMs are usable for transmission. This value is reached at a transmission rate of0.8Gbit/sgiven a total splice loss of 1dB. Increasing the bit rate above the maximal transmission rate used in Fig. 8 makes only sense if we reduce the losses induced through splices and through it modal coupling. Tolerances used for MMF are no longer applicable since our system is not as robust as a MMF under classical operation (all modes contain just one set of information).

The allowed tolerances will rather be similar to the one of a SSMF and we will therefore assume the values given in [16] as a criteria for maximum allowable splice loss, which are given asαspl=0.01dB.This results in a total allowable splice loss of α0.16dB for our transmission link with 16 splices. Using these values we proceed to numerically evaluate a series of fibers guiding various numbers of eigenmodes. The number of guided modes in each of those fibers is listed in Table 2 together with their maximal DGD.

The transmission rate per channel assumed for this MDM transmission is10Gbit/s. Here we point out that these results are an extension of the results already presented in [17] and additionally correct the splice loss values given there. The results presented in Fig. 9 show that the number of usable PMs is very low even if we use the new allowable total splice loss value of 0.16 dB (this curve would be in between of the green and purple curve).

 

Fig. 9 Usable principal modes in a 50 km transmission link, at 10Gbit/s. This implies 16 splicing points. Results represent an enhanced version of the results presented in [17].

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As we increase the number of guided modes, the number of usable PMs decrease monotonically to the point where only one PM is adequate for transmission (right most value in the purple curve).If we reduce the total allowable splice losses even further (blue and red curve) we observe that the number of usable PMs increases as we increase the number of guided modes only up to 21 guided modes and then decreases again monotonically. From these results we can deduce that the turning point of each curve is related to the mode coupling strength induced by the fiber splice points. We can imagine for instance a MMF with no splices. The PMs become the eigenmodes, no modal coupling occurs and no crosstalk is measured at the receiver. As we increase the splice loss, modal coupling appears and for a given frequency range, modal dispersion can be safely ignored. As the frequency range increases for example by using a higher bit rate, modal dispersion cannot be avoided anymore and increased crosstalk is observed, which in turn closes the eye at the receiver. Crosstalk is frequency dependent, total splice loss dependent and PM dependent as shown in Fig. 10 . Here we see the crosstalk values of two random PMs out of the 55 available PMs. This figure shows very clearly that crosstalk is PM dependent and suggests that if the modulation bandwidth of the signal exceeds the MMF bandwidth, in this case roughlyB1/Δtm100MHz, the PMs used for transmission must be carefully selected. If all PMs are to be used for transmission, a multicarrier approach like OFDM as presented for instance in [18], combined with PMs for spatial multiplexing could be used to increase the transmission capacity while keeping the advantage of low signal distortion.

 

Fig. 10 Crosstalk of two PMs. Figure shows very clearly that crosstalk varies between principal modes and in this case a difference of 25 dB can be observed at 0.5 GHz between the dashed red and blue lines.

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5. Conclusion

We presented the concept of using principal modes as carriers for MDM transmission and compared their performance against the concept of using the eigenmodes of the unperturbed MMF as carriers. We showed that MDM using PMs as carriers is essentially possible and modal dispersion is mainly avoided if the product of(ω/2π)ΔτmL1is sustained. If high bit rate transmission using a single frequency carrier is used, splice losses need to be held extremely low, therefore small splice offset tolerances have to be stipulated to minimize the mode coupling in the MMF.

We pointed out that crosstalk at the receiver influences the transmission performance, especially at higher bitrates, since the frequency independence of the PMs does no longer hold and each PM interferes with one another at the frequency independent receiver. This could be well compensated using MIMO equalization as realized in [19] and the questions arises if the receiver complexity can be reduced by using the principal modes as carriers compared to using the eigenmodes of the unperturbed MMF as carriers.

References and links

1. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010). [CrossRef]  

2. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 x 6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012). [CrossRef]  

3. S. Fan and J. M. Kahn, “Principal modes in multimode waveguides,” Opt. Lett. 30(2), 135–137 (2005). [CrossRef]   [PubMed]  

4. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009). [CrossRef]  

5. N. W. Spellmeyer, “Communications performance of a multimode EDFA,” IEEE Photon. Technol. Lett. 12(10), 1337–1339 (2000). [CrossRef]  

6. P. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMH5.

7. G. Stepniak, L. Maksymiuk, and J. Siuzdak, “Binary-Phase Spatial Light Filters for Mode-Selective Excitation of Multimode Fibers,” J. Lightwave Technol. 29(13), 1980–1987 (2011). [CrossRef]  

8. C. P. Tsekrekos and A. M. J. Koonen, “Mode-selective spatial filtering for increased robustness in a mode group diversity multiplexing link,” Opt. Lett. 32(9), 1041–1043 (2007). [CrossRef]   [PubMed]  

9. H.-G. Unger, Planar Optical Waveguides and Fibers (Oxford University Press, 1977).

10. K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” J. Quantum Electron. 16(7), 761–770 (1980). [CrossRef]  

11. N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10 km two-mode fiber with mode coupler,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWA4.

12. N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. R. Ryf, P. J. Winzer, L. L. Buhl, Y. Sun, X. Jiang, and R. Lingle, Jr., “Space-division multiplexing and all-optical MIMO demultiplexing using a photonic integrated circuit,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2012), post-deadline paper PDP5B.

13. E. Voges and K. Petermann, “Vielmodenfaser,” in Optische Kommunikationstechnik, (Springer Verlag 2002) 214–260.

14. J. Carpenter and T. D. Wilkinson, “Precise modal excitation in multimode fiber for control of modal dispersion and mode-group division multiplexing,” in Proceedings of European Conf. Opt. Commun. (2011), paper We.10.P1.

15. A. A. Juarez, S. Warm, C.-A. Bunge, P. Krummrich, and K. Petermann, “Perspectives of principal mode transmission in a multi-mode fiber,” in Proceedings of European Conf. Opt. Commun. (2010), paper P.4.10.

16. R. K. Bocek, J. Hartpence, Y. Qian, and T. Lian OFS. “Ensuring low splice loss with high quality fibers”. [Online] http://stage.ofsinfo.com/resources/splice.pdf (2012).

17. A. A. Juarez, S. Warm, C.-A. Bunge, and K. Petermann, “Number of usable principal modes in a mode division multiplexing transmission for different multi-mode fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JTHA34.

18. A. Li, A. A. Amin, X. Chen, S. Chen, G. Gao, and W. Shieh, “Reception of dual-spatial-mode CO-OFDM signal over a two-mode fiber,” J. Lightwave Technol. 30(4), 634–640 (2012). [CrossRef]  

19. A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007). [CrossRef]  

References

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  1. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [Crossref]
  2. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 x 6 MIMO processing,” J. Lightwave Technol. 30(4), 521–531 (2012).
    [Crossref]
  3. S. Fan and J. M. Kahn, “Principal modes in multimode waveguides,” Opt. Lett. 30(2), 135–137 (2005).
    [Crossref] [PubMed]
  4. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. 27(10), 1248–1261 (2009).
    [Crossref]
  5. N. W. Spellmeyer, “Communications performance of a multimode EDFA,” IEEE Photon. Technol. Lett. 12(10), 1337–1339 (2000).
    [Crossref]
  6. P. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMH5.
  7. G. Stepniak, L. Maksymiuk, and J. Siuzdak, “Binary-Phase Spatial Light Filters for Mode-Selective Excitation of Multimode Fibers,” J. Lightwave Technol. 29(13), 1980–1987 (2011).
    [Crossref]
  8. C. P. Tsekrekos and A. M. J. Koonen, “Mode-selective spatial filtering for increased robustness in a mode group diversity multiplexing link,” Opt. Lett. 32(9), 1041–1043 (2007).
    [Crossref] [PubMed]
  9. H.-G. Unger, Planar Optical Waveguides and Fibers (Oxford University Press, 1977).
  10. K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” J. Quantum Electron. 16(7), 761–770 (1980).
    [Crossref]
  11. N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10 km two-mode fiber with mode coupler,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWA4.
  12. N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. R. Ryf, P. J. Winzer, L. L. Buhl, Y. Sun, X. Jiang, and R. Lingle, Jr., “Space-division multiplexing and all-optical MIMO demultiplexing using a photonic integrated circuit,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2012), post-deadline paper PDP5B.
  13. E. Voges and K. Petermann, “Vielmodenfaser,” in Optische Kommunikationstechnik, (Springer Verlag 2002) 214–260.
  14. J. Carpenter and T. D. Wilkinson, “Precise modal excitation in multimode fiber for control of modal dispersion and mode-group division multiplexing,” in Proceedings of European Conf. Opt. Commun. (2011), paper We.10.P1.
  15. A. A. Juarez, S. Warm, C.-A. Bunge, P. Krummrich, and K. Petermann, “Perspectives of principal mode transmission in a multi-mode fiber,” in Proceedings of European Conf. Opt. Commun. (2010), paper P.4.10.
  16. R. K. Bocek, J. Hartpence, Y. Qian, and T. Lian OFS. “Ensuring low splice loss with high quality fibers”. [Online] http://stage.ofsinfo.com/resources/splice.pdf (2012).
  17. A. A. Juarez, S. Warm, C.-A. Bunge, and K. Petermann, “Number of usable principal modes in a mode division multiplexing transmission for different multi-mode fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JTHA34.
  18. A. Li, A. A. Amin, X. Chen, S. Chen, G. Gao, and W. Shieh, “Reception of dual-spatial-mode CO-OFDM signal over a two-mode fiber,” J. Lightwave Technol. 30(4), 634–640 (2012).
    [Crossref]
  19. A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
    [Crossref]

2012 (2)

2011 (1)

2010 (1)

2009 (1)

2007 (2)

C. P. Tsekrekos and A. M. J. Koonen, “Mode-selective spatial filtering for increased robustness in a mode group diversity multiplexing link,” Opt. Lett. 32(9), 1041–1043 (2007).
[Crossref] [PubMed]

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

2005 (1)

2000 (1)

N. W. Spellmeyer, “Communications performance of a multimode EDFA,” IEEE Photon. Technol. Lett. 12(10), 1337–1339 (2000).
[Crossref]

1980 (1)

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” J. Quantum Electron. 16(7), 761–770 (1980).
[Crossref]

Amin, A. A.

Bolle, C.

Burrows, E. C.

Chen, S.

Chen, X.

Esmaeelpour, M.

Essiambre, R.

Essiambre, R.-J.

Fan, S.

Foschini, G. J.

Gao, G.

Gnauck, A. H.

Goebel, B.

Hsu, R. C. J.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

Jalali, B.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

Kahn, J. M.

Koonen, A. M. J.

Kramer, G.

Li, A.

Lingle, R.

Maksymiuk, L.

Mao, W.

McCurdy, A.

Mumtaz, S.

Panicker, R. A.

Peckham, D. W.

Petermann, K.

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” J. Quantum Electron. 16(7), 761–770 (1980).
[Crossref]

Randel, S.

Ryf, R.

Sayed, A. H.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

Shah, A.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

Shemirani, M. B.

Shieh, W.

Sierra, A.

Siuzdak, J.

Spellmeyer, N. W.

N. W. Spellmeyer, “Communications performance of a multimode EDFA,” IEEE Photon. Technol. Lett. 12(10), 1337–1339 (2000).
[Crossref]

Stepniak, G.

Tarighat, A.

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

Tsekrekos, C. P.

Winzer, P. J.

IEEE Commun. Mag. (1)

A. Tarighat, R. C. J. Hsu, A. Shah, A. H. Sayed, and B. Jalali, “Fundamentals and challenges of optical multiple-input multiple-output multimode fiber links [Topics in Optical Communications],” IEEE Commun. Mag. 45(5), 57–63 (2007).
[Crossref]

IEEE Photon. Technol. Lett. (1)

N. W. Spellmeyer, “Communications performance of a multimode EDFA,” IEEE Photon. Technol. Lett. 12(10), 1337–1339 (2000).
[Crossref]

J. Lightwave Technol. (5)

J. Quantum Electron. (1)

K. Petermann, “Nonlinear distortions and noise in optical communication systems due to fiber connectors,” J. Quantum Electron. 16(7), 761–770 (1980).
[Crossref]

Opt. Lett. (2)

Other (9)

P. Krummrich and K. Petermann, “Evaluation of potential optical amplifier concepts for coherent mode multiplexing,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMH5.

H.-G. Unger, Planar Optical Waveguides and Fibers (Oxford University Press, 1977).

N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, and M. Koshiba, “Demonstration of mode-division multiplexing transmission over 10 km two-mode fiber with mode coupler,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWA4.

N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. R. Ryf, P. J. Winzer, L. L. Buhl, Y. Sun, X. Jiang, and R. Lingle, Jr., “Space-division multiplexing and all-optical MIMO demultiplexing using a photonic integrated circuit,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2012), post-deadline paper PDP5B.

E. Voges and K. Petermann, “Vielmodenfaser,” in Optische Kommunikationstechnik, (Springer Verlag 2002) 214–260.

J. Carpenter and T. D. Wilkinson, “Precise modal excitation in multimode fiber for control of modal dispersion and mode-group division multiplexing,” in Proceedings of European Conf. Opt. Commun. (2011), paper We.10.P1.

A. A. Juarez, S. Warm, C.-A. Bunge, P. Krummrich, and K. Petermann, “Perspectives of principal mode transmission in a multi-mode fiber,” in Proceedings of European Conf. Opt. Commun. (2010), paper P.4.10.

R. K. Bocek, J. Hartpence, Y. Qian, and T. Lian OFS. “Ensuring low splice loss with high quality fibers”. [Online] http://stage.ofsinfo.com/resources/splice.pdf (2012).

A. A. Juarez, S. Warm, C.-A. Bunge, and K. Petermann, “Number of usable principal modes in a mode division multiplexing transmission for different multi-mode fibers,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JTHA34.

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Figures (10)

Fig. 1
Fig. 1

(a) Fiber mismatch in x and y direction, rotated by an angle, (b) Complete fiber is made out of 3 km long segments.

Fig. 2
Fig. 2

Main idea behind the PMs; (a) shows and input vector a p with only three components (Three mode fiber for example) corresponding to a PM at the input of the fiber. As we change the frequency interval from ω 1 ω 2 ... ω n the output vector b p stays unchanged; (b) shows the result if the input vector a is not a PM. As we change the frequency over a small frequency interval ω 1 ω 2 ... ω n the output vector b (ω) changes

Fig. 3
Fig. 3

Exemplary three mode transmission system. optical fields coming out of each modulator (MOD) are encoded and then matched to a desired spatial mode (M-MUX), which can either be a PM or EM. These are then multiplexed into the MMF. At the output of the MMF, the sum of all PMs or EMs is mode-de-multiplexed (M-DE-MUX) and detected at the receiver (Rx). The M-DE-MUX can be realized for instance by a diffractive element or a photodiode array together with a local oscillator to obtain space and phase information.

Fig. 4
Fig. 4

Eye diagram plotted for a PRBS OOK signal at 1 Gbits/s over 50 km MMF; a) using EM as carriers; b) using PMs as carriers.

Fig. 5
Fig. 5

EOP for a three mode fiber under MDM operation at 0.7 Gbit/s. a) EM as carriers and b) PM as carriers. MMF simulation parameters are given in Table 1.

Fig. 6
Fig. 6

EOP of MDM transmission for various bitrates in a three mode fiber; a) Using EM as carriers, b) using PM as carriers.

Fig. 7
Fig. 7

Left: Crosstalk of: (a) LP11 (b) PM 2 and (c) PM 2 at the output of the MMF for three different splice losses. Right: Crosstalk for each guided (a) EM (b) PM and (c) PM using the detection vectors for constant total splice loss of 0.5 dB; Fig. (b) and (c) differ only at the demultiplexing stage at the receiver; (b) uses the conjugate complex of the PMs as mode filter whereas (c) uses the detection vector defined in Eq. (28). Each EM or PM exited at the input of the MMF contains unit power.

Fig. 8
Fig. 8

Number of usable principal modes in a 36 mode fiber for various bitrates. The maximal differential group delay has the value of Δ τ m =134ps/km and limits the maximal transmission rate down to 0.1Gbit/s .

Fig. 9
Fig. 9

Usable principal modes in a 50 km transmission link, at 10Gbit/s . This implies 16 splicing points. Results represent an enhanced version of the results presented in [17].

Fig. 10
Fig. 10

Crosstalk of two PMs. Figure shows very clearly that crosstalk varies between principal modes and in this case a difference of 25 dB can be observed at 0.5 GHz between the dashed red and blue lines.

Tables (2)

Tables Icon

Table 1 Fiber Simulation Parameters Used for MDM Simulation in Three Mode Fiber

Tables Icon

Table 2 Maximal Group Delays for Different MMF

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

E in (x,y)= i=1 I a i E i (x,y) .
1 ξ 2 E i (x,y)· E j * (x,y)·dxdy= δ i,j ,
E out (x,y)= i=1 I b i E i (x,y) .
b = e j ϕ 0 (ω) T(ω)· a .
M m (ω)=( 1 0 0 0 e j(Δ ϕ 1 + ζ 1,m ) 0 0 0 0 0 e j(Δ ϕ I + ζ I,m ) ),
K ij,m = 1 ξ 2 E i,m (x,y)· E j,m+1 * (x´,y´)dxdy.
T(ω)= m=1 M M m (ω)· K m ,
ω b =[ jT(ω) ω ϕ 0 (ω)+ ω T(ω) ] e j ϕ 0 (ω) a ,
ω b =[ jT(ω) ω ϕ 0 (ω)+ ω T(ω) ]T (ω) 1 b .
ω b =[ j τ 0 LI+G(ω) ] b ,
G(ω)· b p = γ p b p .
G(ω) γ p I=0,
ω b p =[ j( τ 0 + τ p )L α p L ] b p ,
ω a =[ j ω ϕ 0 (ω)IF(ω) ] a =0,
[ F(ω) τ p I ] a p =0
n( r )= n 1 ( 12Δ ( r/ r 0 ) 2 ) ,
E l,q (r,φ)= C l,q ( r ξ ) l L q l ( r 2 ξ 2 ) e r 2 2 ξ 2 { sinlφ coslφ .
ξ= r 0 /( k 0 n 1 2Δ ) .
β q,l = n 1 k 0 ( 1 2 2Δ n 1 k 0 r 0 (2q+2l+1) )
τ q,l = N 1 c ( 1+Δ ( 2q+l+1 n 1 k 0 r 0    ) 2 ),
E 01 ( r,φ )= 1 π e r 2  2 ξ 2  
E 11 ( r,φ )= 2/π r e r 2  2 ξ 2 { sinφ cosφ
K m =( e b 2 2 ξ 2 b e b 2 2 ξ 2 ( cos φ 0 +sin φ 0 ) 2 ξ b e b 2 2 ξ 2 ( sin φ 0 cos φ 0 ) 2 ξ b e b 2 2 ξ 2 2 ξ e b 2 2 ξ 2 ( sin φ 0 b 2 +( b 2 2 ξ 2 )cos φ 0 ) 2 ξ 2 e b 2 2 ξ 2 ( ( b 2 2 ξ 2 )sin φ 0 b 2 cos φ 0 ) 2 ξ 2 b e b 2 2 ξ 2 2 ξ e b 2 2 ξ 2 ( cos φ 0 b 2 +( b 2 2 ξ 2 )sin φ 0 ) 2 ξ 2 e b 2 2 ξ 2 ( b 2 sin φ 0 ( b 2 2 ξ 2 )cos φ 0 ) 2 ξ 2 )
B 1 Δ τ m L 0.7GHz.
a T (ω)= i=1 I S i (ω) a i .
EOP=10log( E O BTB EO ),
P j C (ω)= i=1,ij I ( b i · d j )( b i · d j ) * .
P·D=I.

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