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
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 .
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  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 [5–8]. 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 . 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:
Hereis 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:
The relation between the input excitation coefficientand the output coefficientcan be expressed as follows:
Hereanddescribe the column vectorand respectively, is the transmission matrix describing the propagation along the MMF and is a common phase to all modes. Here we have usedas the deviation from the angular center frequency. The common phase is arbitrarily chosen to be the phase of the fundamental mode given as, where is expanded in terms of a Taylor series. The termstands for the group delay per unit lengthis the phase constant and L is the transmission length. The exponential term in (4) is pulled out of the matrixin order to normalize propagation inside the matrix with respect to the fundamental mode to make a slowly varying matrix, which is advantageous for numerical simulations. The transmission matrix is modeled, similar to the model proposed in , 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 matrix. Each of the diagonal elements contains the differential group delay and differential propagation constant, as well as a random phase as follows:
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:
Here is the ith eigenmode of the unperturbed MMF in the segment m and respectively the jth eigenmode of the MMF in segment m + 1. Our transmission matrix is then given by:Eq. (4) with respect to the angular frequency denoted as. This leads to:Eq. (8) and using Eq. (4) we obtain the following expression:
This equation can rearranged such as:Equation (10) tells us that an output field pattern, represented by the output vector, changes with frequency to the first order due to the matrix. Nevertheless, it is possible to find a vectorthat is frequency independent to the first order if the frequency dependent matrixacting oncomplies with the following equation:
This can be rewritten as an eigenvalue equation as follows:Eq. (11) are the PMs at the output of the MMF and we designate them as. Inserting Eq. (11) in Eq. (10) we obtain:Equation (13) shows thatis frequency independent sinceandare scalar values. The eigenvalues of Eq. (12) are in general complex as mentioned earlier since our matrixis not unitary due to losses into radiating modes at splicing points. The imaginary part ofcan be interpreted as the differential group delayof the PM times the transmission length L. The real part ofcan 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:
We emphasize here that Eq. (15) and Eq. (12) are not identical sincewhich is a direct consequence of not being unitary. This means that Eq. (15) and Eq. (12) have different eigenvectors and as a consequence, the principal modes at the inputare not equal to the principal modes at the outputand our equations differ at this point from the derivation presented in . 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 difference. Assuming a weakly guiding MMF, that is, with a parabolic index profile given as :9] as:
Here l describes de circumferential order and q the radial order, the Laguerre polynomial and is given as:Eq. (2). The propagation constant and group delay coefficient are given for the Laguerre Gauss modes according to  by: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  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 . 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.
The transverse field distributions given in Eq. (17) now simplifies for the LP01 mode as:Fig. 1 using Eq. (6) as:
The simulation parameters are given in Table 1 and some values require explanation; represents 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. is the mode field size of the LP01 mode and can be calculated using Eq. (18), is the fiber segment length in which we assume ideal propagation, is the splice mismatch to mode field ratio and induces the splice loss along the MMF; 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  for a fiber with 90% SiO2 concentration andis the maximal differential group delay (DGD) between the fastest and slowest propagating mode.
Using these parameters we can estimate the bandwidth B of the three mode fiber limited by modal dispersion as :
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 . 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 at. Each bit sequence is encoded on one spatial mode and multiplexed into the MMF as:
Here denotes the Fourier transform of the bit sequenceand denotes 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.
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: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.
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.
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 deviationand maximal time delay between modesis less or equal than one, that is if where assuming that mode dependent loss is compensated. Sinceis roughly proportional to the modulation bandwidth of the signal and the maximal time delay is proportional to one over the MMF bandwidth, 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 it
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 , holograms  and diffractive gratings . 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 with a detection vectorThis means for instance, that if we would want to discriminate the LP01 mode at the receiver, we would have to scalar multiply the output vectorwithwhereis 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 with its conjugate complex vectorThis 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 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 :
Hereis the ith output field distribution at the output of the MMF andis 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 vectorsthat are capable of detecting the ith principal modes at the output selectively for the center frequency, where c is the speed of light. They can be estimated solving following equation:
Hereis the matrix containing in its rows the principal modes at the output,is the identity matrix andis 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 for 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.
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 deviationAs 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 atwe 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 outputleads 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 frequencyby 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 radiusof the LP01 mode constant. This value is chosen here to be, which is a value common to the OM4 multimode fiber withand core radius ofThe maximal group delay values for the different MMF are given in Table 2 . Using these values and our previous suggestion that, 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 about, assuming a mode delay ofThis is now numerically verified by simulating a MDM transmission using PMs as carriers for various bit rates.
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 frequency. The results are presented in Fig. 8 .
As we can see the number of usable PMs is constant 36 for the bit rate of. As we increase the bit rate towe 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 of. 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 ofgiven 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  as a criteria for maximum allowable splice loss, which are given asThis results in a total allowable splice loss of 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 is. Here we point out that these results are an extension of the results already presented in  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).
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 roughly, 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 , combined with PMs for spatial multiplexing could be used to increase the transmission capacity while keeping the advantage of low signal distortion.
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 ofis 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  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.
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