We present a closed-form expression for the evaluation of the transfer function of a multimode fiber (MMF) link based on the electric field propagation model. After validating the result we investigate the potential for broadband transmission in regions far from baseband. We find that MMFs offer the potential for broadband ROF transmission in the microwave and millimetre wave regions in short and middle reach distances.
© 2006 Optical Society of America
Multimode fibers (MMFs) have been traditionally employed in short-reach low bandwidth applications . Recently the interest in MMFs has revitalised due to several reasons. On one hand, MMF fibers can be employed to support high speed digital connections such as those required by GbE applications, if these links are complemented with low cost electronic equalizing circuits Ref. . Secondly, it has been recently found that the MMF bandwidth capability over short-reach distances can be expanded by means of different techniques, such as mode group diversity multiplexing Ref. [2–6], optical frequency multiplication Ref. , subcarrier multiplexing (SCM) Ref. , and Ref. [7–8], wavelength division multiplexing (WDM) Ref. , and a combination of SCM and WDM Ref. . In addition, it should be noted that some of these techniques provide the possibility of short-reach transmission in Radio Over Fiber (ROF) applications, such as in-house transmission of millimetre signals Ref. , and Ref. [2–5], and wireless access systems Ref. . Lately as well, a particular way of exploiting the capacity of MMFs by the application of MIMO (Multiple Input Multiple Output) techniques has garnered significant research interest, Ref. .
The potentials of MMFs to support broadband RF, microwave and millimetre wave transmission over short, intermediate and long distances are yet to be fully known. The multimode fiber is more attractive for short-reach applications because is easier to fabricate, to connect and to manipulate than the singlemode fiber. Its principal disadvantage is related to its reduced bandwidth, which is limited, principally by the dispersion in the propagation delays of the guided modes or the so called intermodal dispersion. Overcoming this limitation requires the development of techniques oriented to improve is bandwidth x length product.
This in turn is contingent on the availability of accurate models to describe the signal propagation through multimode fibers.
The most popular technique reported so far for the analysis of signal propagation through MMF fibers is that based on the coupled power-flow equations developed by Gloge Ref. , and Marcuse Ref. , in the early 70’s and later improved by Olshansky Ref. , to account for the propagation and time spreading of digital pulses through MMFs. Most of the published models and subsequent work on the modeling of MMFs, Refs. [14–28], are based on this method in which the MMF power transfer function is solved by means of a numerical procedure like the Crank-Nicholson method, Ref. , Ref. , and Ref. . For example, Yabre, Ref. , has recently used the power flow equations to study the influence of several parameters on the transfer function of MMFs.
The power flow equations are adequate for the description of digital pulse propagation through MMFs but present several limitations either when considering the propagation of analogue signals or when a detailed knowledge of the baseband and RF transfer function is required since in these situations the effect of the signal phase is important. To overcome these limitations it is necessary to employ a method relying on the propagation of electric field signals rather than optical power signals, unfortunately, there are very few of such descriptions available in the literature with the exception of that developed by Saleh and Abdula for digital pulse propagation Ref. .
In this paper we present a closed-form analytic expression to compute the baseband and RF transfer function of a MMF link based on the electric field propagation method described in Ref. . Although the derivation is lengthy and cumbersome, the final result is surprisingly simple. We then proceed to validate our results by comparing to those obtained in Ref. , using the power flow equation. Finally we use our results to evaluate the conditions upon which broadband transmission is possible in RF regions far from baseband. We find that MMFs offer the potential for broadband ROF transmission in the microwave and millimetre wave regions in short (2–5 Km) and middle (10 Km) reach distances.
2. Derivation of the transfer function of a multimode fiber link using an electric field propagation model
We consider an optical transmission system which employs a multimode optical fibre as a transmission medium. The general layout is shown in Fig. 1. Our interest is to determine the end to end linear transfer function of the system under the most general conditions and taking into account as many practical sources of impairment as possible, such as:
• Temporal and spatial source coherence.
• The source chirp.
• Chromatic and intermodal dispersions.
• Mode coupling.
• Input signal coupling to modes at the input of the fiber.
• Coupling between the output signal from the fiber to the detector area.
• Differential mode attenuation.
Our derivation builds upon the model developed in the context of pulse propagation Ref. , which is now reviewed, but is specialized to RF analogue signals.
2.1 The Saleh-Abdulah method revisited
Referring to the system depicted in Fig. 1, the electric field at a point located at a distance z from the fiber origin and in at a point r of its cross section, assuming there aren’t any non linear effects, can be expressed as
where N is the number of guided modes, hµν (t) is the impulse response at z caused by mode ν at the fiber origin over mode µ at z and eν (r) is the modal spatial profile of mode ν.
Therefore, the optical intensity at a point z will be given by:
We assume a linear modulation scheme (valid for direct and external modulation) where the output intensity from the transmitter is proportional to the modulation signal S(t), so the correlation function of the field at the source output is:
where Rs (r̄ 1 r̄ 2) is the source spatial coherence and R(t 1,t 2) is the source temporal coherence given both by autocorrelation. The mode correlation at the fiber input is:
Note that if we suppose an incoherent source, then
and Eq. (5) becomes
The optical intensity at a point z is defined as
in which the external kernel is
and the internal kernel
We will assume that the detector collects the light impinging on the detector area Ar and produces an electrical current proportional to the optical power
Finally, we obtain the next end to end equation for the power collected by de detector
In the above equations:
• The Kernel Q(t′,t″) provides the influence of the source/fiber/detector system.
• Qo (t′,t″) depends on the fiber and the power coupling form the source to the fiber and from the fiber to the detector.
• Cµµ′ provides the effect of the spatial coherence of the source related to the fiber modes.
• Dνν’ is defined as
so in the special case where the detector collects all the incident light, then Dνν′ =δνν′ . The term
which refers to the fiber dispersion and to the mode coupling, must be determined to have a complete system description.
The end to end equation reveals, in general, a nonlinear relationship between the output and input electrical signals. In general, not possible thus to define a transfer function, but under several conditions linearization is possible yielding a linear system with impulse response Q(t), Refs. [29, 30],
We consider the general α-profile multimode fiber with a refractive index variation given by
• r is the offset distance from the core center;
• λ is the free space wavelength of the fiber excitation light;
• α is the refractive index exponent;
• n1 (λ) is the refractive index in the core center;
• Δ(λ) is the refractive index contrast between the core center and the cladding;
• a is the core radius.
The term in Eq. (16), relative to the propagation along the fiber, is composed of two parts, one describing the independent propagation of modes and a second one describing the power coupling between modes. Its determination requires us to consider the coupled mode propagation equations (field amplitudes) in the frequency domain
being the frequency domain electrical field defined as
Kµν =K̃µνf(z) is the coupling factor between modes µ and ν, describing f(z) the actual geometric shape of the core boundary Ref.  and:
Assuming that N modes are propagated along the fiber, we can define the vector
the coupled mode equation transforms then to a vectorial equation
where A(w,z) is a matrix operator whose elements are defined as
Solving Eq. (23) we define the fiber electric field transfer function H(ω):
In practice, the magnitude of Γµ is larger than the magnitude of Kµν and the CME can be solved yielding:
The impulse response terms of the fiber can then be found by inverse Fourier transforming the above matrix elements. Upon substitution into Eq. (17) it is found that P(t) is composed of two terms P(t)=PU (t)+PC (t). The first one corresponds to the case with absence of mode coupling, thus µ=ν and µ′=ν′; while the second one refers to the contribution of modal coupling. The uncoupled part of the received power is given by:
where the uncoupled internal kernel is given by:
The coupled contribution to the total power is given by:
with its kernel:
where Rf (z 1–z 2) is the autocorrelation of the coupling function f(z).
2.2 Kernel calculation assuming second-order dispersion
Further progress in solving the function hµν (t) requires the knowledge of the dependence of Γµ(ω) on ω. If, as in Ref. , we assume that αµ (ω) is independent of ω and that βµ (ω) can be expanded in a Taylor’s series around the central angular frequency ωo and assuming second order-dispersion
Therefore, regarding the uncoupled kernel we get
And regarding the coupled part
If we suppose that
then the coupled part becomes
2.3 Mode-dependent parameters
It should be clarified at this point that the N coupled mode propagation equations, Eq. (19) refer to an N-mode fiber. Each physical mode n can be specified by a pair of numbers (q, l), which, respectively, count the number of radial and azimuthal modal surfaces in the field intensities of that mode. It is customary however, Ref. , to group physical modes in sets where the modes in each set or group have a similar propagation constant. Mode groups are counted using the letter m where the propagation constant of each group βn of depends only on this index which is given by m=2q+l.
The mth group consists of approximately 2·(m+1) degenerate modes where:
for each state of polarization. The total number of mode groups M is found by setting the total number of modes equal to N.
The following relevant parameters have now to be considered:
1) Modal delay: As a consequence of Eq. (48), the delay time τm of a mode depends only on its principal mode number. The delay time of the guided modes can be derived from Eq. (48) using the definition given by Ref.  and Ref. ,
in which ε(λ) is the profile dispersion parameter given by
where N 1 is the material group index defined by:
2) Modal attenuation: The attenuation α µ of a mode µ and the coupling coefficient Kµν between two modes will generally depend on both the principal mode number m and the azimuthal number l, Ref. .
For transitions between modes governed by Δm=±1, rule previously adopted by Gloge Ref. , the power will become distributed among all the modes of each mode group. Consequently, for each mode group, one can define a quantity α(m), which represents an average value of the attenuation of the mth group and a quantity |Km,m ±1|2 which corresponds to an average value of the coupling between a mode of group m+1 and a mode of group m. Modal attenuation originates from conventional mechanisms that are present in a usual fiber, that is, absorption, Rayleigh scattering and loss on reflection at the core-cladding interface. These different loss mechanisms act on each mode in a different manner, which causes the attenuation coefficient to vary from mode to mode. Yabre, in Ref.  and Ref. , suggested using the following empirical formula to describe the power DMA
where αo is the power attenuation of low-order modes (i.e., intrinsic fiber attenuation), Iρ is the ρth-order modified Bessel function of the first kind and η is a weighting constant.
3) Coupling losses from random bends: From the coupled power equations derived by Marcuse Ref. ,
it was found that the power coupling coefficient dµν is related to the coupling coefficient Kµν according to the following expression
in which F(βµ -βν ) is the Fourier transform of the deformation function f(z)
For a graded-index core with power transitions only between mode groups m and m+1, Olshansky Ref. , derived that the coupling coefficient is given by
From Eq. (57) one finds
Finally, it should be mentioned that different autocorrelation functions Rf (u) can be attributed to the core boundary deformation of the multimode fiber. These are needed to compute the value of the coefficient g 2 given by Eq. (34). The main functions studied in the literature are:
• Uncorrelated fluctuations, Ref. ,
• Exponential function, Ref. ,
where the autocorrelation (or coupling) length, D, is the distance at which Rf (u) has decreased to a fraction 1/e of its maximum value and σ2 is the variance of f(z).
3. Closed expression for the transfer function of a multimode fiber
Taking into account the 2m-fold degeneracy of each group of modes, as done in Ref. , we can obtain the total power from the average power per mode of each level, P̄m (t), as
where Pm (t) is the total power distribution of a group m.
Since we have defined the propagation constant and the rest of mode-dependent parameters in function of the mode group number, we can assume from now that the expressions Eq. (14), Eq. (19), Eq. (31) and Eq. (33) refer actually to the sum of the average power per mode of each group. In the final expression of P(t), and thus in the transfer function computation we must take into consideration the factor 2m.
For the determination of the transfer function we assume an electric modulating signal composed of a RF tone (modulation index m), incorporating the source chirp (α), approximated by three terms of its Fourier series
where P is proportional to the average optical power, m represents the modulation index, α the source chirp parameter and Ω the frequency of the RF modulating signal. Moreover, we assume an optical source which has a finite linewidth spectrum (temporal coherence) with a Gaussian time domain autocorrelation function given by:
where: σc≈1/(√2W) is the source RMS coherence time and W is the source RMS linewidth.
The uncoupled part can be divide into a linear and a non linear terms, PU (t)=PUL (t)+PUN (t). For our purpose, we are only interested in the linear part, which results from forcing ν=ν′ in Eq. (30) and Eq. (31) where ν represents the mode group number
Solving the double integration for the terms relative to e -jΩt, we obtain:
The coupled power is also composed of a linear part, which will be the one of interest in the evaluation of the transfer function and a non linear part, which will contribute to the harmonic distortion and intermodulation. Solving the pertinent equation, the linear term of the coupled part Eqs. (32)–(34) is given by
If the second derivative of the propagation constant can be considered to have the same value for all the mode groups, then
and the transfer function becomes
Note that if we assume a photodetector capable of collecting the entire incident light and a source with a uniform current distribution
and the coefficient Gνν is greatly simplified to:
Equation (73) is the central result of this paper and provides a surprisingly simple description of the main factors affecting the RF frequency response of a multimode fiber link. As it can be observed it is the product of three terms or factors. As they appear from left to right in Eq. (73) the first term is a low-pass frequency response term which depends on the first order chromatic dispersion parameter , which is assumed to be equal for all the modes guided by the fiber and the parameter σc≈1/(√2W), which is directly related to the source linewidth W in Hz.
The second term is the well known carrier fading or carrier suppression effect (CSE) that is also present in the transfer function of single mode RF analogue links and that is due to the phase offset between the upper and lower modulation sidebands. Finally, the third term in Eq. (73) represents a microwave photonic transversal filtering effect Ref. , where each sample corresponds to a different mode group ν carried by the fiber. Each sample is time delayed by an amount τν Ref. , which corresponds to the group delay of its mode group and has an amplitude which depends on the modal attenuation αν Ref. , and the sum of the coefficients Cνν and Gνν .
It should be mentioned that polarization effects have not been considered in our derivation of the transfer function.
4. Model validation
The description provided by Eq. (73) is very complete and includes the effects of a considerable number of parameters including the temporal and spatial source coherence, the source chirp, chromatic and intermodal dispersion, mode coupling, signal coupling to modes at the input of the fiber, coupling between the output signal from the fiber and the detector area and the differential mode attenuation. After the lengthy derivation implied to obtain Eq. (73) and the simplicity of the final expression one is worried about the correctness of this result. In order to validate our model, we proceed to study the dispersion characteristics of a Polymer Optical Fiber (POF) with the same characteristics as the one analyzed by Yabre Ref. , using the power flow equation. The fiber in question is an 80/125 µm graded-index fiber with a GeO2-F-SiO2 core and an F-SiO2 cladding, being both regions uniformly doped with fluorine (0.04 mol-%). The core center has a 13.5 mol-% of germanium which is gradually decreased in the lateral direction to form the desired gradient. The refractive indices were approximated using a three-term Sellmeier function for a wavelength of 1300 nm. Fiber modal losses where computed according to Ref. . The parameters relative to the distributed loss were fitted to Ref. , ρ=9, η=7.35 and an intrinsic attenuation of 0.55 dB/Km. The coefficient Gνν was obtained assuming a random coupling process defined by a Gaussian autocorrelation function Ref. , with an rms deviation of σ=0.0009 m and a correlation length of D=90·a, where a is the core radius. We have assumed overfilled launching condition, so the light injection coefficient was set to Cνν =1/M, being M the total number of mode groups. The RMS linewidth of the source was set to 10 MHz and its chirp parameter to zero.
First of all, we evaluated the influence of mode-coupling in absence and in presence of differential mode attenuation. The frequency responses simulated over a fiber length L of 2014 meters and a graded index exponent α=2.02 are displayed in Fig. 2.
Our simulations are in excellent agreement with the results presented by Yabre (see Fig. 7 of Ref. ). For instance, we can see that the filtering effect caused by the DMA is pushed upward in presence of the mode-coupling phenomenon. Moreover, we appreciate that RF bandwidth is increased by mode-coupling while the DMA has little effect on the bandwidth itself.
Frequency responses are displayed in Fig. 3 showing the influence of the graded index exponent on the RF transfer function in absence of differential attenuation and mode-coupling effects. We have performed the simulations for the same graded index exponents as Yabre, α=2, α=2.2, and α=1.8. These results indicate a similar frequency behavior of those presented by Yabre (see Fig. 4 of Ref. ), except for the case of α=2.2 and α=1.8, in which our simulations illustrate the first resonances of the transversal filtering effect at a frequency of 4.29 GHz for α=2.2 and 4.93 GHz for α=1.8.
Finally, we report the influence of the fiber core radius for the values a=30 µm, a=40 µm and a=50 µm, a fixed cladding radius of b=62.5 µm and α=2.02. Figure 4 shows that more baseband bandwidth and especially smaller excursions of the intermediate notches can be gained by strengthening the mode-coupling phenomenon, i.e. by enlarging the core radius. These trends also closely agree with those obtained by Yabre (see Fig. 6 of Ref. ).
In addition to the comparative presented, it has to be noted that the Eq. (73) is extremely similar to that derived in [Ref. , Eq. (23)] for the RF transfer function of a singlemode fiber driven by a multimode laser. We conclude that all these results reinforce our confidence on the correctness of Eq. (73).
5. Applications to radio over fiber systems
Equation (73) can be employed to investigate the potential for broadband ROF systems using multimode fiber and to discover novel features as we shall now do especially in the RF regions far from baseband. As we have shown before, according to Eq. (73), a multimode fiber behaves as a transversal filter whose coefficients depend on the injection coefficient Cνν and on the modal-coupling coefficient, Gνν . Since transversal filters are periodic in frequency it should be in principle possible to consider the use of higher order resonances placed far from baseband to transport RF signals. We therefore will analyze if the 3 dB bandwidth of the high order resonances make MMFs useful to transport RF signals. For this purpose, we have evaluated both the effect of the source linewidth and the carrier suppression effect on a 62.5/125 µm silica graded-index multimode fiber with a SiO2 core doped with a 6.3 mol-% of GeO2 and a pure silica SiO2 cladding. Both the mode-coupling phenomenon and the differential attenuation were taken into account in all the responses presented in this section. The parameters relative to the distributed loss were fitted to Ref. , ρ=9, η=7.35 and an intrinsic attenuation of 0.55 dB/Km. The coefficient Gνν was obtained assuming a random coupling process defined by a Gaussian autocorrelation function Ref. . We have supposed a chromatic dispersion parameter of D=3.5 psec/Km·nm, which leads to =-3.824 (psec)2/Km, typical of the 1300 nm region. Although our model permits simulating different launching conditions, we have assumed overfilled launching condition in all subsequent simulations, so the light injection coefficient was set to Cνν =1/M, being M the total number of mode groups.
The effect caused by the source linewidth is reported in Fig. 5 for a fiber length of 2014 m and a source chirp of zero. In this case, we worked with a parabolic core grading, i. e. α=2. The rms deviation of the coupling autocorrelation function was adjusted to σ=0.001 m and the correlation length to D=140·a, so the maximum value of the factor Gνν was achieved. The frequency response is displayed for a typical distributed feedback laser with an rms linewidth W=10 MHz, a multimode Fabry Perot laser with W=4.5 nm and a LED with W=40 nm. As we can observe, a second resonance featuring a 3 dB bandwidth of around 3.6 GHz in the case of the DFB laser and around 3.3 GHz when we employ a FP laser is obtained at around 12.75 GHz due to the transversal filtering effect, while no resonance is present when a broadband LED source is used. This is due to the fact that in this later case the low pass term in Eq. (73) dominates over the other two. Therefore exploiting the possibility of transmitting broadband signals at high frequencies using a MMF is contingent on the use of low linewidth sources, as presented in Ref. .
Surprisingly Eq. (73) anticipates the interesting possibility of obtaining broad RF spectral regions far from baseband where analogue transmission is potentially possible over middle reach distances. For instance, in Fig. 6, we present the frequency response for a MMF fiber link of 5 Km with a parabolic core grading, from DC to 40 GHz. In particular, a source with a linewidth of 10 MHz is assumed and three cases corresponding to three values of the source chirp parameter are illustrated (0, -2 and -3).
It should be mentioned that the rms deviation of the coupling autocorrelation function was adjusted to σ=0.0015 m to perform simulations for a distance of 5 Km, while it was set to 0.0022 m, 0.003 m and 0.0048 m for distances of 10 Km, 20 Km and 50 Km respectively.
Figure 6 reveals an extremely interesting behavior. If we consider the case represented by the upper curve (no chirp), we can observe that the contrast ratio between the transversal filter resonances and the secondary side-lobes is dramatically reduced as the RF frequency increases. This is due to the fact that the difference in the propagation delays between adjacent mode groups is not a constant value. In other words the MMF link behaves as an imperfect transversal filter Ref. . The practical implication is that the MMF provides the potential for broadband transmission at high frequencies. For instance, in this particular case of Fig. 5, a region featuring small losses is identified for frequencies above 20 GHz. Below this region however RF transmission can also be performed at selected bands corresponding to the filter resonances. For instance in bands centered at 5, 10 and 15 GHz, each one providing a minimum 3 dB bandwidth of around 1 GHz. The intermediate and lower curves of Fig. 6 illustrate the influence of the carrier suppression effect which cannot be neglected. The first notch of the frequency response relative to a chirp=-2 is situated at 35 GHz, while it appears at 29 GHz if we modify the chirp to -3.
The effect of the core graded index profile α for middle reach distances is presented in Fig. 7 for a 10 Km fiber length and a free chirp source with an rms linewidth W=10 MHz. This simulation confirms the results already presented in Fig. 3 since the free spectral range, and thus the 3 dB resonance bandwidth, is smaller in the cases of α=2.2 and α=1.8 than for a parabolic grading. This reduction is a consequence of the increased value of the difference between the modal group delays τν defined by Eq. (50). In conclusion, we can affirm that the optimum profile exponent corresponds to α=2 as it provides the maximum bandwidth for each of the transversal filter resonances.
Figure 8 shows the transfer functions for a MMF link with different lengths (10, 20 and 50 Km), a parabolic core grading and a chirp free source. The rest of the parameters are kept fixed and with the same value of the last simulation. As it can be observed, in principle transmission regions can be identified for L=10 Km while for L=20 and 50 Km the effect of the carrier suppression term cannot be overlooked (but can be avoided using single sideband modulation) and also the presence of intermediate notches. In practice it is to be expected as it actually happens with microwave photonic transversal filters that intermediate notches will present much smaller excursions that those shown by theoretical results thus enabling the possibility of reaching longer distances. This fact however requires further investigation but a possibility relies in the exploitation of mode coupling in the fiber. For instance, in Fig. 9 the influence of the correlation length D of the Gaussian coupling autocorrelation function for a fixed rms deviation σ=0.001 m in the transfer function of a fiber link of 2014 m. The rest of parameters take the same value of the simulation of Fig. 8. The simulations where performed for the case of no mode coupling and for D=1, 10·a, 50·a and 140·a. As it can be observed increased mode coupling results in a broader 3 dB bandwidth of the transfer function resonances but most important in a significant reduction of the filter notches. Same effects were observed fixing D=140·a and increasing rms deviation σ=0.0001, 0.0005 and 0.001 m.
6. Summary and conclusion
We have presented the RF transfer function of a MMF link based on the electric field propagation method described in Ref. . We have validated our results by comparing to those obtained in Ref. , using the power flow equation. We have then used our model to evaluate the conditions upon which broadband transmission is possible in RF regions far from baseband. We have found that MMFs offer the potential for broadband ROF transmission in the microwave and millimetre wave regions in short (2–5 Km) and middle (10 Km) reach distances. Much of this potential is related to the fact that the MMF link behaves like an imperfect microwave photonic transversal filter Ref. , featuring a non-constant delay between adjacent samples. The potential for transmission over longer distances requires further research.
Authors acknowledge financial support from projects TEC2005-08298-C02-01 ADIRA and TEC2004-04754-01/TCM SODICO.
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