Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link

I. Gasulla; J. Capmany

doi:10.1364/OE.15.009366

1. Introduction

Recently different techniques oriented to improve the bandwidth distance product in multimode fibers have been developed for short reach applications, including mode group diversity multiplexing, Ref. [1-2], subcarrier multiplexing (SCM), Ref. [3], wavelength division multiplexing (WDM), Ref. [4-5], and the application of MIMO techniques (Multiple Input Multiple Output), Ref. [6]. To achieve the goal of broadband transmission in these systems, it becomes essential to dispose of accurate models capable of describing the propagation through multimode fibers. The most popular technique reported so far is based on a method in which the coupled power flow equations are solved by means of numerical procedures, Ref. [7]. Although this method is adequate for the description of digital pulse propagation, it presents several limitations when considering the propagation of analogue signals or when a detailed knowledge of the baseband and RF transfer function is required. As we have pointed out recently (Ref. [8]) in these cases it is necessary to employ a method relying on the propagation of electric field signals rather than optical power signals.

In Ref. [8], we have presented a full propagation model for multimode fiber which is capable of providing an analytical expression of the baseband and radiofrequency transfer function of a multimode fiber link based on the electric field propagation method described by Saleh and Abdula in Ref. [9] for digital pulse propagation. The model takes into in account different sources of impairment; such as temporal and spatial source coherence, the optical source chirp parameter, intramodal or 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 and differential mode attenuation.

Simulation results based on the linear transfer function obtained with the reported model have suggested the possibility of using MMFs for broadband radio over fiber (ROF) transmission in the microwave and millimeter wave regions in short (2-5 Km) and middle (10 Km) reach distances. In order to achieve a more complete understanding of the possibility of transmitting analogue radio signals through multimode fiber systems it is essential to evaluate the intermodulation and harmonic distortion that is produced in the link. In this paper we report the results of such analysis showing that composite second order (CSO) values achieved for short and middle reach multimode fiber links can have values below -50 dBc in several regions of the radiofrequency spectrum.

2. Second Order intermodulation distortion model

The potential for broadband ROF transmission was outlined in Ref. [8], based on the results obtained for the transfer function of the multimode fiber link, in other words, from its linear frequency response. For analog radio over fiber signal transmission it is also essential to analyze the contribution of nonlinear harmonic and intermodulation distortions which maybe produced in the multimode fiber optic link. This analysis becomes indispensable in particular when the application of subcarrier multiplexing (SCM) techniques is considered since the modulating signal is composed of several RF tones.

For the determination of the linear transfer function of a multimode fiber link, Ref. [8], we assumed an electric modulating signal composed of a RF tone (modulation index m_o), incorporating the source chirp (α), approximated by three terms of its Fourier series

\sqrt{S (t)} = \sqrt{P} \cdot [1 + \frac{m_{o}}{4} (1 + j α) \cos (Ω t)]

where P is proportional to the average optical power and Ω is the angular frequency of the RF modulating signal. For the transfer function developed in Ref. [8], we were only interested in the linear contribution of the received power and only in one of the two side-bands that resulted from the product $\sqrt{S^{*} (t')} \cdot \sqrt{S (t ″)}$ . However, if we intend to analyze the non linear response we must take into account every term resulting from this product.

Fig. 1. Layout of the Multimode Fiber link.

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With the objective of analyzing the second order harmonic distortion and intermodulation effect produced in a multimode fiber link, we consider a system whose general layout is shown in Fig. 1 and we assume an electrical modulating signal composed of two tones of angular frequencies Ω₁ and Ω₂ respectively:

\sqrt{S (t)} = \sqrt{P} {1 + \frac{m_{o}}{4} (1 + j α) \cdot [\cos (Ω_{1} t) + \cos (Ω_{2} t)]} .

After a lengthy calculation progress which follows the same procedure as that described in Ref. [8], we achieved the following expression for the total electrical received power in the frequency domain:

P (Ω) = \sum_{m = 1}^{M} 2 m \sum_{n = 1}^{M} [U_{mn} \cdot A_{mn} (Ω_{1}, Ω_{2})]

being

U_{mn} = (C_{mn} + G_{mn}) \cdot e^{- (α_{m} + α_{n}) z} \cdot e^{- j (β_{n}^{0} - β_{m}^{0}) z} \cdot e^{\frac{- {(τ_{n} - τ_{m})}^{2}}{2 σ_{c}^{2}}}

where m and n=1 …. M represent the groups of modes and the rest of the parameters have been defined in Ref. [8]. The term A_mn(Ω₁,Ω₂) is composed of the sum of two terms connected to the fundamental tones, A¹ _mn(Ω₁) and A¹ _mn(Ω₂), two terms related to the second order harmonics A² _mn(2Ω₁) and A² _mn(2Ω₂) and two terms that contain the intermodulation products in Ω₂-Ω₁ and Ω₂+Ω₁, A³ _mn(Ω₂-Ω₁) and A³ _mn(Ω₂+Ω₁). Explicitly we have:

A_{mn} (Ω_{1}, Ω_{2}) = A_{mn}^{1} (Ω_{1}) + A_{mn}^{1} (Ω_{2}) + A_{mn}^{2} (2 Ω_{1}) + A_{mn}^{2} (2 Ω_{2}) + A_{mn}^{3} (Ω_{2} - Ω_{1}) + A_{mn}^{3} (Ω_{2} + Ω_{1}),

being

A_{mn}^{1} (Ω') = \frac{m_{o} π}{4} \sqrt{1 + α^{2}} e^{- \frac{{(Ω' β_{o}^{2} z)}^{2}}{2 σ_{c}^{2}}} \cdot e^{\frac{Ω' β_{o}^{2} z (τ_{m} - τ_{n})}{σ_{c}^{2}}} \cdot {e^{j [\arctan (α) \frac{Ω'^{2} β_{o}^{2} z}{2}]} e^{- j Ω' τ_{n}} +

A_{mn}^{2} (2 Ω') = \frac{{m_{o}}^{2} π}{32} (1 + α^{2}) \cdot e^{- \frac{{(2 Ω' β_{o}^{2} z)}^{2}}{2 σ_{c}^{2}}} \cdot e^{\frac{2 Ω' β_{o}^{2} z (τ_{m} - τ_{n})}{σ_{c}^{2}}} \cdot e^{- j Ω' (τ_{m} + τ_{n})} \cdot δ (Ω - 2 Ω'),

where in Eq. (6) and Eq. (7) the angular frequency Ω’ can take the value of Ω₁ and Ω₂, and finally:

A_{mn}^{3} (Ω_{2} \mp Ω_{1}) = \frac{{m_{o}}^{2} π}{32} (1 + α^{2}) \cdot e^{- \frac{{((Ω_{2} \mp Ω_{1}) β_{o}^{2} z)}^{2}}{2 σ_{c}^{2}}} \cdot e^{\frac{(Ω_{2} \mp Ω_{1}) β_{o}^{2} z (τ_{m} - τ_{n})}{σ_{c}^{2}}} \cdot e^{\frac{({Ω_{2}}^{2} - {Ω_{1}}^{2}) j β_{o}^{2} z}{2}} \cdot

It has to be pointed out we have assumed that the first order chromatic dispersion parameter β _o ² has been taken to be equal for all the modes, β ² _m≈β ² _o, ∀m. The analysis of Eq. (6-8) reveals the following points. First of all, we see, from Eq. (6), that the magnitude of the fundamental is linearly proportional to the modulation index m_o, while from Eq. (7) and Eq. (8), the second harmonic distortions are proportional to m ² _o as should be expected. An implication of these observations is that increasing m_o increases the signal to noise ratio but increases also the distortion at a faster rate. Secondly, we must note that forcing Ω₁=Ω₂ in the expression for the intermodulation product centered at Ω₂+Ω₁ yields the same expression as that for the harmonic at 2Ω.

The system response for each tone follows the behavior of a microwave photonic transversal filter (Ref. [10]), where each sample is time delayed by an amount that depends on the group delays τ_n and τ_m, as it is shown in Eq. (6), Eq. (7) and Eq. (8), and has an amplitude whose value depends on the sum of modal attenuations α_n and α_m as well as on the sum of the coefficient of modal injection C_mn and the intermodal coupling coefficient G_mn. All these parameters have been defined in Ref. [8]. If we analyze the term A_mn(Ω₁,Ω₂) for every tone, we observe, from left to right, a first exponential term behaving as a low-pass frequency response term and a second exponential term which introduces an interference effect between the different groups of modes carried by the fiber, depending both on the first order chromatic dispersion parameter β _o ² and on the source RMS coherence time σ _c≈1/(√2W) where W is the source RMS linewidth. The next term, which is only present in the fundamental tones, Eq. (6), and in both intermodulation products, Eq. (8), is the well known carrier suppression effect (CSE), which is only affected by the source chirp parameter in the case of both fundamental tones.

For the evaluation of the impact of second order distortion, we will resort to a common measure often used in cable TV (CATV) and cellular telephone systems, the Composite Second Order intermodulation distortion parameter CSO, which is defined as the electric power ratio between the second order distortion signals and the RF carrier. A typical requisite for CATV applications is CSO<-53 dBc for every channel. In our analysis we will distinguish between the CSO for harmonics and the CSO for intermodulation products, Eq. (9) and Eq. (10), respectively, referring both to the power of the RF carrier of angular frequency Ω₁,

{CSO}_{HD} (Ω) = \frac{{[P^{2} (2 Ω_{2 or 1})]}^{2}}{{[P^{1} (Ω_{1})]}^{2}} = \frac{{\sum_{m = 1}^{M} 2 m \sum_{n = 1}^{M} [U_{mn} \cdot A_{mn}^{2} (2 Ω_{2 or 1})]}^{2}}{{\sum_{m = 1}^{M} 2 m \sum_{n = 1}^{M} [U_{mn} \cdot A_{mn}^{1} (Ω_{1})]}^{2}}

and

{CSO}_{HD} (Ω) = \frac{{[P^{2} (Ω_{2} \mp Ω_{1})]}^{2}}{{[P^{1} (Ω_{1})]}^{2}} = \frac{{\sum_{m = 1}^{M} 2 m \sum_{n = 1}^{M} [U_{mn} \cdot A_{mn}^{2} (Ω_{2} \mp Ω_{1})]}^{2}}{{\sum_{m = 1}^{M} 2 m \sum_{n = 1}^{M} [U_{mn} \cdot A_{mn}^{1} (Ω_{1})]}^{2}}

3. Results

In order to evaluate the impact of the nonlinear terms in the system performance we have evaluated the above expressions for a particular case which is representative of a middle reach multimode fiber optic link. For the sake of comparison we have kept the same values of the parameters related to the differential attenuation, the chromatic dispersion, and the light injection coefficient C_mn as in Ref. [8]. For the correct interpretation of the results it must be taken into account that frequencies of the fundamental tones are taken as multiples of a common frequency f. This is a common practice in the evaluation of the amplitudes and electrical powers of nonlinear RF terms (see Ref. [11]). For instance f₁=f and f₂=4f=4f₁. In this way the results can be plotted in terms of the common frequency f, the value of which can be swept from a minimum to a maximum value. Furthermore, also the amplitudes and electric powers of the harmonics and the intermodulation products can be plotted in terms of f₁.

The frequency response of the received power for every tone can be observed in Fig. 2, for the case of a multimode fiber optic link of 5 Km with a parabolic core grading, i.e. α=2, and a modulation index of m_o=0.01. A typical distributed feedback laser DFB source characterized by an emission wavelength of 1300 nm, an rms linewidth W=10 MHz and chirp=0 has been assumed. As it was pointed out in the previous section, we can see that the spectral response of each one of the six tones considered behaves as microwave photonic imperfect transversal filter with a different Free Spectral Range (FSR) and there is no carrier suppression effect present in the spectral behavior of the harmonics.

Fig. 2. Power received for the fundamental tones, harmonic and intermodulation distortions for a multimode fiber link of 5 Km and a modulating index m_o=0.01.

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In order to analyze more deeply the impact of the nonlinear terms we have plotted the frequency response of the CSO parameter in Fig. 3 for each of the harmonic and intermodulation products under consideration. It can be observed that in the spectral regions corresponding to the filter resonances of the linear frequency response of the fundamental tone at f₁ (5, 10, 15 and 20 GHz), we obtain CSO values well below -60 dBc. This suggests that in these spectral regions the levels of the harmonic and intermodulation products will have a negligible impact on the systems performance. In other words, intermodulation and harmonic distortions will not affect the response of the fundamental tones if the modulating frequency is placed in one of the resonance frequency bands of the system’s linear frequency response. It must be noted that the free spectral range (FSR), and thus the 3 dB bandwidth and location of the resonances, depends on the number of mode groups propagated through the fiber and on the modal delay spread. Consequently, the FSR can be modified by varying the link length, by changing the core graded index profile α, by tuning the emission wavelength of the optical source or by modifying the light injection condition. In addition, it can be observed that outside the spectral regions corresponding to the resonances a reasonably good margin of the Composite Second Order intermodulation distortion (CSO), between -40 and -60 dBc, is still obtained which can be enough for certain applications. Although not shown here we have computed results for other typical modulation index values. For instance for m_o=0.05 a CSO margin between -30 and -70 dBc was obtained.

Fig. 3. CSO relative to the harmonic and intermodulation distortions for a link of 5 Km.

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4. Summary and conclusions

We have analyzed the impact of second order harmonic and intermodulation distortion produced when transmitting a modulating signal composed of two radiofrequency tones of different frequencies over a multimode fiber link. For typical modulation index values (m_o=0.01 and m_o=0.05) and typical fiber parameters the results show a CSO margin between -40 and -80 dBc, and -30 and -70 dBc, the minimum values corresponding to the transmission in the frequency bands centered at the resonances of the linear frequency response of the link and the maxima corresponding to off resonance transmission. It is expected therefore that in the first case the intermodulation and harmonic distortion will have a negligible impact in ROF transmission with high requirements (CSO<-50 dBc). These results confirm the potential feasibility of implementing subcarrier modulation techniques (SCM) with the objective of improving the transmission capacity of a multimode fiber optic link for short and middle reach distances.

References and links

1. A. M. J. Koonen, A. Ng’Oma, H. P. A. van den Boom, I. Tafur Monroy, and G. D. Khoe, “New techniques for extending the capabilities of multimode fibre networks,” in Proceedings of NOC, (2003), pp. 204–211.

2. H. R. Stuart, “Dispersive multiplexing in multimode fiber,” Science 289, 305–307 (2000). [CrossRef]

3. S. Kanprachar and I. Jacobs, “Diversity of coding for subcarrier multiplexing on multimode fibers,” IEEE Trans. Commun. 51, 1546–1553 (2003). [CrossRef]

4. X. J. Gu, W. Mohammed, and P. W. Smith, “Demonstration of all-fiber WDM for multimode fiber local area networks,” IEEE Photon. Technol. Lett. 18, 244–246 (2006). [CrossRef]

5. E. J. Tyler, P. Kourtessis, M. Webster, E. Rochart, T. Quinlan, S. E. M. Dudley, S. D. Walker, R. V. Penty, and I. H. White, “Toward terabit-per-second capacities over multimode fiber links using SCM/WDM techniques,” J. Lightwave Technol. 21, 3237–3243 (2003). [CrossRef]

6. A. R. Shah, R. C. J. Hsu, A. Tarighat, A. H. Sayed, and B. Jalali, “Coherent Optical MIMO (COMIMO),” J. Lightwave Technol. 23, 2410–2419 (2005). [CrossRef]

7. G. Yabre, “Comprehensive Theory of Dispersion in Graded-Index Optical Fibers,” J. Lightwave Technol. 18, 166–177 (2000). [CrossRef]

8. I. Gasulla and J. Capmany, “Transfer function of multimode fiber links using an electric field propagation model: Application to Radio over Fibre Systems,” Opt. Express 14, 9051–9070 (2006). [CrossRef] [PubMed]

9. B. E. A. Saleh and R. M. Abdula, “Optical Interference and Pulse Propagation in Multimode Fibers,” Fiber Integr. Opt. 5, 161–201 (1985). [CrossRef]

10. J. Capmany, B. Ortega, D. Pastor, and S. Sales, “Discrete-time optical processing of microwave signals,” J.Lightwave Technol. 23, 702–723 (2005). [CrossRef]

11. C. S. Ih and W. Gu, “Fiber Induced Distortions in a Subcarrier Multiplexed Lightwave System,” IEEE J. Sel. Areas Commun. 8, 1296–1303 (1990). [CrossRef]

Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link

Abstract

1. Introduction

2. Second Order intermodulation distortion model

3. Results

4. Summary and conclusions

References and links

Cited By

Figures (3)

Equations (12)

Optics Express