Transfer function of radio over fiber multimode fiber optic links considering third-order dispersion

I. Gasulla; J. Capmany

doi:10.1364/OE.15.010591

1. Introduction

The continuing rise in demand for local-area networks capable of 10 Gbit/s transmission has motivated the recent development of different novel techniques oriented to improved the bandwidth distance product in short-reach MMF optic links. In this context, it is necessary to dispose of accurate propagation models capable of describe the performance of broadband radio over fiber transmission under the most general conditions. With that objective, the authors have previously reported an optical field propagation model that presents an analytical expression for the Transfer Function of a graded-index MMF, Ref. [1], and for the non linear harmonic and intermodulation distortion, Ref. [2], assuming in both a material dispersion phenomenon characterized by a second-order propagation constant βµ(ω) approximation.

At most operating wavelengths, chromatic dispersion is described adequately by the second derivative d²β_µ(ω)/dω²=β²_µ of the propagation constant β_µ (ω), related to the µ-th mode carried by the fiber. However, for MMF fiber links operating in the wavelength region around the 1300 nm region for silica fibers, the second derivative is minimum, i.e. essentially vanishes, d²β_µ(ω)/dω²≈0 and it is, thus, necessary to consider the third derivative d³β_µ(ω)/dω³=β³_µ in order to have a precise description of the MMF link transfer function. The effects of the third-order dispersion have been previously studied in detail in the context of singlemode digital fiber optic links links, Ref. [3–6], and different techniques for its compensation have been proposed, Ref. [7,8]. In the case of multimode fiber links the main interest is the characterization of the link transfer function and so far, to the best of our knowledge no model has been reported.

In this paper we provide such a model which is based of a previously one published by the authors in Ref. [1], for multimode fibers under the effect of second-order chromatic dispersion. Our results will allow us to obtain the desired MMF link transfer function, identify the relevant parameters affecting such a response and also to compare the broadband (0–20 GHz) radio over fiber system performance in presence and in absence of the third-order dispersion when an analog modulation procedure is considered.

2. Third-order dispersion analytical model

We expand the propagation constant β_µ(ω) of the µ-th mode propagated by the MMF in a third-order Taylor series around the central angular frequency of the light source, ω₀

β_{μ} (ω) \approx β_{μ} (ω_{o}) + \frac{d β_{μ} (ω)}{d ω} ∣_{ω = ω_{o}} (ω - ω_{o}) + \frac{1}{2!} \frac{d β_{μ}^{2} (ω)}{d ω^{2}} ∣_{ω = ω_{o}} {(ω - ω_{o})}^{2} + \frac{1}{3!} \frac{d β_{μ}^{3} (ω)}{d ω^{3}} {∣_{ω = ω_{o}} (ω - ω_{o})}^{3}

If we are working at a wavelength for which β²_µ vanishes, the field transfer functions associated to the multimode fiber propagation in presence of modal attenuation, become from the model previously reported in Ref. [1], first for the uncoupled propagated optical field

H_{μ μ} (ω) = e^{- Γ_{μ} (ω) z} = e^{- α_{μ}^{0} z} \cdot e^{- j [β_{μ}^{0} + β_{μ}^{1} (ω - ω_{o}) + \frac{β_{μ}^{3}}{6} {(ω - ω_{o})}^{3}] z}

while for the coupled propagated optical field

H_{μ ν} (ω) = {\hat{K}}_{μ ν} [\int_{0}^{z} f (z') d z'] Φ_{μ ν} (ω), μ \neq ν

where K̂µν is the modal coupling factor, f(z) is a function describing the actual geometric shape of the core boundary and

Φ_{μ ν} (ω) = \frac{e^{- Γ_{μ} (ω) z} - e^{- Γ_{ν} (ω) z}}{[Γ_{ν} (ω) - Γ_{μ} (ω)] \cdot z} \approx \frac{e^{- Γ_{μ} (ω) z} - e^{- Γ_{ν} (ω) z}}{[α_{ν}^{0} - α_{μ}^{0} + j (β_{ν}^{0} - β_{μ}^{0})] \cdot z}

In Eq. (4) we have assumed that the term (β¹ν-β¹_µ)·(ω-ω₀) is negligible as compared to the rest of parameters included in the imaginary part of the term Γν(ω)-Γ_µ(ω) and that β³_µ takes the same value for all the modes carried by the fiber, β³_µ≈β₀³. This last assumption is supported by the fact that the chromatic dispersion in multimode fibers is given by the composite chromatic dispersion parameter, which is defined as the average chromatic dispersion on a graded-index MMF observed using a light source that uniformly overfills the fiber, Ref. [9]. Under this definition, the Dispersion Slope parameter S₀ at the zero-dispersion wavelength λ₀, which is defined as

S_{0} = {(\frac{2 π c}{{λ_{0}}^{2}})}^{2} β_{0}^{3},

takes the same value for all the modes propagated through the fiber, as well as the zero-dispersion wavelength, Ref. [10]. Furthermore, corrections to these global parameters can be directly derived for particular launching conditions from Eqs. (21) and (22) of Ref. [10]. As a consequence, for a given launch condition, it can be assumed that the third derivative β_µ³ is independent of the mode order.

The relative impulse responses will be given by their respective inverse Fourier Transforms as

h_{μ μ} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} H_{μ μ} (ω) \cdot e^{j ω t} d ω and Φ_{μ ν} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} Φ_{μ ν} (ω) \cdot e^{j ω t} d ω

The contribution of uncoupled power can be obtained from

P^{U} (t) = \sum_{ν = 1}^{N} \sum_{ν' = 1}^{N} C_{ν ν'} D_{ν ν'} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \sqrt{S^{*} (t')} \sqrt{S (t ″)} \cdot R (t', t ″) \cdot h_{ν ν}^{*} (t - t') h_{ν' ν'} (t - t ″) \cdot dt' dt ″

where R(t’, t”) is the source temporal autocorrelation function, which will be assumed Gaussian, C_νν’ is the light injection coefficient, D_νν’ the detector coupling coefficient, N the number of propagated modes and S(t) the modulating radiofrequency signal composed of a RF tone

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

with m_o the modulation index, α the source chirp parameter, P proportional to the average optical input power and Ω the angular frequency of the modulating signal.

On the other hand, the contribution of coupled power is given by

P^{C} (t) = g^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \sqrt{S^{*} (t')} \sqrt{S (t ″)} \cdot R (t', t ″) .

being the coefficient g² related to the function f (z) and properly defined in Ref. [1].

It must be noted that for the evaluation of the transfer function, we are interested in the linear part of the received power which results from forcing ν=ν’ in Eq. (7) and µ=µ’, ν=ν’ in Eq. (9). After the lengthy calculation of the double integral for both contributions, we obtained the following expression for the transfer function of the MMF link assuming third-order chromatic dispersion in terms of the modal groups m=1 … M carried by the fiber:

H_{3 rd} (Ω) = \sqrt{1 + α^{2}} \cdot {[1 + {(\frac{Ω β_{o}^{2} z}{σ_{c}^{2}})}^{2}]}^{- 1 ⁄ 4} \cdot e^{\frac{Ω^{4}}{8} \cdot \frac{σ_{c}^{2}}{Ω^{2} + {(\frac{σ_{c}^{2}}{β_{o}^{3} z})}^{2}}} \cdot e^{- j \frac{arctg (\frac{Ω β_{o}^{3} z}{σ_{c}^{2}})}{2}} \cdot e^{- \frac{j Ω^{3} β_{o}^{3} z}{24} \cdot \frac{1 + 4 {(\frac{σ_{c}^{2}}{Ω β_{o}^{3} z})}^{2}}{1 + {(\frac{σ_{c}^{2}}{Ω β_{o}^{3} z})}^{2}}} .

where σ_c≈1/(√2W) is the source RMS coherence time, W the optical source RMS linewidth and the definition of the rest of parameters is the same one as in Ref. [1].

In order to evaluate the above expression we will perform a comparative analysis against with the transfer function model described in Ref. [1] for second-order dispersion:

H_{2 nd} (Ω) = \sqrt{1 + α^{2}} \cdot e^{\frac{1}{2} {(\frac{β_{o}^{2} z Ω}{σ_{c}})}^{2}} \cdot \cos (\frac{β_{o}^{2} z Ω^{2}}{2} + \arctan (α)) \cdot \sum_{m = 1}^{M} 2 m \cdot (C_{mm} + G_{mm}) \cdot e^{- 2 α_{m} z} \cdot e^{- j Ω τ_{m}}

As it has been reported, the first term of Eq. (11) is a low-pass frequency response term which depends on the temporal coherence of the optical source, the next term is the Carrier Suppression Effect (CSE) while the third term represents a microwave photonic transversal filtering effect. If we concentrate only in the magnitude of the third-order dispersion transfer function, Eq. (10), we can see that the first two terms introduce a low-pass frequency effect similar to that found for the second-order dispersion response, performing the second one a deeper effect than the exponential one; while no carrier suppression effect is present in the third-order dispersion instance. As it was expected, the same transversal filtering effect mainly characterizes the response for both cases of material dispersion. In addition, it must be noted that the first term of the third-order dispersion response is identical to the first term of the fiber baseband frequency response reported in Ref. [6] for SMFs.

3. Results and discussion

We have simulated the frequency response of a silica graded-index MMF link for a laser diode emitting at a wavelength for which the second-order dispersion is negligible so the third-order dispersion plays an important role in the performance of the link for broadband radio over fiber transmission. The fiber in question is a 62.5/125 µm graded-index fiber with a SiO₂ core doped with a 6.3 mol-% of GeO₂ and a pure silica SiO₂ cladding. The refractive indices were approximated using a three-term Sellmeier function for a wavelength of 1300 nm, obtaining n₁=1.4558 for the core and n₂=1.4472 for the cladding, which leads to a relative index difference Δ=0.0059. We have performed a comparison between a link affected only by second-order dispersion and another one merely affected by third-order dispersion. We have supposed a Dispersion Slope parameter of S=0.05 psec/(Km·nm²), which leads to a third derivative β³_o=0.0402 (psec)³/Km.

Fiber modal losses where computed according to Ref. [1], (see Eq. 53). The parameters relative to the distributed loss were fitted to ρ=9, η=7.35 and an intrinsic attenuation of 0.55 dB/Km. It must also be taken into account that we have computed a parabolic fiber core grading, i. e. α=2, and assumed a uniform launching distribution, so the light injection coefficient was set to C_νν=1/M, being M=12 the total number of mode groups, (see Eq. 49 of Ref. [1]). The coupling coefficient G_νν was obtained assuming a random coupling process defined by a Gaussian autocorrelation function, (see Eq. 68 of Ref. [1]), where the rms deviation and the correlation length were adjusted in order to achieve the maximum value of G_νν. The group delay τ_m of the mode group m was computed following Eq. 50 of Ref. [1].

Fig. 1. Influence of the temporal coherence of the source on the frequency response of a 5 Km MMF link for second and third order dispersions.

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Fig. 2. Evaluation of the transfer function for different link lengths for second and third order dispersions.

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Figure 1 shows the magnitude of the simulated transfer function for a 5 Km link and three different free chirp optical sources: a typical distributed feedback (DFB) laser with an RMS linewidth W=10 MHz, a multimode Fabry Perot (FP) laser with W=4.5 nm and a LED with W=40 nm. The simulations show that no difference is present when considering a DFB laser while a deeper low-pass effect is achieved in the case of the FP laser for the third-order dispersion. As it can be observed, both curves feature the same transversal filtering behavior previously reported in Ref. [1] but presenting a notable magnitude difference in the filter resonances which diminishes for increasing electrical frequencies: 17, 15, 11 and 5 dB respectively for the resonances placed at 5, 10, 15 and 20 GHz. Another important point is observed when raising the source linewidth above the 4.5 nm of the FP laser since no change is produced in the frequency response when computing the third-order dispersion. This effect can be clearly observed in the situation of the LED source where, in contrast, no resonance is present for the case of the second-order dispersion.

Fig. 3. Evaluation of the transfer function for different source chirps for second and third order dispersions.

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The analysis of the MMF link frequency response for different distances, (10, 20 and 50 Km), and a free chirp DFB laser with W=10 MHz is plotted in Fig. 2, while the simulation for different source chirp parameters and a 10 Km link is shown in Fig. 3. As it was expected from the theory, see Eqs. (10) and (11), both figures illustrate that only the second-order dispersion case is affected by the Carrier Suppression Effect. We can conclude that working at the minimum-dispersion wavelength region results in a better performance for broadband radio over fiber transmission, even for middle and long reach distances, as long as a low-linewidth laser diode is employed. Figure 3 shows, following the same concept, that no influence of the source chirp parameter is produced when we are considering the third-order dispersion phenomenon.

4. Conclusions

Third-order dispersion effects in a graded-index MMF optic link have been analytically analyzed for the first time. Following the optical field propagation model described previously by the authors, we reported the transfer function of the link for minimum-dispersion wavelength, β²_µ≈0, under the most general conditions and taking into account as many practical sources of impairment as possible. The simulated comparison between second and third-order material dispersions illustrate several important points. Firstly, it has been found that third-order dispersion introduces a stronger low-pass frequency effect for source linewidths up to a certain value above which this effect disappears. Secondly, it has been corroborated that no Carrier Suppression Effect is produced when the second-order dispersion is assumed negligible. As a consequence, the performance of broadband radio over fiber transmission through middle-reach distances can be improved by working at the minimum-dispersion provided that low-linewidth lasers are employed.

References and links

1. 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]

2. I. Gasulla and J. Capmany, “Analysis of the harmonic and intermodulation distortion in a multimode fiber optic link,” Opt. Express 15, 9366–9371 (2007). [CrossRef] [PubMed]

3. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion,” Appl. Opt. 18, 678–682 (1979). [CrossRef] [PubMed]

4. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode optical fiber due to third-order dispersion: effect of optical source bandwidth,” Appl. Opt. 18, 2237–2240 (1979). [CrossRef] [PubMed]

5. D. Marcuse, “Pulse Distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980). [CrossRef] [PubMed]

6. L. G. Cohen, W. L. Mammel, and S. Lumish, “Dispersion and Bandwidth Spectra in Single-Mode Fibers,” IEEE J. Quantum Electron. 18, 49–53 (1982). [CrossRef]

7. E. Hellström, H. Sunnerud, M. Westlund, and M. Karlsson, “Third-Order Dispersion Compensation Using a Phase Modulator,” J. Lightwave Technol. 21, 1188–1197 (2003). [CrossRef]

8. S. Kumar, “Compensation of third-order dispersion using time reversal in optical transmission systems,” Opt. Lett. 32, 346–348 (2007). [CrossRef] [PubMed]

9. TIA/EIA-455-168, “Chromatic dispersion measurement of multimode graded-index and single-mode optical fibers by spectral group delay measurement in the time domain.”

10. G. D. Brown, “Chromatic Dispersion Measurement in Graded-Index Multimode Optical Fibers,” J. Lightwave Technol. 12, 1907–1909 (1994). [CrossRef]

Transfer function of radio over fiber multimode fiber optic links considering third-order dispersion

Abstract

1. Introduction

2. Third-order dispersion analytical model

3. Results and discussion

4. Conclusions

References and links

Cited By

Figures (3)

Equations (14)

Optics Express