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## 誌謝

承蒙溫盛發老師的悉心指導、教誨與鼓勵，使我得以順利完成 碩士論文與學業，並且給了我正確的處事態度，僅致上我最誠摯的 感謝與敬意。同時也要感謝交通大學副校長祁甡教授，和吳俊傑博 士在口試時對本論文提供的寶貴意見。

感謝學長石英男、熊克原、呂林聲，同學蕭忠志、陳裕文、莊 朝欽、翁志勳、郭永富、俞志明，學弟張耀祖、李泓毅，陪我走過 最坎坷的一段時間，感謝學長石英男教我 ZEMAX，感謝吳國書及石 英男教我 BCB 程式，感謝學弟劉文豪、陳俊嘉、黃明同、陳振中、

徐啟明，林欽豪、在論文程式及其他方面的協助。感謝公司同仁對 我論文的關心。

最後謹以本論文獻給我最親愛的父母，家人以及關心我的朋友 們。

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### 摘要

本論文研究拉曼雜訊對於波長多工光通信系統的影響我們以 25 個頻道的超長距離的波長多工光通訊系統為例，結果發現且在超 長距離的情況下，拉曼雜訊對於波長多工光通信系統的影響。拉曼 雜訊對於設計超長距離波長多工光通訊系統是不可忽略的。拉曼雜 訊降低系統 Q 值和降低量隨訊號功率而增加，對所考慮的例子而 言，顯示 Q 值降低大約 1.0。可以預期對於更多頻道的系統 Q 值降 低將會更加嚴重。

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**ABSTRACT **

The impact of Raman noise on a long-haul 25-channel WDM system is numerically studied.

It is shown that Raman noise is not negligible for designing the long-distance WDM optical communications. Raman noise degraded system Q factors and the degradation increases with signal power. The results show that Q factors are degraded by about 1.0. For the system with more channel number, one can expect more serious degradation.

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## Contents

**I. ** **INTRODUCTION ………..5 **

**II. ** **THEORETICAL MODEL……….6 **

**III. SYSTEM CONFIGURATION………..….8 **

**IV. ** **NEUMERICAL RESULTS………9 **

**V. ** **CONCLUSIONS………..9 **

**REFERENCES………...11 **

**FIGURE CAPTIONS………...14 **

5

**I. INTRODUCTION **

Fiber nonlinearities may deteriorate the signal quality of wavelength-division-multiplexing (WDM) optical communication systems [1-3]. Kerr nonlinearity introduces self-phase modulation, cross-phase modulation, and four-wave mixing (FWM) to the channels. The use of chirped-return-to-zero (CRZ) signal may reduce the impairment effects due to Kerr nonlinearity [4-8]. The pre-chirping of the signal can be used to enhance pulse broadening for the case of high residual dispersion so that the effects of fiber nonlinearities can be reduced. In long-haul wideband WDM systems, usually there is high residual dispersion even if inline dispersion compensation techniques are adopted. The broadened pulse shape can be recovered by post- dispersion compensation (PDC) at receiver [9].

The frequency downshift of light by the scattering with the medium determined by molecular vibrational levels is known as Raman effect. The frequency-shifted radiation is called the Stoke wave. The phenomena that Stokes waves grow rapidly with high intensity pump waves is known as stimulated Raman scattering (SRS). Raman fiber laser and Raman fiber amplifier are the applications of SRS. On the other hand Raman effect introduces signal crosstalk in WDM optical communication systems. The signal channel of short wavelength transfers its power to the signal channel of long wavelength through Raman effect, which is known as Raman crosstalk [10-13]. The Raman crosstalk can be compensated with a passive tilt compensator or a Raman crosstalk compensation optical amplifier (RCCOA) [14-19]. For long haul transmission systems, periodic amplification is used for fiber loss compensation. Usually, the fiber loss compensation period (LCP) is several tenths of kilometers. The fiber loss compensation optical amplifier (LCOA) is designed to be gain equalized, that is its gain spectrum is flat in the operating wavelength regime. RCCOA can be designed to compensate for both fiber loss and Raman

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crosstalk. Therefore, Raman crosstalk can be compensated for every LCP with the gain-tilted RCCOA instead of the gain-equalized LCOA.

There is Raman noise accompanied with Raman effect [20-24]. The Raman noise in Raman fiber amplifier has been extensively studied. The coupled equations of pump, signal, and noise powers are usually used as the theoretical model for describing their power evolutions along optical fiber. Such a model assumes that the interacting waves are CW. It was found that Raman noise enhances soliton self-frequency shift [23] and limits the squeezing of subpicosecond solitons [21], in which the wave equations describing the interacting fields are used in stead of the coupled power evolution equations. In this paper, we consider the Q factor degradation of long-distance WDM optical communication system in the presence of Raman noise.

**II. THEORETICAL MODEL **

Signal propagating in an optical fiber follows the wave equation

( ) ( )^{exp}( ) [ *n*] ^{0}

*i* *q* *q* *i t d* *N N q*

*z* ^{∞}η

−∞

∂ + Ω Ω − Ω Ω + + =

∂

### ∫

^{%}

^{, }

^{ }

^{(1) }

where ^{q}^{%}

### ( )

^{Ω}

*is the spectrum of electric field q and*Ω is angular frequency;

^{η}

### ( )

Ω is fiber dispersion and loss spectrum( ) 2 ^{2} 3 ^{3} ( )

1 1

2 6 *i*

η Ω = β Ω + βΩ + α Ω , (2)

where

β2 and

β3 relate to the second and third order fiber dispersions respectively; and α( )Ω is fiber loss spectrum. In Eq.(1) the nonlinear term

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( )^{2} ^{(} ^{)}

*t*

*N* α γ*R* *q t* *R t t dt*

−∞

′ ′ ′

=

### ∫

−^{, }

^{(3) }

*where γ is the coefficient of Kerr nonlinearity; and R(t) is the response function including both *
the electronic and vibrational (Raman) contributions.

( ) ^{(1} *R*^{) ( )} *R R*^{( )}

*R t* = − *f* δ *t* + *f h t* , (4)

where ( )δ *t* * is delta function and the fraction f**R** comes from the Raman response function h**R**(t). *

( ) 4 0 ( ) sin(2 )

*R R* *R*

*f h t* *g* *t d*

γ =

### ∫

^{∞}Δν π νΔ Δν

^{, }

^{(5) }

where *g** _{R}*(Δ is the Raman gain spectrum and ν) Δ is the frequency difference between pump ν
and stoke waves [12]. Figure 1 shows the Raman gain and fiber loss spectra used in this paper.

The maximum spectral gain is at Δν_{max}=13.2 THz and the corresponding maximum Raman
power gain coefficient is 4πγ*T** _{R}*Δν

_{max}

*, where T*

*R*relates to the slope of the Raman gain spectrum

*[3]. Following the method given in [25], we can obtain f*

*R*

*= 0.219 for the given Raman gain*spectrum. In Eq.(1), the Raman noise source term

( ) * ( ) ( )

*t*

*n* *R* *R*

*N* *f* γ *s t q t h t t dt*

−∞

′ ′ ′ ′

=

### ∫

−^{, }

^{(6)}

where the correlation of noise filed or the spectral density of the Langevin noise source is

( , ) * ( , ) [ * _{th}*( 1] ( ) ( )

*s z* *s* *z*^{′ ′} *h n*ν δ *z z*^{′}δ ^{′}

<% Ω % Ω >= Ω + − Ω − Ω . (7)

In Eq.(7), the thermal distribution

### ( )

^{1}

exp( / ) 1

*th*

*B*

*n* Ω = *k T*

Ω −

h , (8)

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where *k and T are Boltzmann constant and absolute temperature respectively. *_{B}

**III. SYSTEM CONFIGURATION **

Figure 2 shows the considered system configuration and dispersion map. Nonzero
dispersion-shifted fiber (NZDSF) is used as the transmission fiber. At 1550 nm, its dispersion
parameter and dispersion slopes are 4.5 ps/km/nm and 0.05 ps/km/nm^{2 }respectively. Fiber loss is
taken as 0.24 dB/km and is periodically compensated by the LCOA of 5 dB noise figure. LCP is
taken as 50 km. Fiber dispersion is compensated by the dispersion compensation fiber (DCF)
of –95-ps/km/nm dispersion parameter and –0.24-ps/km/nm^{2} dispersion slope at 1550 nm. Fiber
loss of DCF is taken as 0.5 dB/km. Symmetric dispersion map is used, where the dispersion
compensation period (DCP) is 500 km and an inline DCF (IDCF) is placed at the middle of a
dispersion compensation period. The length of an IDCF is taken as 23.5 km so that fiber
dispersion is nearly compensated at 1500 nm. Post dispersion compensation fiber (PDCF) before
the receiver is used to tailor the received pulse shape. RCCP is chosen to be the multiple of LCP
so that a RCCOA compensates for the fiber loss of an LCP together with the Raman crosstalk of
a RCCP. The noise figure of RCCOA is also assumed to be 5 dB. γ= 1.83 W^{-1}km^{-1 }and 3.5
W^{-1}km^{-1 }*for NZDSF and DCF respectively. T*R= 3.65 fsec for both fibers.

The WDM system of 25 channels spaced 100 GHz is considered. The central channel is at 1550.4 nm. The carrier frequencies of the WDM signals follow ITU grid. The bit rate of a

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*channel B= 10 Gb/s. We use split-step Fourier numerical method [3] to solve Eq.(1) with the *
initial condition representing the input signals of the WDM channels

( 0, ) * _{n}*( ) exp(

*)*

_{n}*n*

*q z*= *t* =

### ∑

φ*t*− Δ

*i*ω

*t*

^{, }

^{(9) }

where φ* _{n}*( )

*t*and Δω

*are the pulse shape and angular frequency deviation from the carrier*

_{n}*frequency assumed in Eq.(1) for the n-th channel. It is noticed that during the transmission of the*signal channels along the fiber links, they walk off each other owing to fiber dispersion and the nonlinear interactions among the signal channels depend on the walk off effect. CRZ signal of raised cosine pulse shape is used. Cosine phase modulation is applied to the signal. The field envelope of a CRZ signal pulse can be written as

( ) ( )

0 1 cos 2 exp cos 2 2

*P* *Bt* *i* *Bt*

φ= ^{⎡}⎣ + π ^{⎤}⎦ ^{⎡}⎣η π ^{⎤}⎦ . (10)

*where P**0** is peak power and η is chirping parameter. The average power P**avg**= 0.5P**0* for the same
*probability of “ONE” and “ZERO” bits. P**avg**= 0.5 mW and η= 2.5 are taken for every channel. *

At receiver, the signals are filtered with a second order 80-GHz Butterworth optical filter and a
*second order 10-GHz Butterworth electrical filter. System performance is evaluated by Q factor *
[26]. 40 simulation runs are taken for every considered WDM system. In each simulation run,
every channel comprises 32 pseudo random bits and the bit sequences of all channels are
different. The numbers of “ONE” and “ZERO” bits are the same for a pseudo random bit
*sequence. The bit sequences are also different for all simulation runs. The Q factor of a given *
channel is evaluated from 40×32=1280 pseudo random bits.

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**IV NEUMERICAL RESULTS **

Figure 3 shows the Q factors versus input signal peak power for the cases without Raman noise and with Raman noise. One can see that in the presence of Raman noise, Q factors are degraded by about 1.0. The degradation increases with signal power that is due to larger Raman noise. Without Raman noise, the optimal signal powers for the maximum Q factors are about 1.75 mW, 1.75 mW, and 1.38 mW for 6000-km, 7500-km, and 9000-km transmission distances, respectively. In the presence of Raman noise, the optimal signal powers are about 1.5 mW, 1.38 mW, and 1.5 mW for 6000-km, 7500-km, and 9000-km transmission distances, respectively.

**V. CONCLUSIONS **

The impact of Raman noise on a long-haul 25-channel WDM system is numerically studied.

It is shown that Raman noise is not negligible for designing the long-distance WDM optical communications. Raman noise degraded system Q factors and the degradation increases with signal power. The results show that Q factors are degraded by about 1.0. For the system with more channel number, one can expect more serious degradation.

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**REFERENCES **

[1] F. M. Madani and K. Kikuchi,” Design theory of long-distance WDM dispersion-managed transmission system,” J. Lightwave Technol., vol. 17, pp. 1326-1335, 1999.

[2] E. Ciaramells,” Nonlinear impairments in extremely dense WDM systems,” IEEE Photon.

Technol. Lett., vol. 14, pp. 804-806, 2002.

*[3] G. P. Agrawal, Nonlinear Fiber Optics, 3*^{rd} ed., Academic Press, London, 2001.

[4] M. Zitelli, F. Matera, and M. Settembre,” Single-channel transmission in dispersion management links in conditions of very strong pulse broadening: application to 40 Gb/s signals on step-index fibers,” J. Lightwave Technol., vol. 17, pp. 2498-2505, 1999.

[5] A. Mecozzi, C. B. Clausen, M. Shtaif,” Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett., vol. 12, pp. 392-394, 2000.

[6] S.-G. Park, A. H. Gnuauck, J. M. Wiesenfeld, and L. D. Garrett,” 40-Gb/s transmission over multiple 120-km spans of conventional single-mode fiber using highly dispersed pulses,”

IEEE Photon. Technol. Lett., vol. 12, pp.1085-1087, 2000.

[7] M. J. Ablowitz and T. Hirooka,” Nonlinear effects in quasi-linear dispersion-managed pulse transmission,” IEEE Photon. Technol. Lett., vol. 13, pp.1082-1084, 2001.

[8] R.-M. Mu, T. Yu, V. S. Grigoryan, nad C. R. Menyuk,” Dynamics of the chirped return-to- zero modulation format”, J. Lightwave Technol., vol. 20, pp. 47-57, 2002.

[9] S. Wen and T.-K. Lin,” Ultralong lightwave systems with incomplete dispersion compensations," IEEE J. Lightwave Technol., vo. 19, pp. 471-479, 2001.

[10] K.-P. Ho,” Statistical properties of stimulated Raman crosstalk in WDM systems,” J.

Lightwave Technol., vol. 18, pp. 915-921, 2000.

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[11] V. J. Mazurczyk, G. Shaulov, and E. A. Golovchenko, ”Accumulation of gain tilt in WDM amplified systems due to Raman crosstalk,” IEEE Photon. Technol. Lett., vo. 12, pp. 1573- 1575, 2000.

[12] W. Ding, Z. Chen, D. Wu, and A. Xu,” Asymmetry of Raman crosstalk in wavelength division multiplexing transmission systems,” Electron. Lett., vol. 38, pp. 1265-1267, 2002.

[13] T. Yamamoto, and S. Norimatsu, ”Statistical analysis on stimulated Raman crosstalk in dispersion-managed fiber links,” J. Lightwave Technol., vol. 21, pp. 2229–2239, 2003.

[14] H. S. Seo, K. Oh, W. Shin, U. C. Ryu, and U. C. Paek,” Compensation of Raman-induced crosstalk using a lumped germanosilicate fiber Raman amplifier in the 1.571-1.591-μ m region,” IEEE Photon. Technol. Lett., vol. 13, pp. 28-30, 2001.

[15]H. S. Seo, K. Oh, and U. C. Paek, ”Gain optimization of germanosilicate fiber Raman amplifier and its applications in the compensation of Raman-induced crosstalk among wavelength division multiplexing channels,” IEEE J. Quantum Electron., vol. 37, pp. 1110- 1116, 2001.

[16] S. Namiki, and Y. Emori,” Ultrabroad-band Raman amplifiers pumped and gain equalized by wavelength-division-multiplexed high-power laser diodes,” IEEE J. Selected Topics in Quantum Electron., vol. 7, pp. 3-16, 2001.

[17] H. S. Seo, K. Oh, and U. C. Paek,” Simultaneous amplification and channel equalization using Raman amplifier for 30 channels in 1.3-μm band,” J. Lightwave Technol., vol. 19, pp.

391–397, 2001.

[18] V. E. Perlin and H. G. Winful,” Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol., vol. 20, pp. 250–254, 2002.

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[19] X. Liu and B. Lee,” Optimal design for ultra-broad-band amplifier,” J. Lightwave Technol., vol. 21, pp. 3446-3455, 2003.

[20] J. Auyeung and A. Yariv,” Spontaneous and stimulated Raman scattering in long low loss fibers,” IEEE J. Quantum Electron., QE-14, pp. 347-352, 1978.

[21] F. X. Kartner, D. J. Dougherty, H. A. Haus, and E. P. Ippen,” Raman noise and soliton squeezing,” J. Opt. Soc. Am. B, 11, pp. 1267-1276, 1994.

[22] P. D. Drummond and J. F. Corney,” Quantum noise in optical fibers. I. Stochastic equations,” J. Opt. Soc. Am. B, 18, pp. 139-152, 2001.

[23] P. D. Drummond and J. F. Corney,” Quantum noise in optical fibers. II. Raman jitter in soliton communications,” J. Opt. Soc. Am. B, 18, pp. 153-161, 2001.

[24] A. Kobyakov, M. Vasilyev, S. Tsuda, G. Giudice, and S. Ten,” Analytical model for Raman noise figure in dispersion-managed fibers,” IEEE Photon. Technol. Lett., 15, pp. 30-32, 2003.

[25] R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus,” Raman response function of silica-core fibers,” J. Opt. Soc. Am. B, vol. 6, pp. 1159-1166, 1989.

[26] C. D. Anderson and J. A. Lyle,” Technique for evaluating system performance using Q in numerical simulations exhibiting intersymbol interference”, Electron. Lett., vol. 143, pp. 203- 208, 1997.

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**FIGURE CAPTIONS **

**Figure 1: Raman gain spectrum ***g** _{R}*(Δ . ν)

**Figure 2: System configuration and dispersion map, where D***NZDSF** and D**DCF* are the dispersion
parameters of the transmission fiber and dispersion compensation fiber respectively.

**Figure 3: Q factors versus input signal peak power for the cases without Raman noise and with **
Raman noise.

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**0** **5** **10** **15** **20** **25** **30** **35** **40**

**0.0**
**0.2**
**0.4**
**0.6**
**0.8**
**1.0**

**Raman Gain g** **R****(Δυ) (a. u.)**

**Frequency Difference **Δυ** (THz)**

**Figure 1 **

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**Figure 2 **

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