nonlinear phase noise in coherent optical ofdm transmission systems

Nonlinear phase noise in coherent optical OFDM

transmission systems

Xianming Zhu1,*

and Shiva Kumar1,2

1Corning Incorporated, SP-TD-01-1, Corning, NY 14831 USA

2On leave from Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada

*[email protected]

Abstract: We derive an analytical formula to estimate the variance of

nonlinear phase noise caused by the interaction of amplified spontaneous

emission (ASE) noise with fiber nonlinearity such as self-phase modulation

(SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in

coherent orthogonal frequency division multiplexing (OFDM) systems. The

analytical results agree very well with numerical simulations, enabling the

study of the nonlinear penalties in long-haul coherent OFDM systems

without extensive numerical simulation. Our results show that the nonlinear

phase noise induced by FWM is significantly larger than that induced by

SPM and XPM, which is in contrast to traditional WDM systems where

ASE-FWM interaction is negligible in quasi-linear systems. We also found

that fiber chromatic dispersion can reduce the nonlinear phase noise. The

variance of the total phase noise increases linearly with the bit rate, and

does not depend significantly on the number of subcarriers for systems with

moderate fiber chromatic dispersion.

©2010 Optical Society of America

OCIS codes: (060.2330) Fiber optics communications; (190.4380) Nonlinear optics, four-wave

mixing

References and links

1. W. Shieh, and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett.

42(10), 587–588 (2006).

2. A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave

systems,” J. Lightwave Technol. 25(1), 131–138 (2007).

3. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).

4. A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara,

and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J.

Lightwave Technol. 27(16), 3705–3713 (2009).

5. S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral

efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009).

6. Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s

coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).

7. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical

OFDM systems,” Opt. Express 15(20), 13282–13287 (2007).

8. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky,

“Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express

16(20), 15777–15810 (2008).

9. A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt.

Express 15(20), 12965–12970 (2007).

10. L. B. Du, and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission

using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008).

11. X. Liu, and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical

OFDM,” Opt. Express 16(26), 21944–21957 (2008).

12. X. Liu, F. Buchali, and R. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-

OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009).

13. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859

(2008).

#123422 - $15.00 USD Received 27 Jan 2010; revised 8 Mar 2010; accepted 11 Mar 2010; published 24 Mar 2010(C) 2010 OSA 29 March 2010 / Vol. 18, No. 7 / OPTICS EXPRESS 7347

14. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM

transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888

(2008).

15. E. Ip, and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J.


16. E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K.

Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km

transmission of 111-Gb/s no-guard-interval co-OFDM,” ECOC 2009, paper 9.4.6.

17. J. Gordon, and L. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt.

Lett. 15(23), 1351–1353 (1990).

18. P. Winzer, and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J.


19. A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line

amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).

20. K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B 20(9), 1875–1879 (2003).

21. K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,” Opt. Lett. 28(15), 1350–1352 (2003).

22. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).

23. A. G. Green, P. P. Mitra, and L. G. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt.

Lett. 28(24), 2455–2457 (2003).

24. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24),

3278–3280 (2005).

25. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton

systems,” Opt. Lett. 27(21), 1887–1889 (2002).

26. C. McKinstrie, and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel.

Top. Quantum Electron. 8(3), 616–625 (2002).

27. M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed

systems by the use of the moment method,” J. Opt. Soc. Am. B 21(1), 24–28 (2004).

28. K.-P. Ho, and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. 31(14), 2109–2111

(2006).

29. K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE

J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).

30. X. Zhu, S. Kumar, and X. Li, “Analysis and comparison of impairments in differential phase-shift keying and

on-off keying transmission systems based on the error probability,” Appl. Opt. 45(26), 6812–6822 (2006).

31. A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–

2032 (2007).

32. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J.


33. G. P. Agrawal, Nonlinear Fiber optics, New York: Academic, 3rd Edition, 2001.

34. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical

amplifiers,” Opt. Lett. 17(11), 801–803 (1992).

35. J. G. Proakis, Digital Communications, New York: McGraw-Hill, 4th Edition, 2000, pp. 269–274.

1. Introduction

Coherent optical orthogonal frequency division multiplexing (OFDM) has drawn significant

attention in optical communications due to its high spectral efficiency using hundreds of

subcarriers with higher-order modulation formats and its robustness to fiber chromatic

dispersion and polarization mode dispersion [1–5]. However, due to the large number of

subcarriers, OFDM is believed to suffer from high peak-to-average power ratio, which makes

it less suitable for legacy optical communication systems with periodic inline chromatic

dispersion compensation fibers [6]. In Ref [7], a simple formula for estimating the

deterministic distortions caused by four-wave mixing (FWM) is developed, and it is found

that the nonlinear limit in OFDM systems is independent on the number of OFDM subcarriers

in the absence of dispersion. Ref [8]. analytically studied the combined effect of dispersion

and FWM in OFDM multi-span systems and concluded that dispersion could significantly

reduce the amount of FWM. Recently, significant research effort has been put in nonlinear

compensation for coherent OFDM systems [9–16]. Of particular interest is the digital

backward propagation [14–16], a technique in which the signal is propagated backwards in

distance using digital signal processing (DSP) so that the deterministic linear and nonlinear

impairments can be compensated. However, the nonlinear phase noise caused by the

interaction between amplified spontaneous emissions (ASE) noise and fiber Kerr nonlinearity,


also known as Gordon-Mollenauer effects [17], cannot be compensated using digital

backward propagation. Nonlinear phase noise has been studied extensively for single-carrier

systems [17–32]; however, to the best of our knowledge, nonlinear phase noise effects have

not been investigated for OFDM systems.

In this paper, we derive an analytical formula to calculate the nonlinear phase noise

induced by the interaction of ASE with SPM, XPM and FWM in coherent OFDM

transmission systems. The analytical model is verified with numerical simulation results,

enabling the study of the nonlinear phase noise in coherent OFDM systems without lengthy

simulations. With the analytical model, we quantitatively compare the nonlinear phase noise

induced by SPM, XPM and FWM, separately, and find that the nonlinear phase noise induced

by FWM is dominant compared to that induced by SPM and XPM. This is in contrast to the

results of Ref [27]. for WDM systems, in which it is found that ASE-FWM interaction is

negligible in quasi-linear systems. This difference is likely due to the sub-carriers of OFDM

systems interacting coherently, since they are derived from the same laser source. We also

study the effects of fiber chromatic dispersion and the bit rate on the total phase noise in

coherent OFDM systems and find that, the total phase noise decreases as fiber chromatic

dispersion increases, achieving the limit of linear phase noise. The total phase noise scales up

as bit rate increases.

The remainder of the paper is organized as follows. Section 2 describes the mathematical

analysis of the nonlinear phase noise. In section 3 we show the validation of the mathematical

model with numerical simulation of coherent OFDM systems, and study the impact of fiber

chromatic dispersion, number of subcarriers, and bit rate on the variance of the total phase

noise. Section 4 gives the conclusion.

2. Mathematical analysis for the nonlinear phase noise in coherent OFDM systems

The nonlinear Schrödinger equation governing light propagation in optical fiber is [33]

2

22

2

( )( , ) ( , )exp[ ( )] ( , ) ( , ) 0,

2

u t u tj w u t u t

t

βγ

∂ ∂− + − =

∂ ∂

zz zz z z

z (1)

where 2( )β z is the dispersion profile, γ is the nonlinear coefficient,

0( ) ( )w s dsα= ∫

z

z ,

and ( )α z is the fiber loss/amplifier gain profile.

There are large numbers of subcarriers in OFDM systems, making each subcarrier a quasi-

cw wave due to low bit rate. The OFDM signal can be described as [8]

( )/ 2 1

/2

( , ) ( , )exp ,N

l l

l N

u t u t j tω−

=−

= ∑z z (2)

where N is the total number of subcarriers, ( , )l

u t z is the slowly varying field envelope, and

2l block

l Tω π= is the frequency offset from a reference, with block

T as the OFDM symbol

time. First we derive the analytical formula for the variance of nonlinear phase noise

including the interaction of ASE noise with SPM and XPM. Second, we include the nonlinear

phase noise variance induced by FWM.

2.1 SPM and XPM induced nonlinear phase noise

Inserting Eq. (2) into (1) and considering the effects of SPM and XPM only, we have


2

22 2

2 2

2 2

2 2

exp[ ( )] 2 .

l l l

l l l l

l m l

m l

u u uj u

t t

w z u u u

β ββ ω ω ω

γ≠

∂ ∂ ∂ − − + = ∂ ∂ ∂

− − +

∑

z (3)

Within each OFDM block, l

u is constant; therefore, the first and second order derivative

of l

u with respect of time, appearing in Eq. (3), can be ignored. Now the exact solution of Eq.

(3) can be written as

2 222( , ) ( ,0)exp ( ) 2 ,

2l l l eff l m

m l

u t u t j j L u uβ

ω γ≠

= + +

∑z z z (4)

where

1 exp( )

( ) .eff

Lα

α− −

=z

z (5)

We define s

L as the fiber span length, and the signal is periodically amplified by

amplifiers located at s

Lκ , κ = 1, 2, …, M, where M is the total number of fiber spans.

Consider the noise added by the amplifier located at s

Lκ . Let us expand the noise field as a

discrete Fourier transform

/ 2 1

/2

( ) exp( ).N

l l

l N

n t n j tω−

=−

= ∑ (6)

Strictly speaking, the noise field should be expressed as a Fourier transform instead of a

discrete Fourier transform. In other words, we have approximated the noise as a field with 2N

degrees of freedom (DOF) instead of infinite degrees of freedom. For a linear system that

employs matched filter at the receiver, 2N DOFs accurately describe the noise process [32]. In

Ref [17]. it is argued that 2 DOFs per carrier (or total 2N DOFs) is sufficient to describe the

noise field even in a nonlinear system. The total field immediately after the amplifier located

at s

Lκ is

( )/ 2 1

/ 2

( , ) ( , ) exp( ).N

s l s l l

l N

u t L u t L n j tκ κ ω−

+

=−

= +∑ (7)

Let

[ ] 2 22

2

( , ) ( , )

( ,0) ' exp / 2 ( ) 2 .

l s l s l

l l l s eff s l m

m l

u t L u t L n

u t n j L j L L u u

κ κ

β ω κ γ κ

+

≠

= +

= + ⋅ + + ∑

(8)

We assume that the ASE noise is a white noise process with power spectral density ASE

ρ ,

from which it follows that,

* *

,' ' ,

' ' 0,

ASE

l k l k l k

block

l k

n n n nT

n n

ρδ⋅ = ⋅ =

⋅ =

(9)

where ,l k

δ is the Kronecker delta function,


,

1 if.

0 otherwise.l k

l kδ

==

(10)

Now treating ( , )l s

u t Lκ+ as the initial field, Eq. (3) is solved to obtain the field at the end

of the optical system, located at s

ML=z , as

[ ] ( ){ }* *( , ) ' exp ( ) ( ) ' ' ,l s l l D eff s l l l l

u t ML u n j j M L L u n u nγ κ+ = + Φ + ⋅ − ⋅ + (11)

where D

Φ is the deterministic phase shift caused by dispersion, SPM and XPM, which has no

impact on the nonlinear phase noise, and is expressed as

2 22

2 / 2 ( ) 2 .D l s eff s l m

m l

ML ML L u uβ ω γ≠

Φ = + +

∑ (12)

The linear phase noise is embedded in the term 'l l

u n+ , and the nonlinear phase noise

caused by SPM and XPM by the amplifier located at s

Lκ=z is:

( ) ( )* *

, ( ) ' ' .SPM XPM eff s l l l lM L L u n u nκδ γ κ+Φ = ⋅ − ⋅ + (13)

Squaring Eq. (13) and making use of Eq. (9), we obtain the variance of the nonlinear

phase noise caused by SPM and XPM

2 22 2 2 2

, 2 ( ) ( ) 2 .ASE

SPM XPM eff s l m

m l block

M L L u uT

κ

ρδ γ κ+

≠

Φ = ⋅ − +

∑ (14)

Assuming that the number of subcarriers carrying data is e

N (equivalently the

oversampling factor is /e

N N ) and each subcarrier has equal power, summing Eq. (14) over

all amplifiers, we obtain the nonlinear phase noise variance caused by SPM and XPM

( )2 2 21( 1)(2 1) ( ) 2 1 ,

3

ASE

SPM XPM eff s e sc

block

M M M L L N PT

ρδ γ+Φ = − − ⋅ − (15)

where sc

P is the power per subcarrier. Equation (15) is our final equation for the nonlinear

phase noise variance taking into account the interaction of ASE with SPM and XPM. The

analytical model will be validated in the next section.

2.2 FWM induced nonlinear phase noise

Substituting Eq. (2) in Eq. (1), and considering only the FWM effect, we obtain the following

equation with the quasi-cw assumption

[ ] ( )2 * 2 2 22 2

,

exp ( ) exp .2 2

l

l l p q r p q r

p q r lp l q r

uj u j w u u u jβ β

ω γ ω ω ω+ − =≠ ≠

∂ − = − ⋅ − + −

∂ ∑z z

z(16)

The solution to Eq. (16) for the field l

u is as follows

0 0 0 0

0

0 0 0 0

' *

, , , , , , ,

*

, , , , , , , 0

( ) ' exp ( ') '

' ( , ),

sML

l s l p p r p q r l

p q r l

l p q r p q r l s

p q r l

u ML u j e u u u j d

u j u u u Y ML

αγ β−

+ − =

+ − =

= + − ∆

= +

∑ ∫

∑

z

z z z z

z

z z z z

z z

z

(17)

where


0 0

22

, , 0' exp( ),2

l l lu u jβ

ω= −z z z (18)

with 0, 0( )l lu u=z z .

, , ,( )

p q r lβ∆ z is the phase-mismatch factor given by

2 2 2 2 2

, , , ( ) ( ) .2

p q r l p q r l

ββ ω ω ω ω∆ = + − −

zz (19)

and

0

'

, , , 0 , , ,( , ) exp ( ') '.

sML

p q r l s p q r l

z

Y ML e j dαγ β− = − ∆ ∫ zz z z (20)

To obtain Eq. (17), we have ignored the depletion of FWM pumps appearing on the right

hand side (RHS) of Eq. (16), which is known as the un-depleted pump approximation [34].

We have also assumed that the chromatic dispersion has been completely compensated using

digital backward propagation [14–16] at the end of the system.

Now consider the noise added by the amplifier located at s

Lκ . The optical field

immediately after the amplifier is shown in Eq. (7). Equation (17) is solved using the initial

condition of Eq. (7). Replacing 0,l zu in Eq. (17) with ( )

l su Lκ+ , we obtain the optical field at

the end of the fiber span as

( )

2 *2, , , , , ,

22

,

* *

, , , , , ,

( , ) ( ) exp( ) ( , )2

exp( )2

( )( )( ) ( , ).

l s l s l s p q r p q r l s s

p q r l

l l l s

p p q q r r p q r l s s

p q r l

u ML u L j L j u u u Y L ML

u n j L

j u n u n u n Y L ML

κ κ κ

κ

κ κ κ

βκ κ ω κ κ

βω κ

κ

+ + + + +

+ − =

+ − =

= − +

= + − +

+ + +

∑

∑

(21)

Ignoring the higher order term of l

n , we have

( ) 22

,

* * * *

, , , , , , , , , , ,,

( , ) exp( )2

( ) ( , ).

l s l l l s

p q p q r q p r r p q p q r l s s

p q r l

u ML u n j L

j u u u n u u n u u n u u Y L MLr

κ

κ κ κ κ κ κ κ κκ

βκ ω κ

κ

+

+ − =

≈ + − +

+ + +∑(22)

From Eq. (22), we have

22

, ,( , ) exp( ) ( , ),2

l s l l s FWM l su ML u j L u u MLκ κ

βκ ω κ δ κ+ = − + + (23)

where ,FWM

u κ is the deterministic distortion caused by FWM, expressed as

*

, , , , , ,,( , ).FWM p q p q r l s s

p q r l

u j u u u Y L MLrκ κ κ κ κ+ − =

= ∑ (24)

This distortion can be compensated with the digital backward propagation, and thus it has

no impact on the nonlinear phase noise. The third term on the RHS of Eq. (23) ( , )l

u Mδ κ

describes the ASE-FWM interaction and can be expanded as

( )2 *2

1

( , ) exp( ) ,2

N

l s l l s q q q q

q

u ML n j L j n A n Bβ

δ κ ω κ=

= − + +∑ (25)

where


/ 2 1

*

, , , ,

/ 2

2 ( , ), ,N

q p l q p q p l q p l s s

p N

A u u Y L ML p q l p l qκ κ−

+ − + −=−

= ≠ ≠ + −∑ (26)

/ 2 1

, , , ,

/ 2

( , ), ,N

q q l p p q l p p q l s s

p N

B u u Y L ML p q l q l pκ κ−

+ − + −=−

= ≠ ≠ + −∑ (27)

From Eq. (25), we have

( )2 2 22 2

1

,N

l l q q q

q

u n n A Bδ=

= + +∑ (28)

/ 2 1

222 22

/ 2

2 exp( ) 2 .2

N

l l l l s q q q

q N

u j n B j L n A Bβ

δ ω κ−

=−

= − − ∑ (29)

After the digital backward propagation removes the deterministic distortions, the phase

noise of the received field is

*

,

Im( ),

2

l l l

l

l l

u u u

u j uκ

δ δ δδ

−Φ ≈ = (30)

where Im( )⋅ denotes the imaginary part. As , 0l κδΦ = , we can calculate the variance of the

phase noise as

( ) ( )22 2 *2*

2

, 2 2

2.

4 4

l l ll l

l

l l

u u uu u

u uκ

δ δ δδ δδ

− +−Φ ≈ − = (31)

Insert Eqs. (28) and (29) into (31) and use Eq. (9), we have:

2

2 22

,

1

2 4 Im exp( ) .2 4 2

NASE ASE

l q q l l s

qsc block sc block

A B B j LP T P T

κ

ρ ρ βδ ω κ

=

Φ = + + + −

∑ (32)

The first term on the RHS of Eq. (32) is the variance of the linear phase noise, and the second

term on the RHS of Eq. (32) is the variance of the nonlinear phase noise caused by FWM.

Summing Eq. (32) over all amplifiers in the fiber system, we obtain the phase noise variance

caused by linear phase noise and FWM as follows:

2 2

, , ,

1

.2

MASE

linear l linear l

sc block

MP T

κκ

ρδ δ

=

Φ = Φ =∑ (33)

22 2

, , ,

1 1 1

22

1

24

4Im exp( ) .2

M M NASE

FWM l FWM l q q

qsc block

M

l l s

A BP T

B j L

κκ κ

κ

ρδ δ

βω κ

= = =

=

Φ = Φ = +

+ −

∑ ∑∑

∑ (34)

The first term on the RHS of Eq. (34) is the nonlinear phase noise induced by FWM, and the

second term on the RHS of Eq. (34) is the interaction between the linear and nonlinear phase

noise.


2.3 Total phase noise

In summary, the total phase noise for the lth

subcarrier in an OFDM system including the

linear phase noise and nonlinear phase noise (induced by interaction between ASE and SPM,

XPM and FWM) is as follows:

2 2 2 2

, , ,l linear l SPM XPM FWM lδ δ δ δ+Φ = Φ + Φ + Φ (35)

where the first, second, and third terms on the RHS of Eq. (35) are given by Eqs. (33), (15),

and (34), respectively.

3. Results and discussions

In this section, we validate our analytical model for the variance of the total phase noise in

OFDM systems given by Eq. (35) with numerical simulation. The following parameters are

used throughout the paper unless otherwise specified: the bit rate is 10 Gb/s, the amplifier

spacing is 100 km, and the noise figure (NF) is 6 dB. A single type of fiber is used between

amplifiers. To separate the deterministic (although bit pattern dependent) distortions due to

nonlinear effects from the ASE-induced nonlinear noise effects, we use digital backward

propagation [14–16]. Since digital backward propagation compensates for both dispersion and

deterministic nonlinear effects, we do not use the cyclic prefix. 2048 OFDM frames are used

to get a good Monte Carlo statistics. Each OFDM subcarrier is modulated with binary-phase-

shift-keying (BPSK) data. Figure 1 shows the coherent OFDM system structure in our

simulation.

Fig. 1. Structure of coherent OFDM transmission systems

For Figs. 2-3, we choose a fiber dispersion D of 1 ps/nm/km and a total launch power of 0

dBm. Here we use only one subcarrier (Ne = 1) to carry data while the total number of

subcarriers is 8 (8th-folder oversampling), so that the nonlinear phase noise model that

includes SPM effects alone can be validated. The subcarrier carrying data is located at the

central of the OFDM spectrum, corresponding to the first subcarrier U0 in Fig. 1 due to FFT

operation. The signal spectrum before entering into the fiber span is shown in Fig. 2. And in

Fig. 3, the solid lines show the analytical linear phase noise and nonlinear phase noise

variance induced by SPM only, the dashed line with circles show the numerical simulation

results for the variance of linear phase noise and SPM induced nonlinear phase noise, as a

function of fiber propagation distance. As can be seen, the agreement is quite good.


Fig. 2. OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is

8, with one subcarrier carrying data.

Fig. 3. Variance of the total phase noise as a function of propagation distance for SPM effect

only. Total number of subcarrier is 8 with only one subcarrier carrying data.

In order to validate the nonlinear phase noise model including the ASE interaction with

SPM, XPM, and FWM effects in Eq. (35), we use 8 subcarriers to carry data for an OFDM

system with 64 subcarriers. The subcarrier carrying data is located at the central of the OFDM

spectrum, corresponding to the subcarriers U0~U3 and U60~U63 in Fig. 1. Figure 4 shows the

OFDM signal spectrum, and Fig. 5 shows the variance of the linear phase noise and nonlinear

phase noise for numerical simulation (dashed line with circles) and analytical calculation

(solid line), respectively. We see that very good agreement is achieved, which validates our

model for the nonlinear phase noise considering SPM, XPM, and FWM effects.



64, with 8 subcarriers carrying data.

Fig. 5. Variance of the total phase noise as a function of propagation distance considering the

ASE interaction with SPM, XPM and FWM effects. Total number of subcarriers is 64 with 8

subcarriers carrying data.

In Ref [7], the authors showed that the nonlinear degradation due to FWM effects in OFDM

systems is nearly independent of the number of ODFM subcarriers used in the system in the

absence of chromatic dispersion, while in Ref [8]. the authors studied the chromatic

dispersion effects on the FWM and showed that chromatic dispersion could decrease the

FWM effects significantly. However, both of these analyses focused on the deterministic

nonlinear effects. In this section, we will study the dependence of the nonlinear phase noise

effects on fiber dispersion and bit rate in an OFDM system with digital backward propagation.

In Figs. 6-8, we fix the transmission distance to be 1000 km, the total number of

subcarriers is 128 with 64 subcarriers carrying data (two-fold oversampling). Figure 6 shows

the OFDM signal spectrum. Figure 7 shows the variance of the total phase noise (linear +

nonlinear phase noise) as a function of the launch power, for D = 17 ps/nm/km and D = 0

ps/nm/km. Solid lines and circles show the analytical results and the numerical simulation

results, respectively. As can be seen from Fig. 7, the nonlinear tolerance increases

significantly as the fiber chromatic dispersion parameter increases. It is also shown in Fig. 7

that the total variance of the phase noise initially decreases with launch power since the linear

phase noise is dominant at low launch powers. However, as the launch power increases

beyond −2 dBm, the variance increases with D = 0 ps/nm/km since nonlinear phase noise

becomes dominant at higher powers.



128 with 2-folder oversampling (64 subcarriers carrying data).

Fig. 7. Variance of the total phase noise as a function of channel power. The total number of

subcarriers is 128 with two-fold oversampling, bit rate is 10 Gb/s, and transmission distance is

1000 km.

In Fig. 8, we show the impact of the bit rate on the total phase noise for a transmission

fiber with D = 17 ps/nm/km and D = 0 ps/nm/km. The total launch power is −3 dBm. Solid

lines show the analytical results, while filled circles show the numerical simulation results.

From Fig. 8, we see that the variance of the total phase noise scales linearly with the bit rate.

This could be explained by the fact that, with the increase of the bit rate, the OFDM symbol

time block

T decreases, which leads to the increase of the total phase noise as described in Eqs.

(15), (33) and (34). The qualitative explanation for the increase in phase noise when the bit

rate increases is as follows: as the bit rate increases, OSNR requirement for a given BER

increases. This is because the receiver filter bandwidth scales with bit rate, which leads to the

increase of the total noise within the receiver bandwidth. The same thing happens with phase

noise: the total phase noise within the receiver bandwidth increases as the bit rate increases.


Fig. 8. Variance of the total phase noise as a function of bit rate in Gb/s. The total number of

subcarriers is 128 with two-fold oversampling, total channel power is −3 dBm, and

transmission distance is 1000 km.

For a BPSK system with coherent detection, from [35], one would obtain the bit error rate

BER as a function of the phase noise variance 2σ as

2

1 1,

2 2BER erfc

σ

=

(36)

where ( )erfc ⋅ is the complementary error function [35]. From Eq. (36), we would get that, for

a BER of 51 10−× , the required phase noise variance would be 25.5 10−× . However, this result

would only be valid when linear phase noise is dominant. In the presence of nonlinear phase

noise, it is hard to evaluate the BER without knowing the probability density function of the

nonlinear phase noise, which would be a subject for future investigation.

In Fig. 9, we show the impact of the number of subcarriers on the variance of total phase

noise, obtained analytically using Eq. (35). Two-folder oversampling is used in the

simulation. The total launch power is −3 dBm, the bit rate is 10 Gb/s. Figure 9 shows that, in

the absence of dispersion, the variance of total phase noise scales linearly with the number of

subcarriers, while with moderate levels of dispersion, the variance of total phase noise is

almost constant because the linear phase noise is dominant for such systems.


Fig. 9. Variance of the total phase noise as a function of number of subcarriers, obtained

analytically. Two-folder oversampling is used in the simulation. Bit rate is 10 Gb/s, total

channel power is −3 dBm, and transmission distance is 1000 km.

Finally Fig. 10 shows the variance of the nonlinear phase noise as a function of

propagation distance for SPM induced nonlinear phase noise alone (dashed line), XPM

induced nonlinear phase noise alone (dashed line with ‘x’), and FWM induced nonlinear

phase noise alone for D = 0 ps/nm/km (solid line with circles), D = 10 ps/nm/km (solid line

with triangles) and D = 17 ps/nm/km (solid line with diamonds), obtained analytically using

Eqs. (15) and (34). From Fig. 10, we see that, for an OFDM system with large number of

subcarriers, nonlinear phase noise induced by FWM is significantly larger than that induced

by SPM and XPM. This is in contrast to the results of Ref [27]. for WDM systems, in which it

is found that ASE-FWM interaction is negligible in quasi-linear systems. This difference is

likely due to the fact that the subcarriers of OFDM system are derived from the same laser

source and interact coherently. We also see that, with moderate levels of fiber chromatic

dispersion, the nonlinear phase noise induced by FWM decreases since the phase matching

becomes more difficult.

Fig. 10. Variance of the nonlinear phase noise due to separate effects of SPM, XPM, and

FWM, as a function of propagation distance, obtained analytically. Total number of subcarriers

is 128 with 2-folder oversampling. Bit rate is 10 Gb/s with −3 dBm launch power.

4. Conclusions

In summary, we derive an analytical formula for the variance of the nonlinear phase noise in

an OFDM system, taking into account the interaction of ASE noise with SPM, XPM and


FWM effects. Our analytical results agree well with the numerical simulation. In addition, we

quantitatively compared the nonlinear phase noise induced by SPM, XPM and FWM, and

found that the nonlinear phase noise induced by FWM is the dominant nonlinear effect in

OFDM systems with digital backward propagation. We also studied the effect of dispersion

and bit rate on the nonlinear phase noise. Our results show that nonlinear phase noise can be

suppressed with fiber chromatic dispersion, approaching the limit of linear phase noise for

large dispersion values. Finally, we show that for OFDM systems, the variance of the total

phase noise scales linearly with the channel bit rate.


nonlinear phase noise in coherent optical ofdm transmission systems

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