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Alleviation of additional phase noise in fiber

optical parametric amplifier based signal

regenerator

Lei Jin,* Bo Xu, and Shinji Yamashita

Department of Electronic Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

*jinlei@cntp.t.u-tokyo.ac.jp

Abstract: We theoretically and numerically explain the power saturation

and the additional phase noise brought by the fiber optical parametric

amplifier (FOPA). An equation to calculate an approximation to the

saturated signal output power is presented. We also propose a scheme for

alleviating the phase noise brought by the FOPA at the saturated state. In

simulation, by controlling the decisive factor dispersion difference term Δk

of the FOPA, amplitude-noise and additional phase noise reduction of

quadrature phase shift keying (QPSK) based on the saturated FOPA is

studied, which can provide promising performance to deal with PSK

signals.

©2012 Optical Society of America

OCIS codes: (060.0060) Nonlinear Optics and optical communications; (191.4380) Nonlinear

optics, four-wave mixing; (200.6015) Signal regeneration.

References and links

1. M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron. 18(2), 738–

752 (2012).

2. M. Gao, J. Kurumida, and S. Namiki, “Wide range operation of regenerative optical parametric wavelength

converter using ASE-degraded 43-Gb/s RZ-DPSK signals,” Opt. Express 19(23), 23258–23270 (2011).

3. G. K. P. Lei, C. Shu, and H. K. Tsang, “Amplitude noise reduction, pulse format conversion, and wavelength

multicast of PSK signal in a fiber optical parametric amplifier,” National Fiber Optics Engineers Conference

(NFOEC), JW2A.79, Mar. 2012.

4. C. S. Brès, A. O. J. Wiberg, J. Coles, and S. Radic, “160-Gb/s optical time division multiplexing and

multicasting in parametric amplifiers,” Opt. Express 16(21), 16609–16615 (2008).

5. P. O. Hedekvist and P. A. Anderson, “Noise characteristics of fiber-based optical phase conjugators,” J.

Lightwave Technol. 17(1), 74–79 (1999).

6. P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Noise characteristics of fiber

optical parametric amplifiers,” J. Lightwave Technol. 22(2), 409–416 (2004).

7. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one-

and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7),

957–959 (2003).

8. M. Matsumoto, “Phase noise generation in an amplitude limiter using saturation of a fiber-optic parametric

amplifier,” Opt. Lett. 33(15), 1638–1640 (2008).

9. P. Kylemark, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Semi-analytic saturation theory of fiber optical

parametric amplifiers,” J. Lightwave Technol. 24(9), 3471–3479 (2006).

10. S. Watanabe, F. Futami, R. Okabe, R. Ludwig, C. Schmidt-Langhorst, B. Huettl, C. Schubert, and H. Weber,

“An optical parametric amplified fiber switch for optical signal processing and regeneration,” J. Sel. Top.

Quantum Electron. 14(3), 674–680 (2008).

11. M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of

DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express 16(9), 5974–5982 (2008).

12. G. Agrawal, Nonlinear Fiber Optics, 4th ed.(Aademic Press, 2007) Chap. 10.

13. G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: exact solutions and spatial

instability effects,” J. Opt. Soc. Am. B 8(4), 824–838 (1991).

14. R. Elschner and K. Petermann, “Impact of pump-induced nonlinear phase noise on parametric amplification and

wavelength conversion of phase modulated signals,” in Proc. Eur. Conf. Opt. Commun. (ECOC), Sep. 2009,

Paper.

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Received 11 Jul 2012; revised 31 Aug 2012; accepted 1 Sep 2012; published 19 Nov 2012

19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27254

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1. Introduction

Current optical communications technologies rest upon the principle of repeaterless optical

amplification, a passive form of inline signal processing. This is in sharp contrast with earlier

systems, where signals were periodically passed through electronic repeaters. A key question

is to what extent active inline signal processing, such as all-optical signal regeneration [1],

could prove beneficial in future developments, both in terms of system performance and

economic return. Fully transparent features in both the time and frequency domain are

required for optical signal regenerators because future wavelength-division multiplexed

(WDM) networks will utilize ultra-broad bandwidths over 100 nm and data rates will be 160

Gb/s or even higher, and all-optical signal regeneration is an efficient method to extend reach

of high-speed optical signal transmission without relying on optical-electric- optical

conversion and signal processing in the electric domain. Optical parametric regeneration,

which is based on nonlinear optical principles, and also fiber optical parametric amplifiers

(FOPA) has been known for years, but only recently has their potential in optical signal

processing been recognized, with experiments conducted utilizing them in applications

including optical regeneration [2], wavelength conversion and multicast [3], and optical time

domain multiplexing (OTDM) [4].

One promising property of the FOPA is its potential to have phase-insensitive, quantum-

limited amplification [5–7]. Recent research has applied this property as a signal regenerator,

which takes advantage of the instantaneous (femtosecond) saturation properties of FOPA

operating within the gain-limited regime. Regeneration in FOPAs works on the principle of

ultrafast power-dependent gain saturation, which can be utilized to suppress intensity

variation in signals. This suppression can increase the signal-to-noise ratio of signals as well

as equalize the optical power level. Matsumoto [8] and Kylemark et al. [9] discussed the

noise properties of the saturated FOPAs. Watanabe, et al. have attempted to process phase-

shift keying (PSK) and on-off keying (OOK) signals with saturated FOPAs [10]. All the

results are focused on the performance of the saturated FOPAs as amplitude limiters.

However, in their discussion, it is difficult to get an expression for the output signal power,

which is not convenient for real applications.

Fig. 1. The additional phase noise introduced by the saturated FOPA working as an amplitude

limiter.

Another problem is that if we only consider the amplitude limitation properties, the phase

fluctuation introduced by the FOPA would be a crucial factor that will degrade the system

performance. This phenomenon is illustrated in Fig. 1. Some researchers have mentioned this

problem [8,11], but the origin of this additional phase noise was not explained clearly. In this

paper, we discuss the saturation in the FOPA and derive a useful expression for estimating the

output signal power. The reason for the additional phase noise in the saturated FOPA is also

discussed, and we propose a method to cancel or alleviate this problem.

2. Signal output power in a saturated FOPA

For an unsaturated FOPA, the theory and applications can be found in many textbooks and

papers [12]. The classical theory is mainly based on these assumptions: the signal power is

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(C) 2012 OSA

Received 11 Jul 2012; revised 31 Aug 2012; accepted 1 Sep 2012; published 19 Nov 2012

19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27255

Page 3

much less than the pump power; the pump power is unchanged during the amplification

process; and the phase-matching condition is always perfect. Under these hypotheses the

complex amplitudes of the three waves after propagation in the FOPA can be calculated [9].

However, in the discussion of a saturated FOPA, this theory is no longer suitable. A

theoretical analysis without approximation of pump depletion is essential. Usually, the three

coupled-mode equations for the degenerate four-wave mixing can be expressed as

222

*

0

0

012012

2()2 exp( ),

dE

dz

iEEEE i E E E

γ

i kz

Δ

γ

=+++

(1a)

222

*

2

*

0

1

10210

2() exp( ),

dE

dz

iEEEE i E E E

γ

i kz

− Δ

γ

=+++

(1b)

222

*

1

*

0

2

21020

2() exp( ).

dE

dz

iEEEE i E E E

γ

i kz

− Δ

γ

=+++

(1c)

E0, E1, and E2 are the electric field complex amplitudes of the pump power, and two sideband

amplitudes, respectively. Δk, γ, and z are the propagation constant mismatch, third order

nonlinear parameter, and propagated distance in the fiber, where Δk = k1 + k2−2k0. Since the

total power P = |E0|2 + |E1|2 + |E2|2 is conserved as we ignore the loss in the fiber, Eq. (1) can

be conveniently rewritten in terms of normalized dimensionless amplitude variables. To this

end, the normalized pump power η(z)≡|E0|2/P and the normalized signal and idler amplitudes

a1,2≡|E1,2|/P1/2 are introduced. The phase matching condition is defined as φ(z) = Δkz + φ1(z) +

φ2(z)−2φ0(z). Kylemark et al. [9] and Cappellini et al. [13] studied the analytical saturation

theory for FOPAs. In their discussion, the process of FWM can be expressed by the

normalized coupled equations:

1 2

aa

η

d

d

4 sin ,

η

ξ

φ = −

(2a)

1

2

d

d

sin ,

a

ξ

a

ηφ=

(2b)

2

1

d

d

sin ,

a

ξ

a

ηφ=

(2c)

2

1

2

2

0

1 2

aa

d

d

2()2cos ,aa

φ

ξ

ηφ=+++

(2d)

2

1

2

2

12

1

d

d

2() cos ,

η

a

a

aa

φ

ξ

ηφ=+++

(2e)

2

2

2

1

21

2

d

d

2()cos ,

η

a

a

aa

φ

ξ

ηφ=+++

(2f)

2

1

2

2

12

1 2

aa

21

d

d

2( ) [(

+

)

η

4 ]cos .

a

a

a

a

aa

φ

ξ

κη

φ=+−++−

(2g)

η, a1, and a2 are the normalized pump power, and the two normalized sideband amplitudes,

respectively. ξ = zγP is the normalized propagation length, and κ = Δk/γP is the normalized

phase-matching term. In Ref [13], the results are expressed in the form of normalized pump

power as

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Received 11 Jul 2012; revised 31 Aug 2012; accepted 1 Sep 2012; published 19 Nov 2012

19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27256

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2

2

1 21 2

(0)

( ) sn [ (( 7 2)

)sn [ (( 7 2)

2

)(

a c b

−

)/

)/

] (

] (

−

)

)

( ),

(

d

η

where,.

[()] [ ( )]

f

b

b c a

c

−

g

g

a c b

a c

η

′

′

b

g

d

η

ξ

ξ

δ

δ+

η ξ

δ

−±

±

+−−

−

=

==

−

(3)

In this work, the mathematical notation is the same as in Ref [10]. unless noted otherwise.

Function sn stands for the elliptic sine function, which is a periodic function. The parameters

a, b, c, and d, are the roots of equation dη/dξ = 0 (Eq. (2a) in Ref [10].)), which are ordered

so that a > b > η(z) ≥ c > d. From Eq. (3) it can be known that the pump power at the output

is a periodic function of the normalized propagated length ξ with periodicity of the sn

function. When the total power P is unchanged in the fiber, the power will be transferred

between the signal/idler and the pump periodically.

However, Eq. (3) does not lend itself to a clear and intelligible understanding of the

properties, as the analytical expressions of a, b, c, and d are too complicated to calculate the

saturated signal output power from this equation. However, in many applications of saturated

phase-insensitive FOPA as an amplitude limiter, the results can be simplified, where the input

pump power is very large compared with the power of the input signal. In this paper, we

choose fiber with γ = 12/W/km and dispersion slope dD/dλ = 0.03 ps/nm2/km. The difference

between the pump and the zero-dispersion wave-length is λp−λ0 = 3 nm. Pump power P0 is

400 mW. The effect of fiber loss is neglected. Signal frequency is set at the peak position in

the unsaturated gain spectra, Δν = (−2γP0/k2)1/2/(2π), where Δν and k2 are the frequency

separation between the signal and the pump and the group-velocity dispersion (GVD)

coefficient at the pump wavelength, respectively.

When the input signal power is comparable with the input pump power, an additional

frequency component is considered as a higher order FWM process takes place. In this case,

Eq. (1) yields an inaccurate solution. In this section, however, we proceed with Eq. (1)

assuming an input pump with large enough power.

Fig. 2. Evolution of signal power in the fiber for different input powers. The dashed, full, and

dot lines are corresponding to the input signal power 0.35mW, 0.17mW and 0.05mW,

respectively.

Under such an assumption, we use a numerical method for solving Eq. (1) to show the

evolution of the signal power, and the results are shown in Fig. 2. From Fig. 2 we can see that

the signal propagates in the fiber periodically with a period dependent upon the input power.

However, at a certain specific position the powers of each signal will have nearly the same

value. For example, if we choose a fiber with proper length (1000 m in Fig. 1), then the

signals would have nearly the same output power at this position, as shown in Fig. 3.

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Received 11 Jul 2012; revised 31 Aug 2012; accepted 1 Sep 2012; published 19 Nov 2012

19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27257

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Fig. 3. The Saturation Behavior in the FOPA.

In the case of small input signal, the power difference term α = a1

signal and idler approaches zero, and a, b, c, and d can be simplified to

2−a1

2 between the input

{}

2 1 2

]

1

7

( 3) [(

+

3) 141,aH

κκ

=−−−−≈

(4a)

2 1 2

] 1 [(

+ −

1)2 1,bH

κκ=++≈

(4b)

{}

2 1 2

]

1

7

1

7

( 3) [(

−

3) 14(21),cH

κκκ

=−−−− ≈ −+

(4c)

2 1 2

] 1 [(

+ +

1)22 1.dH

κκκ=++≈+

(4d)

H = 4ηa1a2cosφ−(κ−1)η−3η2/2 is the Hamiltonian of the degenerate FWM system, which is

determined by the initial conditions. Because the pump power only changes in the range of bP

and cP, and c corresponds to the saturated length, the saturated output signal power can be

written as

2

1

1

2

44

7

( )L ( )L (1),

77

signal

P

k

γ

Ac PPP

κ

+Δ

==−≈=+

(5)

where P is the system power, Δk is the dispersion difference, and γ is the third-order nonlinear

parameter.

3. Nonlinear phase shift and phase noise

Much research has been done concerning the satuation regime of the FOPA discussing

applications in optical transmission systems to processing a PSK signal. Essential to this

application is the characterization of the phase noise in FOPAs. For example, Ref [14]. bases

its calculation of the phase noise characteristics of the FOPA on the classical theory of the

unsaturated FOPA. This analysis, however, is not valid for the saturated regime, as the phase

noise cannot be solely attributed to self-phase modulation (SPM) or cross-phase modulation

(XPM), as in the classical theory. In this section, we discuss the phase noise introduced by the

saturated FOPA.

We still use the approximation that the input signal power is small enough compared with

pump power. The subscripts 1, 2, and 0 stand for signal, idler, and pump, respectively.

Generally, based on the discussion concerning Eq. (2), the signal phase in the FOPA can be

expressed as a differential equation by

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Received 11 Jul 2012; revised 31 Aug 2012; accepted 1 Sep 2012; published 19 Nov 2012

19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27258