Page 1

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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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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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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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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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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2222

2

1

1020

1

22

0

2

2

2

0

2

1

1

d

d

[ 2() cos ]

[ 2() cos ].

E

E

EEEE

z

A

A

AAAA

φ

γφ

γφ

=+++

=+++

(6)

When the amplifier is a linear unsaturated one, the phase-matching term φ is equal to π/2 by

the classical theory, and cosφ = 0. If the signal is amplified to a power comparable (tens of

mWs) with that of the pump, the last term with cosφ cannot be ignored. The FWM affects not

only the power of signal, but the phase of it. We can rewrite Eq. (6) thus as

22

0

12

1

1

d

d

[2 cos ].

φ

A

A

PAA

z

φ

γ=−+

(7)

From this equation, we can see that the nonlinear phase shift can be considered as the

equivalent of a background phase shift (BPS) with the value 2γPL, and a combination of SPM

and FWM.

The analytical form of φ1 by integrating Eq. (7) is too complicated to be expressed.

However, because of the peoriodicity in such a system with perturbation, there is a simple

method for us to analysis the phase noise introduced by the power fluctuation from input

signals. For example, as shown in Fig. 4(a), consider the power evolution of the signals with

input power 0.35 mW (red solid line) and input power 0.05 mW (green dashed line) in the

fiber mentioned above. If we move the dashed line to the left, it can be found that the two

lines coincide with each other except the lengths in the right-red and left-green rectangle

parts, which are defined as LphaseG and LphaseR respectively (shown in Fig. 4(b)). It is obvious

that LphaseG = LphaseR. Because the pump power is the same (400 mW) and much larger than the

signal (0.05 mW and 0.35 mW), the two FWM processes with the same signal power (the

coincided part in Fig. 4(b)) will have nearly same pump power and idler power. During the

distance where all the powers (pump, signal, and idler) are the same, the FWM processes will

bring the same nonlinear phase shift to the signals. Under such simplification, we can

calculate the nonlinear phase difference at the end of the fiber.

Fig. 4. (a) The power evolution of the signals with input power 0.35 mW (red solid line) and

input power 0.05 mW (green dashed line). (b) Illustration of the concided part and uncoincide

part of the two signals by moving the dashed line along the length axis.

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If there is a small power modulation ΔP1 = ΔP in the input signal power, the phase shift

difference Δφ1 can be expressed as

1

22

0

22

0

22

11

11

dd

(22 cos )d

φ

(2cos )d .

φ

phaseRphaseG

redgreen

LL

A

A

A

A

PPAAzPAAz

φφφ

γγ

Δ=−

= + Δ−+−−+

(8)

Because of the deep saturation over the length LphaseR, the energy transfer direction will

change at this area. The phase-matching term φ is near zero, so cosφ ≈1, A2 ≈A1. In the LphaseG

part, the phase is matched, so cosφ ≈0; and compared with P, A1 ≈0 and A2 ≈0. Using these

approximations, the Δφ1 can be rewritten as

22

011

(2 )d .

phaseR

L

PAAz

φγ

Δ≈Δ−+

(9)

In Eq. (9), the A1 and A0 can be further simplified as A1(L) and A0(L) further, where L is

the length of the fiber. Here, it is easy to see that the nonlinear phase noise is still composed

of a background phase noise, and the noise from SPM and FWM.

4. Phase noise alleviation

Usually, the design considerations for FOPAs aims to maximize the signal gain or entirely

convert the pump. When designing the regenerator for a PSK signal, we are interested in how

to limit the fluctuation of amplitude without introducing additional phase noise. Here, we find

a condition for the FOPA under which the phase noise is alleviated in the saturation regime.

We first consider a 1000 m FOPA with nonlinearity 0.012/(W⋅m). For a sufficiently high

pump power (400 mW in our simulation), the signal is amplified in a highly nonlinear fiber

(HNLF) and the SOP of the signal approaches to that of the pump. If we consider this as a

linear FOPA without pump depletion, the phase-matching condition Δk = −2γP0 ≈−0.096 /m

is perfect. The nonlinear phase shift is defined by the difference between the phase of the

signal after and before the FOPA (assuming that the input signal phase is zero). Under this

parametric gain saturation condition, the FOPA works as a limiter amplifier, suppressing the

amplitude noise of the signal. However, as the red dashed line shown in Fig. 5, the nonlinear

phase shift from the FOPA increases with the input power.

Fig. 5. Transmission relations of input signal power and output power (full line)/output

nonlinear phase shift (dashed line).

From the analysis of Eq. (8) and Fig. 9, it can be found that if we want to alleviate the

phase noise from the FOPA, Δφ1 should approach 0. By the analysis in Section 2 and 3,

wihout integration, we try to find the condition under which Δφ1 = 0.

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In fact, Eq. (9) can be rewritten in the form of inequlity as

22

0

22

0

22

111

11

2

0

2

1

(22 cos )d

φ

(2 cos )d

φ

( ( )

L

( ))

L L

.

phaseRphaseG

LL

phaseR

A

A

A

A

PPAAzPAAz

AA

φγγ

γ

Δ= + Δ−+−−+

<−

(10)

Substituting Eq. (5) into Eq. (10) and using the energy relation between the signal and the

pump, we get the relation

1

1(3

7

5).

phaseR

PL

φκγ

Δ < −+

(11)

If there is a case that Δφ1 = 0, the term 3κ + 5 should be larger than zero, κ < −5/3.

Furthermore, from Eq. (4c), there is a ouput power of the signal only if κ > −4.

The discussion above shows that the alleviating condition relates to the the normalized

phase-matching term κ. If the phase noise can be alleviated, κ should be in a interval that

5

3

4.

κ − < < −

(12)

This told us that, by changing the pump frequency to control Δk, we can find a specific

condition that alleviating the phase noise while limiting the amplitude at the same time.

By numerical analysis, we can find the optimized condition. Figure 6 shows resulting

transmission relation for the optimized Δk ≈−0.0175 /m.

Fig. 6. Transmission relations of input signal power and output power (full line)/output

nonlinear phase shift (dashed line).

From Fig. 6, we can see that if the input signal power is around 0.54 mW, both the output

signal power and phase shift will be at the saturated state. We can see the relation between the

signal-to-noise ratio (SNR) of the output signal and the phase noise of output signal

introduced by FOPA from Fig. 7. The introduced phase noise is calculated by the difference

of output and input signal phase fluctuation, only considering the phase noise introduced by

the signal power fluctuation at the input. It is clear that the additional phase noise introduced

by the FOPA can be reduced by adjusting the dispersion relation.

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Fig. 7. The relation between input signal SNR and output phase noise before (blue)/after (red)

the optimization.

5. Simulation

To test the theoretical analysis and illustrate the effect on the PSK signal more intuitively, we

analyze the signal constellation using the system shown in Fig. 8. The FOPA consists of a

1000 m HNLF, a polarizer, and an optical bandpass filter to select signals at the wavelength

λS. The signal and pump were input into the HNLF via an optical coupler with their state of

polarization (SOP) properly controlled by automatic polarization controllers. The HNLF has a

zero-dispersion wavelength at λ0 = 1559 nm, dispersion slope dD/dλ = 0.03ps/nm2/km. and a

nonlinear coefficient of γ = 12/W/km. The wavelength of the pump is set to λP = λ0 + 3nm,

while a 100 GSymbols/s QPSK signal has wavelength λS = 1550 nm. The signal passes

through an optical signal-to-noise ratio (OSNR) controller comprising an optical attenuator

and an EDFA. Then, the signal enters the FOPA together with the pump, which is a

continuous wave. (We can also use the 100 GHz single polarization pulse train as the pump,

which is synchronized with the signal). The output signal of the FOPA was transmitted over a

1nm filter and is detected by the coherent receiver. In this paper, the simulation is based on

the fully numerical solution of the nonlinear Schrödinger equation (NLSE). The numerical

calculation assumes a non-return-to-zero (NRZ) PSK format for the input signal, and white

Gaussian noise is added to the complex amplitudes of both signal and pump.

Fig. 8. FOPA configuration for numerical simulation.

This scheme of phase alleviating is much more suitable for dealing with higher order M-

PSK signal, especially for the signal with low SNR. We compare our scheme with the theory

in Ref [6], where the energy conversion ratio is near 100%. Figure 9 shows the simulation

results of a QPSK signal with Gaussian noise at SNR = 20dB and Fig. 10 is for SNR = 15dB.

or each figure, the (a), (b), and (c) parts correspond to the input signal, the output signal

without optimization, and the output signal with optimization by changing the pump

frequency into 1564.5nm.

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Fig. 9. Constellations of the signal (SNR 20dB) before (a), after (b) (c) the FOPA, (c) is the

optimized result.

Fig. 10. Constellations of the signal (SNR 15dB) before (a), after (b) (c) the FOPA, (c) is the

optimized result.

The angles in the constellations are the objects of reference for the phase noise. They are

equal to the spread of the phase noise of the signal before entering the FOPA. From Fig. 9 and

Fig. 10, we can find that when the SNR is at a low level, the amplitude limiting introduces

very large phase noise. Furthermore, when the SNR is 15dB, the limiting capability itself no

longer performs well. This noise leads to a higher bit errors rate. However, if we use the

phase alleviating scheme, not only the amplitude is limited but the phase noise brought by

FOPA is compressed.

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5. Conclusion

The theory of the saturated FOPA is discussed. Based on the theory, we explain the power

saturation and the additional phase noise brought by the FOPA, and gave an equation to

approximately calculate the saturated signal output power. We proposed a scheme for

alleviating the phase noise brought by the FOPA at the saturated state. Amplitude-noise and

additional phase noise reduction of BPSK and QPSK based on the saturated FOPA was

numerically studied. The dispersion difference term Δk is one decisive factor for the saturated

FOPA. The FOPA can be optimized by controlling the dispersion relation between the pump,

signal and idler, and this provides the means to deal with PSK signals.

#172245 - $15.00 USD

(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 27264