Theoretical study of gain distortions in dual-pump fiber optical parametric amplifiers

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Optics Communications 267 (2006) 244–252

Theoretical study of gain distortions in dual-pump fiberoptical parametric amplifiers

Armand Vedadi *, Arnaud Mussot 1, Eric Lantz, Herve Maillotte, Thibaut Sylvestre

Departement d’Optique P.M. Duffieux, Institut FEMTO-ST, Universite de Franche-Comte, CNRS UMR 6174, F-25030 Besancon Cedex, France

Received 9 March 2006; received in revised form 25 May 2006; accepted 31 May 2006

Abstract

We study analytically and numerically the small signal gain in dual-pump fiber optical parametric amplifiers by including the phasemodulation of the pump waves needed for practically increasing the stimulated Brillouin scattering threshold. As for the single-pumpcase, we show that large signal gain distortions are generated under co-phase modulation, which depend on the rise/fall time of the phasemodulation and on the fiber dispersion slope. However, it is clearly confirmed that the counter-phase modulation scheme allows to effi-ciently suppress these gain distortions over the whole flat gain region. In addition, we demonstrate through realistic numerical simula-tions that this useful technique overcomes the additional impact of pump-phase modulation to amplitude modulation conversion andzero-dispersion wavelength variations.� 2006 Published by Elsevier B.V.

Keywords: Fiber optical parametric amplifier; Four-wave mixing; phase modulation

1. Introduction

As well as being an optical amplifier, a fiber optical para-metric amplifier (FOPA) is versatile and can be used for avariety of all-optical signal processing techniques for futureultra-fast optical networks such as wavelength conversion,optical multiplexing, sampling, limiting, switching, noiseand dispersion monitoring [1–3]. In all FOPAs, it is neces-sary to employ schemes in order to avoid stimulated Brill-ouin scattering (SBS) of the high-power continuous-wavepumps. One approach is to broaden the Brillouin gain spec-trum by varying parameters of the fiber such as tempera-ture, strain, doping or geometrical properties [4–6]. Themost commonly used approach, however, is to increasethe SBS threshold by phase modulation of the pumps,

0030-4018/$ - see front matter � 2006 Published by Elsevier B.V.

doi:10.1016/j.optcom.2006.05.074

* Corresponding author.E-mail addresses: [email protected] (A. Vedadi), thibaut.

[email protected] (T. Sylvestre).1 Present address: Laboratoire de Physique des Lasers, Atomes et

Molecules, Universite des Sciences et Technologies de Lille, CNRS UMR8523, France.

which in turn induces other limitations on the FOPA per-formance. First, phase modulation has a detrimental effecton the coherently-coupled idler wave generated by four-wave mixing (FWM), by inducing its spectral broadeningby twice that of the phase-modulated pump wave. Second,as it has been recently demonstrated theoretically [7], thephase modulation of the pump wave can induce large signalgain distortions which depend both on the rise/fall time ofthe phase modulator and on the dispersion slope of theamplifying fiber. Its impact on system performances hasrecently been observed through bit-error rate and Q penal-ties measurements [8,9]. Here we must stress that the lattereffect is highly detrimental because it impairs all potentialapplications of FOPA-based optical devices.

To avoid the undesirable effects of phase modulation inFOPA-based wavelength converters, various techniqueshave been proposed and demonstrated [10–14]. Idler spec-tral broadening has been partially or totally cancelled byusing either a binary-phase shift keying phase modulationor a dual-pumping out-of-phase scheme, respectively. Inthe latter case, the counter-phase modulation scheme wasemployed so that the frequency chirp induced on the idler

ωIωS ω2ω1 ω0

AP2

AS AI

AP1

Ω

ωSB1 ωSB2

ASB2ASB1 Δωs

ΔωP

Fig. 1. Exact model of a two-pump fiber optical parametric amplifier.

A. Vedadi et al. / Optics Communications 267 (2006) 244–252 245

wave by one pump is exactly balanced by an opposite fre-quency chirp induced by the second pump, paving the wayfor fully transparent parametric wavelength conversion[10]. The same suppression technique for the signal gaindistortions induced by the pump-phase modulation hasbeen recently suggested through simple considerationsand quantified with an analytical expression [9,15,16].The purpose of the present paper is to investigate this effectboth analytically and numerically to clearly show the effec-tiveness of counter-phase modulation technique. To thisend, we present a theoretical analysis of the small signalgain in a two-pump (2P) FOPA that fully takes intoaccount the time-dependent phase of the pump waves.Our model is able to predict the large gain distortionsinduced by the phase modulation of the pump waves. Asfor the single-pump case, it is shown that these gain fluctu-ations depend both on the fiber dispersion slope and on therise/fall time of the phase modulator. As in Refs. [9,15,16],our results show that these gain distortions can indeed becancelled over the whole flat gain region when the twopump waves are counter-phase modulated. The results ofour analytical approach are shown to be in good agreementwith realistic numerical simulations of the non-linearSchrodinger equation with a NRZ 10 Gbit/s small signaland a pseudo-random bit sequence (PRBS) phasemodulation for the pump waves. In addition, our numeri-cal results show that the cancellation technique is robustagainst the additional impairment of phase modulation(PM) to amplitude modulation (AM) conversion of thepump waves, recently predicted by Yaman et al. [15] for2P-FOPA. Finally, the influence of zero-dispersion wave-length (ZDW) variations on the impact of pump-phasemodulation is also studied.

2. Analytical model

Parametric amplification driven by two pump waves in asilica fiber is based on non-degenerate four-wave mixing(FWM) whereby two pump waves at frequencies (x1,x2)around the zero-dispersion wavelength (ZDW) amplify afrequency-detuned signal xS and generate a phase-conju-gated idler wave xI [17]. Unlike for the single-pump case,the dual-pump case additionally leads to the generationof two sidebands (xSB1,xSB2) that result from degenerateFWM of the signal and idler waves with one pump alone,as it is depicted on Fig. 1. Depending on the frequencydetuning between the pumps and signal, these non-phasematched waves can have more and less influence on theparametric gain, by shrinking the gain bandwidth nearthe pump waves [18,19]. To assess the impact of pump-phase modulation on the parametric gain, let us first con-sider the total electric field that can be written as

Eðt; x; y; zÞ ¼ Aðt; zÞF ðx; yÞe�jðx0t�b0zÞ; ð1Þwhere x0 = (x1 + x2)/2 is the mean pump frequency andb0 = b(x0) the propagation constant. F(x,y) is the trans-verse field distribution and A(t,z), which will be denoted

A for simplicity, is the slowly varying amplitude of thefield. Taking into account the above mentioned interactingwaves and neglecting F(x,y) dependency on the frequency,A can be expressed as

A ¼ AP 1þ AP 2

þ AS þ AI þ ASB1 þ ASB2: ð2ÞThe wave amplitudes Am are connected to the Fourier

transforms A(xm,z) by

Am ¼ Aðxm; zÞe�j½ðxm�x0Þt�ðbðxmÞ�b0Þz�; ð3ÞA is found to verify the non-linear Schrodinger equation(NLSE). Neglecting the fiber loss and including higher-orderdispersion coefficients, NLSE takes the following form:

oAoz¼ j

X1k¼2

jk bk

k!

okAoskþ cjAj2A

( ); ð4Þ

where s = t � z/vg is the time expressed in a reference framemoving at the group velocity of the mean pump frequencyx0. bk is the kth-order dispersion coefficients, and c thenon-linear coefficient. Inserting Eq. (2) in Eq. (4) and assum-ing that all amplitudes are negligible with respect to thepumps, we obtain the following equations for the pumps:

oAP i

oz¼X1k¼2

jbk

k!ðxi � x0ÞkAP i

þ jcðjAP i j2 þ 2jAP 3�i j

2ÞAP i ði ¼ 1; 2Þ: ð5Þ

Note also that we assume higher-order harmonics thatcould be generated from FWM of the two pump wavesat frequencies (2x2 � x1, 2x1 � x2) are negligible. Forthe signal wave As, Eq. (4) leads to the following equation:

oAS

oz¼ j

X1k¼2

bk

k!ðxS � x0Þk þ 2cðjAP 1

j2 þ jAP 2j2Þ

!AS

þ 2jcAP 1AP 2

A�I þ 2jcAP 1A�P 2

ASB2 þ 2jcA2P 1

A�SB1: ð6Þ

Similar equations can be easily derived for the idler waveAI, and the non-phase matched waves ASB1 and ASB2. Now,we assume a time-dependent phase ui(s) that accounts forthe phase modulation of the two pump waves. This phaseterm will indeed induce an instantaneous frequency chirpon both pump waves that can be approximated byxi(s) = xi + ui,s, with ui;s ¼ ouiðsÞ

os the first-order time deriv-ative of the phase. The phase matching conditions for allparametric processes will be modified, thus shifting theidler and sidebands frequencies,

246 A. Vedadi et al. / Optics Communications 267 (2006) 244–252

x0I ¼ xI þ u1;s þ u2;s;

x0SB1 ¼ xSB1 þ 2u1;s;

x0SB2 ¼ xSB2 þ u2;s � u1;s:

ð7Þ

It is also convenient to make the following phase rota-tion of the field variables

A0i ¼ Aie�jP1k¼2

ð�1Þkbkk!ðDxPþu1;sÞk z

ði ¼ P 1; S; SB1Þ;

A0i ¼ Aie�jP1k¼2

bkk!ðDxPþu2;sÞkz

ði ¼ P 2; I; SB2Þ;

ð8Þ

where DxP = x2 � x0 is the pump frequency detuning.Thus, Eq. (5) rewrites

oA0P i

oz¼ jcðjA0P i

j2 þ 2jAP 03�ij2ÞA0P i

ði ¼ 1; 2Þ: ð9Þ

In the undepleted pump approximation, Eq. (9) admitsthe following steady-state CW solutions for the two pumpwaves

A0P i¼

ffiffiffiffiffiP i

pexpðjcðP i þ 2P 3�iÞzÞ ði ¼ 1; 2Þ; ð10Þ

where Pi is the power of the pump i (i = 1, 2) at the fiberinput. Substituting Eq. (10) and (8) into Eq. (6), one getsfor the signal wave A0SoA0Soz¼X1k¼2

jbk

k!ð�1Þk Dxk

S � ðDxP � u1;sÞk

h iA0S

þ 2jcðP 1 þ P 2ÞA0S þ 2jcrA0�I e3jcðP 1þP 2Þz

þ 2jcrA0SB2ejcðP 2�P 1Þz þ 2jcP 1A0�SB1e2jcðP 1þ2P 2Þz; ð11Þwhere r ¼ 2c

ffiffiffiffiffiffiffiffiffiffiP 1P 2

pand DxS = xI � x0 is the signal fre-

quency detuning. Similar equations can be also derivedfor A0I;A

0SB1 and A0SB2. A second transformation of the fields

is necessary in order to eliminate the z dependence

A00i ¼ A0ie�jcðP 1þ2P 2Þz ði ¼ S; SB1Þ;

A00i ¼ A0ie�jcð2P 1þP 2Þz ði ¼ I; SB2Þ:

ð12Þ

Inserting Eq. (12) in Eq. (11), we obtain a set of fourequations that can be written as

�joA00

SB1

oz ¼P4k¼2

ð�1Þk bkk!ðDxSB � 2u1;sÞ

k � ðDxP � u1;sÞk

h iA00SB1

þcP 1ðA00SB1 þ A00S�Þ þ rðA00I þ A00�SB2Þ;

�joA00

S

oz ¼P4k¼2

ð�1Þk bkk!

DxkS � ðDxP � u1;sÞ

kh i

A00S

þcP 1ðA00�SB1 þ A00SÞ þ rðA00�I þ A00SB2Þ;

�joA00

I

oz ¼P4k¼2

bkk!ðDxS þ u1;s þ u2;sÞ

k � ðDxP þ u2;sÞk

h iA00I

þcP 2ðA00I þ A0�SB2Þ þ rðA00SB1 þ A00S�Þ;

�joA00

SB2

oz ¼P4k¼2

bkk!ðDxSB þ u2;s � u1;sÞ

k � ðDxP þ u2;sÞk

h iA00SB2

þcP 2ðA00�I þ A00SB2Þ þ rðA00�SB1 þ A00SÞ;

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ð13Þ

with DxSB = 2DxP � DxS. The parametric gain is then ex-tracted by solving numerically the above set of equations.Nevertheless, a straightforward analytical formula of the

parametric gain can be derived by taking only into accountthe signal and the idler waves. Note that this truncatedmodel is valid only for signal and idler far enough fromthe pumps, so that the impact of the sidebands xSB1 andxSB2 on the parametric gain becomes negligible [18]. Theset of equations therefore reduces to

�joA00

S

oz ¼P4k¼2

ð�1Þk bkk!

DxkS � ðDxP � u1;sÞ

kh i

þ cP 1

� �A00S þ rA00�I ;

�joA00

I

oz ¼P4k¼2

bkk!ðDxS þ u1;s þ u2;sÞ

kh�

�ðDxP þ u2;sÞkiA00I þ cP 2

�A00I þ rA00S

�:

8>>>>>>>><>>>>>>>>:

ð14Þ

The parametric gain g for this truncated model takes thefollowing form:

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4c2P 1P 2 �

jþ dj2

� �2s

; ð15Þ

where j is the standard phase mismatch

j ¼ cðP 1 þ P 2Þ þ b2ðDx2S � Dx2

PÞ þb4

12ðDx4

S � Dx4PÞ ð16Þ

and dj the instantaneous phase mismatch due to pump-phase modulation. Assuming that jui,sj � (DxP � DxS),which is valid for common shifts used in phase modulationtechniques, dj can be simplified to

dj ¼ b3

2ðDx2

S � Dx2PÞðu1;s þ u2;sÞ: ð17Þ

Eqs. (15) and (17) together show that the parametric gain de-pends now both on the fiber dispersion slope and on the fre-quency chirp induced by the pump-phase modulation. Notethat if we set dj = 0 in (15), we retrieve the usual expressionof the parametric gain for the case of non-modulated pumps[17]. Note also that the instantaneous phase mismatch has anegligible dependency on b2 and b4. We can also readily de-duce from Eq. (15) that the gain becomes time-independentagain when the two pumps are in phase opposition, i.e.,u1,s = �u2,s, as expected. Finally, the net signal gain at the2P-FOPA’s output with length L is given by

G ¼ 1þ 2cffiffiffiffiffiffiffiffiffiffiP 1P 2

p

gsinhðgLÞ

� �2

: ð18Þ

To demonstrate the cancellation technique, we studyand compare two 2P-FOPA configurations based on a300 m long highly non-linear fiber (HNLF) with b3 =1.2 · 10�40 s3 m�1, b4 = 2.85 · 10�55 s4 m�1 and c = 18W�1 km�1. HNLFs indeed allow for the achievement ofultra-wide and flat parametric gain band in the 1.55 lmregion, depending on an accurate tuning of the pumpfrequency separation DxP with respect to both b2 and b4

dispersion coefficients. The first 2P-FOPA has a mean pumpfrequency at the exact ZDW in order to achieve a flat 45 nm(5.6 THz) gain bandwidth. The second 2P-FOPA has beenoptimized to generate an ultrawide 74 nm (9.2 THz)

-300 -200 -100 0 100 200 300

14

16

18

Time (ps)

-300 -200 -100 0 100 200 300

14

16

18

Gai

n (d

B)

-200-1

-0.5

0

0.5

1x 10

11

-300 -100 0 100 200 3000

1

2

3

ϕ i,τ

s.dar(-1

)

ϕ i (

rad.

)

a

b

c

Gai

n (d

B)

ϕ1(τ) = −ϕ2(τ)ϕ1(τ) = ϕ2(τ)

ϕ1(τ) = −ϕ2(τ)

ϕ1(τ) = ϕ2(τ)

Fig. 2. (a) Phase of the pump waves (solid line, right scale) and its associatedtime derivative (dashed line, left scale). (b,c) Instantaneous parametric gainof a CW signal located at DxS = 0.5 THz for the two 2P-FOPAs (dashedlines: u1 = u2), (solid lines: u1 = �u2). Parameters are (b) b2 = 0, (c) b2(x0)= �2 · 10�29 s2 m�1, b3 = 1.2 · 10�40 s3 m�1, b4 = 2.85 · 10�55 s4 m�1,c = 18 W�1 km�1, P1 = P2 = 250 mW and L = 300 m.

A. Vedadi et al. / Optics Communications 267 (2006) 244–252 247

bandwidth with ripples of less than 0.2 dB, using theChebyshev polynomial method of Ref. [17]. Note that arelatively high value of b3 was chosen compared to standardHNLFs in order to stress the impact of phase modulation ongain distortions. Nevertheless, this value is standard for con-ventional dispersion shifted fibers (DSFs), which haveshown to be good candidates for 2P-FOPAs in terms ofcrosstalk for WDM systems [20]. In addition, we assume thatthe two pumps are p-binary phase-shift-keying phase-modu-lated by a pseudo-random-bit sequence (PRBS) operating ata frequency of 3-GHz and with a realistic rise-fall time of27-ps. This modulation scheme induces a uniform broaden-ing of the pumps spectra by the frequency modulation andhas been demonstrated for both efficient SBS suppressionand idler spectral broadening reduction [14]. Fig. 2(a) depictsone bit of the phase sequence (solid line) and its associatedtime derivative (dashed line). Figs. 3(a) and (b) show theinstantaneous parametric gain spectrum of 2P-FOPAs forthe two extreme values of the phase derivative u1,s = ±1 ·1011 rad s�1 when both pumps are in phase, and withoutphase modulation (solid line), derived from Eq. (18) of thetruncated model. For comparison, the parametric gain bandobtained from the exact solutions of (Eq. (13)) are also plot-ted in Figs. 3(c) and (d). First, we see on Figs. 3(a)–(d) notice-able deterioration of the gain spectrum for both extrema ofthe frequency chirp, meaning that the flatness of the 2P-FOPA is not conserved during each phase jump. Indeed,the phase modulation of the pump waves induces an instan-taneous pump frequency chirp (or dithering) that will modifythe phase-matching condition through the term dj in Eq.(17), and subsequently the parametric gain. This detrimentaleffect is so much marked in the second FOPA configuration

Frequency Shift Δωs/2 (THz)

Para

met

ric

Gai

n G

(dB

)

-6 -4 -2 0 2 4 60

5

10

15

20

-6 -4 -2 0 2 4 60

5

10

15

20

-6 -4 -2 0 2 4 60

5

10

15

20

-6 -4 -2 0 2 4 60

5

10

15

20a b

c d

Fig. 3. Parametric gain of the two 2P-FOPAs during the rise time (dashed lines) and the fall time (dotted lines) of the phase modulation when u1 = u2 andwithout phase modulation (solid lines), with the truncated model (a,b), and with exact solutions (c,d). Pump frequency detunings are (a, c)DxP = 2p · 4 THz, (b,d) DxP = 2p · 5.3 THz.

248 A. Vedadi et al. / Optics Communications 267 (2006) 244–252

that the signal gain exhibits a strong gap at the center of theband. Note also that in all cases the maximum parametricgain is lowered during both the rise time and the fall timeof the phase modulation. Second, the direct comparisonbetween Figs. 3(a) and (b) and Figs. 3(c) and (d) shows thatthe gain spectrum has significantly shrunk near the pump fre-quencies when the two external sidebands ASB1 and ASB2

generated by FWM are taken into account. These wavesindeed are not phase-matched and therefore reduce the gainbandwidth, as it has been previously demonstrated numeri-cally [18]. This comparison indicates that the standard modelof FWM is incomplete and gives incorrect results when thesignal frequency detuning DxS is comparable to the pumpfrequency detuning DxP. In Fig. 2(b) and (c) is illustratedthe impact of the pump-phase modulation on the instanta-neous parametric gain for a signal located at DxS = 0.5 THzfor both 2P-FOPA configurations. As it can be seen, thephase modulation of the two pump waves induces large gaindistortions during both the leading and the falling edges ofthe phase. Unlike for the single-pump case [7], the parametricgain is always reduced by the phase modulation in the dual-pumps case, in good accordance with the gain spectra plottedin Fig. 3. In the worst case, the gain drops by more than 3 dBas shown by the dotted curve in Fig. 2(c). When the pump-phases have opposite signs, i.e., u1 = �u2, the gain distor-tions are totally cancelled, as shown by the solid lines ofFig. 2(b) and (c).

3. Numerical simulations

To further study the impact of phase modulation on theparametric gain, we perform numerical simulations of the

7 80

50

100

150

0

1

2

3

9

Time (n

Pow

er (

µW)

0

1

2

3

0

50

100

150

7 8 9

1

1

1

1

a

c

Fig. 4. (a,b) Optical, and (c,d) electrical traces showing how a bit sequence isDashed line, phase of the pumps. The input signal power is 2 lW.

2P-FOPAs with a numerical integration of NLSE Eq. (4).We consider the parametric amplification of a non-return-to-zero (NRZ) data modulation format at a bit rateof R = 10 Gbit/s, in the same operating conditions as inSection 2. The two pump waves are phase-modulated witha 27-1 PRBS sequence operating at 3-GHz with commonrise/fall time of 27 ps. The small signal is shifted byDxS = 0.5 THz from the mean pump frequency. Addition-ally, the receiver is modelled in a realistic manner as in Ref.[7]. We used an optical Fabry–Perot filter (band-width = 4R) combined with an electrical square detectorand a second-order Butterworth filter (bandwidth = 0.8R)[21]. Note also that no noise source term has been addedin the simulation. We then compare the system perfor-mances of our two broadband and flat amplifiers studiedin Section 2. The results of our numerical simulations arepresented in Fig. 4(a) and (b) that show the output ampli-fied NRZ signal when the two pumps are co-phase modu-lated. The phase of both pumps are also plotted in dashedlines in Fig. 4(a) and (b). Clearly, the signal undergoessharp distortions during each phase jump. These resultsare in quite good agreement with the analytical study ofSection 2. As an example, for the first 2P-FOPA configura-tion (Fig. 4(a)), the gain is lowered by 17% (0.8 dB) and12.5% (0.6 dB) on a rising and a falling edge, respectively,in accordance with the analytical results of Fig. 2 (1 and0.8 dB respectively). Figs. 4(c) and (d) illustrate the electri-cal traces after the bandpass filter. The distortions seen onthe NRZ signal are lowered because of the bandpass filter-ing, but they still remain present. It is important to empha-size that such ultra-fast signal distortions may be difficultto observe directly in a practical system. This is why indi-

s)

ϕ(τ)

(rad

.)

0

50

00

50

1 1.5 2 2.5 30

1

2

3

0

50

00

50

1 1.5 2 2.5 30

1

2

3

b

d

ϕ(τ)

(rad

.)

deteriorated by the phase modulation when the two pumps are in-phase.

A. Vedadi et al. / Optics Communications 267 (2006) 244–252 249

rect observations have been performed recently throughbit-error rate and Q penalties measurements [8,9]. Whenthe two pumps are synchronously counter-phase modulated,as illustrated on Fig. 5, the optical and electrical tracesshow that the signal distortions are almost cancelled, ingood agreement with the analytical prediction of Section 2.Other numerical results, presented in Figs. 6(a)–(d), showthe eye patterns at the FOPA’s input and output when

Pow

er (

μW)

Time (n

1

1c

04 4.5 5 5.5 6

0

1

2

3

50

100

150

a

04 4.5 5 5.5 6

0

1

2

3

50

100

150

1

1

Fig. 5. (a,b) Optical, and (c,d) electrical traces showing how the signal distortphase of the pumps.

0 50 100 150 2000

0.5

1

1.5

2

2.5

0 50 100 150 2000

50

100

150

Tim

Pow

er (

μA)

a

c

Fig. 6. Electrical eye patterns of the signal located at 0.5 THz from x0, (a)respectively, when the pumps are in-phase modulated. (d) Amplifier’s output

the pump are co-phase and counter-phase modulated,respectively. As it can be seen, the eye pattern is severelydistorted at the FOPA’s output, especially with the second2P-FOPA under study (Fig. 6(c)), while it becomes as openas at the amplifier’s input when counter-phase modulationis used. The residual small signal distortions seen inFig. 5 can be attributed to additional impairment of thephase modulation (PM) to amplitude modulation (AM)

ϕ 1,2

(τ)(

rad.

)

s)

ϕ 1,2

(τ)(

rad.

)

d

04 4.5 5 5.5 6

0

1

2

3

50

00

50

04 4.5 5 5.5 6

0

1

2

3

50

00

50 b

ions can be cancelled when the two pumps are out-of-phase. Dashed line,

0 50 100 150 2000

50

100

150

e (ps)

0 50 100 150 2000

50

100

150

b

d

Amplifier’s input. (b,c) Output of first and second 2P-FOPA under testwhen the pumps are out-of-phase modulated.

246

248

250

252

4.5 5 5.5 60

1

2

3

246

248

250

252

4 4.5 5 5.50

1

2

3

246

248

250

252

1 1.5 2 2.50

1

2

3

246

248

250

252

1 1.5 2 2.50

1

2

3Pum

p Po

wer

(m

W)

Time (ns)

a b

c d

P1

P2

P2

P1

Pum

p Ph

ase

(rad

.)

4

Fig. 7. Pump powers (P1, P2, solid lines) and their phases (u1, u2, dashed lines) at the FOPA output showing the small PM to AM conversion with thesame parameters as in Fig. 2. (a, c) 1st 2P-FOPA, (b,d) 2nd 2P-FOPA.

250 A. Vedadi et al. / Optics Communications 267 (2006) 244–252

conversion because of fiber dispersion, recently predictedfor FOPA by Yaman et al. [15]. As demonstrated theoret-ically, the PM to AM conversion also leads to signal distor-tions via the parametric gain, even when a counter-phasemodulation scheme is implemented. We numerically inves-tigated the impact of PM to AM conversion on both thepump waves and the signal gain using NLSE integration.Fig. 7(a)–(d) shows the temporal evolution of the pumppowers (P1, P2, solid lines) and their phases (u1, u2, dashedlines) at the FOPA’s output for both 2P-FOPA configura-tions. We can see the PM to AM conversion of the pumpwaves which exhibit opposite amplitude modulations dur-ing a rise/fall time of the phase modulation because of theiropposite group-velocity dispersion with respect to theZDW. Consequently, the PM to AM conversion for thetwo pump waves cannot balance each other out to equalizethe parametric gain. However, on Fig. 7(a)–(d), the PM toAM conversion induces very low pump power fluctuations,

0 100 200 300 -1.5

-1

-0.5

0

0.5

Fiber length (m)

ZD

W s

hift

(nm

) a

Fig. 8. (a) Zero-dispersion wavelength (ZDW) fluctuations map (solid line) andwith ZDW variations (solid line) and without ZDW variations (dashed line).

of less than 0.5% of the total continuous pump power. Thisnumerical result clearly shows that the counter-phasepumping scheme does not suffer from PM to AM conver-sion of the pump waves for realistic rise/fall time com-monly used to efficiently suppress SBS. However, becauseof its dependency on the second derivative of the pumpsphase and on the pumps dispersion, PM to AM conversioncan be detrimental for sharp rise/fall time of pump-phasemodulation or wide amplifier bandwidth [15].

4. Influence of dispersion fluctuations

In practice, the ZDW of single-mode optical fiberslongitudinally fluctuates due to random variations inopto-geometric parameters of the preform and drawingprocesses during fabrication. Because of the phase-match-ing conditions of FOPAs, the ZDW variations induceparametric gain ripple and reduce the gain bandwidth

-6 -4 -2 0 2 4 60

5

10

15

20

Gai

n (d

B)

Frequency Shift (THz)

b

mean ZDW (dashed line). (b) Parametric gain spectrum of the 2P-FOPAsDxP = 2p · 4 THz.

-2.5 -2 -1.5 -10

20

40

0

1

2

3

-0.5

a

0

20

40

200

c

7 7.5 8 8.50

20

40

0

1

2

3

9

b

0 50 100 1500

20

40

200

d

Time (ns)

Time (ps)

Pow

er (

μW)

Pow

er (

μA)

ϕ 1,2

1,2

(τ)

(rad

.)

0 50 100 150

Fig. 9. (a,c) Optical trace and electrical eye pattern of a signal at the output of the 2P-FOPA with ZDW fluctuations when the two pumps are in-phasemodulated, as indicated by the dashed lines, and (b,d) out-of-phase modulated.

A. Vedadi et al. / Optics Communications 267 (2006) 244–252 251

[22,23]. It has been shown in Ref. [9] that the ZDWfluctuations can strongly reduce Q penalties caused bypump-phase modulation in FOPAs. While it is foreseeablefrom Eqs. (15)–(18) that the counter-phase modulationtechnique should remain robust against ZDW fluctua-tions, the fact that the optimized gain is not achieved inthis case may lead to behaviors different of the last sec-tion. It is therefore important to assess the gain distor-tions due to pump-phase modulation with randomfluctuations of ZDW. We therefore consider a 300-m longHNLF whose ZDW variations was mapped in Ref. [24],as plotted in Fig. 8(a). In this section, we restrict ourstudy to the first 2P-FOPA configuration of Sections 2and 3 for which the mean pump frequency is exactly atthe ZDW. Fig. 8(b) shows the gain spectrum obtainedthrough a numerical integration of the NLSE with thesame parameters as in Fig. 2(b). We then perform numer-ical simulations of a FOPA with a small signal located atDxS = 0.5 THz from the mean pump frequency. Asshown in Fig. 9(a), when the two pumps are co-phasemodulated, the signal undergoes substantial distortionsdue to pump-phase modulation. In particular, in additionto dips, one can also see spikes on the falling edges of thepumps phases. Unlike the former case of Section 3 whenZDW was considered constant and dj could only inducea gain decrease, the pump-phase shift may in some caseenhance the phase-matching conditions thus increasingthe instantaneous signal gain. When the counter-phasemodulation is used, the temporal optical trace and theelectrical eye diagrams of Figs. 9(b) and (d) show that

the gain distortions are almost cancelled, meaning thatthe counter-phase modulation technique is robust againstZDW variations.

5. Conclusion

In this work, we have revisited theoretically the smallsignal gain of a broadband and flat dual-pump fiber opti-cal parametric amplifier by including the phase modula-tion of the pumps, which is implemented in practice toavoid stimulated Brillouin back-scattering. It has beenshown both analytically and numerically that the dual-pumping counter-phased scheme in parametric amplifierscan in principle totally cancel not only the idler spectralbroadening, but also the large gain distortions inducedby pump-phase modulation. Moreover, our numericalresults have shown that this useful technique does not sig-nificantly suffer from the additional impairment of pump-phase modulation to amplitude modulation conversion.Our results suggest that the dual-pumping counter phasedscheme in parametric amplifiers is clearly more suitablefor practical applications than single-pump scheme, aslong as phase modulation is used to suppress stimulatedBrillouin scattering.

Acknowledgements

The authors thank the Conseil Regional de Franche-Comte for financial support and A. Durecu from Alcatelfor helpful discussions.

252 A. Vedadi et al. / Optics Communications 267 (2006) 244–252

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