October 1, 2008 / Vol. 33, No. 19 / OPTICS LETTERS 2203

Investigation of gain ripple in two-pump fiberoptical parametric amplifiers

Armand Vedadi,1,* Michel E. Marhic,1 Eric Lantz,2 Hervé Maillotte,2 and Thibaut Sylvestre2

1Institute of Advanced Telecommunications, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, UK2Département d’Optique P.M. Duffieux, Institut FEMTO-ST, Centre National de la Recherche Scientifique

UMR 6174, Université de Franche-Comté, 25030 Besançon, France*Corresponding author: a.vedadi@swansea.ac.uk

Received May 21, 2008; accepted August 6, 2008;posted August 25, 2008 (Doc. ID 96530); published September 25, 2008

By using the four-sideband theory, we analyze the gain spectrum in wideband two-pump fiber optical para-metric amplifiers and predict gain ripples over the flat gain region. We derive an approximation of theirpseudo-periods and discuss methods for reducing their amplitudes. © 2008 Optical Society of America

OCIS codes: 060.0060, 060.4370.

Fiber-optical parametric amplifiers (FOPAs) are be-ing developed for efficient ultrahigh bandwidth andultrafast signal processing technologies such aswavelength conversion, multiplexing, sampling, buff-ering, etc. [1,2]. Thanks to both the ultrafast Kerrnonlinearity and the flexibility in the design of the fi-ber’s dispersion curve, large and broadband paramet-ric gain can be achieved in the 1.55 �m region by us-ing one or two pumps [1,2]. The latter architecture isof interest for telecommunication applications: It al-lows cancellation of signal distortion and idler broad-ening, improvement of polarization insensitivity, andthe design of wideband amplifiers with flat gain spec-tra [3–7]. Indeed, using a standard four-wave mixingtheory involving the two pumps, the signal, and thegenerated idler waves, it is possible to optimize theposition of the two pumps so as to achieve widebandflat gain between the two pumps [6,7]. However, thismodel is not valid close to the pumps, as one has totake into account the two other sidebands that aregenerated in the region outside the two pumps by

k=2

0146-9592/08/192203-3/$15.00 ©

Bragg scattering (BS) and modulation instability(MI). The interaction among all four sidebands leadsto a gain reduction in the vicinity of the two pumps[8–10]. Recently, we showed that by taking into ac-count the four sidebands, one can predict smallripples on the flat gain region that can lead to impair-ments for telecommunication applications [11]. Inthis Letter, we review these preliminary resultslinked to a recent experimental demonstration of awideband and flat-gain two-pump FOPA (2P-FOPA)[7]. We explain in detail the mechanism of the ob-served ripples and discuss guidelines to minimizethose ripples when designing flat-gain 2P-FOPAs.

Following notations similar to [12], we denote by�1 ,�2 the angular frequencies of the two pumps andby �3, �4, �5, and �6 those of signal, idler, and thesidebands generated outside the two pumps (see Fig.1). We refer to this model as the four-sideband (4S)model. Assuming that the pumps are nondepletedand following the steps in [5], it leads to a set of fourcoupled equations:

�1�

This equation has the form �Y /�z= jMY. P1 ,P2 arethe pump powers, and A3, A4, A5, and A6 are the side-bands’ slowly varying amplitudes (SVAs). � is the fi-ber nonlinearity coefficient and r=2��P1P2. Theterms ��i�i=3,4,5,6� in M represent linear wave-vector mismatches between sideband i and its closestpump. They can be expanded in Taylor series about�C= ��1+�2� /2:

��i = ���i� − ���g� = ��

�− 1�k�i�k

k!���i

k − ��Pk�, �2�

where ��P= ��2−�1� /2, g= ��−1��+3� /2 , ��i=��S= ��3,4−�C� for i= �3,4�, and ��i=��SB=�C−�5=2��P−��S for i= �5,6�. �k are the higher-order dis-persion coefficients at �C. In practice, it is sufficientto consider �2, �3, and �4 to account for mostdispersion-related behaviors.

M does not depend on z; hence the solution can bereadily derived as follows:

Y�z� = exp�jMz�Y�0�. �3�

By setting the appropriate sideband amplitude to 1

and the others to 0 as initial conditions, the gain can

2008 Optical Society of America

2204 OPTICS LETTERS / Vol. 33, No. 19 / October 1, 2008

be numerically calculated for signals located in ]�1−��P, �2+��P[.

To obtain a closed-form analytical solution for theparametric gain, one would need to calculate the ei-genvalues and eigenvectors of M. This has been donefor signals located in the vicinity of one pump [12],but the calculation remains complex for the wholespectrum. It is therefore convenient to neglect side-bands A5 and A6. Using this two-sideband (2S) model,M is reduced to the 2�2 matrix framed in dotted linein Eq. (1). The signal output SVA can then be derivedas in Section A1.2 of [1], with the following result:

Ai = exp��− 1�ij���P2 − P1�/2 + ��3/6����S3 − ��P

3��z

� cosh�gz� − j�

2gsinh�gz�� �i = 3,4�, �4�

where �=��P1+P2�+2�k=1� �2k / �2k�!���S2k−��P

2k� isthe total wave-vector mismatch between the four in-teracting waves, g=�r2− �� /2�2, and r=2��P1P2. Forhigh gain, one can neglect the decaying exponentialin Eq. (4):

Ai � 1

2− j

�

4g� � exp���− 1�ij���P2 − P1�/2

+ ��3/6����S3 − ��P

3�� + gz� �i = 3,4�. �5�

The signal gain is also straightforwardly derivedfrom Eq. (4) as G=1+ ��r /g� sinh�gL��2.

Using the 2S and 4S models, we analyzed an ex-perimental 2P-FOPA with flat gain spectrum overmore than 70 nm demonstrated by Chavez Boggio etal. in a highly nonlinear fiber of 243 m effectivelength [7]. The results are shown in Fig. 2. The 2Smodel, which was used to design a flat-gain FOPA,shows good agreement with the experimental resultsfor signals that are separated by more than 10 nmfrom the closest pump. In particular, the flat-gain re-

Fig. 1. Four-sideband model with pumps, signal, andthree generated idlers.

Fig. 2. Spectrum of the 2P-FOPA gain. Squares, experi-mental results; solid curve, 4S analysis; dashed curve,2S analysis (standard model). The parameters are �2=6.4�10−30 s2 m−1, �3=0.65�10−40 s3 m−1, �4=−1.6

−55 4 −1 −1 −1 13

�10 s m , �=8 W m , and ��P=3.25�10 rad/s.

gion of 70 nm is fairly well fitted. As expected, the re-duction of the parametric gain in the vicinity of thepumps can be fitted by using the 4S model. The dis-crepancy observed for wavelengths larger than 2 isdue to Raman gain. However, when zooming in onthe edges of the flat region of the spectrum, Fig. 3shows that the 4S model predicts gain ripples.Clearly, these ripples occur in a region where both MIand BS processes are negligible. Their existence wasverified by simulation of the nonlinear Schrödingerequation. The pseudo-periods of the ripples rangefrom 0.5 to 1 nm, and their amplitudes can be largerthan 1.5 dB, which is sufficient to impair telecommu-nication applications [11].

Equation (3) gives very little insight about the un-derlying mechanisms of these ripples, as it can besolved only by numerical means. Instead, we willlimit our discussion to qualitative considerationswithin the flat-gain region (where the approximationof the two sidebands model is good). The four complexeigenvalues of M have the form 3,4���S�=p1���S�± jg1���S� and 5,6���S�=p2���S�± jg2���S�. The output field amplitude of the signal isa linear combination of four exponentials exp�jiz� �i=3,4,5,6�, with weights that depend on the initialfield amplitudes. Neglecting all decaying exponen-tials, the output SVA is approximated by

A � a exp�jp1z + g1z� + b exp�jp2z + g2z�, �6�

and thus the parametric gain by

G�z� � �a�2e2g1z + �b�2e2g2z + 2�ab�e�g1+g2�z

�cos��p1 − p2�z + �, �7�

where =arg�ab*� and a, b are weights. This latterexpression shows how the different eigenvalues com-bine to give the output parametric gain. It also showsthat G contains an oscillating term whose pseudo-period depends on z. In the region of flat gain that isof interest for telecommunications, Eq. (5) also showsthat the output field is approximated by a single ex-ponential. Because the exponentials are a basis forfunction decomposition, we infer that one of theterms [say a exp�jp1z+g1z�] in Eq. (6) should be domi-nant and equal to Ai of Eq. (5), while the other termis a small perturbation ��a�� �b��. In Eq. (7), the sec-ond term could therefore be neglected, and the thirdterm would account for the small ripples that are ob-served. By assumption, we also get p1���S�= �−1�i���P2−P1� /2+ ��3 /6����S

3 −��P3��z �i=3,4� and

g1=g. We note that ��3,4���S�=��5,6���SB�. Supposethat 3���S� is an eigenvalue of M with the associ-

Fig. 3. (a), (b) Zoom in on the edges of the flat-gain region.

Dashed curve, plot of Eq. (9).

October 1, 2008 / Vol. 33, No. 19 / OPTICS LETTERS 2205

ated eigenvector T�v5���S�v3���S�v4���S�v6���S��.From the second row of M it then comes that

���3���SB� + �P1� � v3���SB� + �P1 � v5���SB�

+ r � v6���SB� + r � v4���SB�

= ���5���S� + �P1� � v3���SB� + �P1 � v5���SB�

+ r � v6���SB� + r � v4���SB�. �8�

Similar expressions hold with all other lines of M.Thus, we deduce that −3

*���SB� is another eigen-value of M with associated eigenvectorT�v3���SB�v5���SB�v6���SB�v4���SB��. Identifyingthe real and imaginary parts with 5���S�, we de-duce p2���S�=−p1���SB� and g2���S�=g1���SB�.Hence

cos��p1 − p2�z� � cos��3��P���P − ��S�2z

+ ��P2 − P1�z�. �9�

We have plotted cos��3��P���P−��S�2z� with an ad-justed phase in Fig. 3 as a dashed curve. It shows afairly good agreement with the pseudo-periods of theripples within the flat region. Equation (9) also showsthe influence of the fiber dispersion slope and lengthon the ripples pseudo-period. In this region where MIand BS processes are negligible, the relative ampli-tude of the ripples depends on the ratio2�b /a�exp��g���SB�−g���S��z. Although b /a is verysmall, this latter expression shows that noticeableripples may occur if g in the inner band between thetwo pumps (at ±��S) is smaller than or equal to g fora signal located at the symmetrical frequency rela-tively to the closest pump [at ±��SB= ± �2��P−��S�].

The maxima of g correspond to the roots of �=0.When �4�0, one can choose �2 and ��P such that �can be written in terms of a fourth-order Chebyshevpolynomial [6] whose roots are all located betweenthe two pumps. As a consequence, g is always greaterin the inner band between the two pumps than in theouter band and the ripples become negligible. When�4 0, two roots ±�1 of �=0 are always located in theouter band [7]. To minimize the ripples, one shouldtherefore design the FOPA spectrum such that � pos-sesses two other roots located close to ±2��P−�1. Toverify these assumptions, we have plotted in Fig. 4(a)the spectra of two different 2P-FOPAs: one with posi-tive �4 and the other one with a negative �4 and anoptimized �2. All other parameters are the same asfor the FOPA studied in Figs. 2 and 3. In both cases,the parametric gain is increased in the flat regionand the ripples are reduced compared with the FOPAstudied in Figs. 2 and 3. To quantify the reduction ofthe ripples amplitudes, we have numerically calcu-lated �a�2e2g1L and subtracted that amount from G�L�.The results are shown on Fig. 4(b) and show that therelative ripples amplitudes are significantly reducedfor the FOPAs of Fig. 4(a) compared with the FOPA of

Figs. 2 and 3.

In conclusion, we have investigated the nature ofripples in the flat-gain region of 2P FOPAs. Theseripples may be large enough to lead to impairmentsin telecommunication systems. By looking at the ei-genvalues, we were able to predict their pseudo-periods. To reduce the ripples amplitudes, we haveshown that the 2P-FOPA should be designed suchthat the gain within the flat region between the twopumps is larger than the gain in the outer band. Withpositive �4 fibers this design is straightforward. How-ever, most conventional fibers have negative �4. Inthis case, the gain between the two pumps should bemaximized in order to reduce the ripples.

This work has been funded by the Conseil Regionalde Franche-Comté, the European Regional Develop-ment Fund (ERDF), and the UK’s Engineering andPhysical Sciences Research Center (EPSRC). The au-thors gratefully thank J. M. Chavez Boggio and co-workers for sharing their experimental results.

References

1. M. E. Marhic, Fiber Optical Parametric Amplifiers,Oscillators and Related Devices (Cambridge U. Press,2007).

2. C. J. McKinstrie, S. Radic, and A. H. Gnauck, Opt.Photon. News 18(3), 34 (2007).

3. M. C. Ho, M. E. Marhic, K. K. Y. Wong, and L. G.Kazovsky, J. Lightwave Technol. 20, 469 (2002).

4. A. Vedadi, A. Mussot, E. Lantz, H. Maillotte, and T.Sylvestre, Opt. Commun. 267, 244 (2006).

5. M. E. Marhic, K. K.-Y. Wong, and L. G. Kazovsky, J.Opt. Soc. Am. B 20, 2425 (2003).

6. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky,Opt. Lett. 21, 1354 (1996).

7. J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, andH. L. Fragnito, Opt. Express 15, 5288 (2007).

8. M. Yu, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev.E 48, 2178 (1993).

9. L. Provino, H. Maillotte, E. Lantz, T. Sylvestre, and J.M. Dudley, in Conference on Lasers and Electro-OpticsEurope, IEEE Conference Digest (IEEE, 2000), paperCtuL4.

10. C. J. McKinstrie and S. Radic, Opt. Lett. 27, 1138(2002).

11. A. Vedadi, E. Lantz, H. Maillotte, and T. Sylvestre, in2008 IEEE/LEOS Winter Topical Meeting Series(IEEE, 2008), paper TuB 1.2.

12. M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, J. Opt.

Fig. 4. (a) Gain spectra of 2P-FOPAs with �2=6.3�10−30 s2 m−1, �4=1.6�10−55 s4 m−1 (solid curve) and �2=5�10−29 s2 m−1, �4=−1.6�10−55 s4 m−1 (dotted curve).(b) Ripple relative amplitudes associated with the FOPA ofFig. 2 (dashed curve) and the two FOPAs of Fig. 3(b) withpositive �4 (solid line) and negative �4 (dotted curve).

Soc. Am. B 25, 22 (2008).