## Abstract

A new two-pump fiber optical parametric amplifier (FOPA) is presented, which is composed of two-section high nonlinear fibers (HNLFs). Genetic algorithm (GA), a multivariate stochastic optimization algorithm is applied to optimize parameters of two fiber segments, such as the length and dispersion coefficient of fiber. A broadband FOPA with two-section HNLFs is obtained using optimum design parameters, which theoretically provides a uniform gain of 20.3 dB with 0.2-dB uniformity over a 346-nm bandwidth.

©2004 Optical Society of America

## 1. Introduction

At present, the rapid development of optical fiber communications along with large-capacity broadband transmission systems has led to the growth of requirement to super-broadband optical amplifiers. But the traditional erbium-doped fiber amplifiers (EDFA) and fiber Raman amplifiers (FRA) can’t provide such a wide gain bandwidth that the low loss windows (1250–1650 nm) of all-wave fiber can’t be utilized fully. A new type of broadband amplifier, fiber optical parametric amplifier (FOPA) appears, which utilizes the principle of four-wave mixing (FWM) to amplify the signals and offers a broadband amplification at arbitrary wavelengths [1–6]. Moreover, FOPA can also offer a small noise figure that is lower than the 3-dB quantum limit, and is used as broadband wavelength converters, pulse generator, optical time division multiplexed switch and all-optical sampler, etc [7].

Although simple single-pump FOPA could experimentally offer a gain bandwidth of more than 200 nm, the gain spectrum was not uniform over the amplifier bandwidth but had a difference as high as 15 dB between the lowest and the highest value [8]. Such a large gain ripple makes it difficult to equalize the power of the various channels in wavelength-division multiplexing (WDM) systems. For this reason, much research has been done to flatten the gain spectrum of FOPA. An approach to efficiently widen and flatten the gain spectrum is to use a single-pump FOPA with multi-section high nonlinear fibers (HNLFs) arrangement [9,10]. The relatively uniform parametric gain with a broad bandwidth of 300 nm has been theoretically obtained using a four-section HNLFs configure [10] and a continuous-wave (CW) FOPA with 60-dB gain using a two-section HNLFs design was experimentally achieved [11].

However, several inherent problems exist in single-pump FOPA with multi-section HNLFs configure. Firstly, the pump frequency overlaps the signal band, which makes it difficult to filter out the pump. Secondly, when single-pump FOPA is operated in CW mode, the idler spectrum is broadened due to the required pump dithering, which degrades the bit error rate (BER) when FOPA is used as wavelength converter. Hence two-pump FOPA with a single-section fiber allocation was introduced [12,13], which could cancel idler broadening induced by the pump dithering [14]. Furthermore, two-pump FOPA can also provide a polarization-independent performance by using two pumps with orthogonal polarization states [15], but the gain ripple still exists.

In this paper, we put forward for the first time a new two-pump broadband FOPA with two-section HNLFs, which can not only keep preceding attractive features but also solve the problem of gain ripple. As the structure parameters of FOPA, such as the length and dispersion coefficient of fiber, have a great effect on the performance of an amplifier, it is necessary to find an effective optimization method for designing two-pump multi-section FOPA. Genetic algorithm (GA), a multivariate stochastic optimization algorithm is proposed in the paper, which has been applied to design broadband Raman amplifiers [16] and multistage erbium-doped fiber-amplifier [17]. As a result, a new two-pump FOPA with two-section HNLFs is achieved using optimum parameters calculated by GA, which theoretically provides a uniform gain of 20.3 dB with 0.2-dB uniformity over a 346-nm bandwidth.

## 2. Theory and scheme of FOPA

Maxwell’s equations in the fiber take the form of the following coupled wave equations [18]:

Where *A*
_{1} and *A*
_{2} are the pumps amplitudes; *A*
_{3} and *A*
_{4} are the signal and idler amplitudes; *z* is the distance from the beginning of the first HNLF segment; *γ* is the nonlinear coefficient of fiber; Δ*β* is the linear propagation-constant mismatch.

Assume that the pumps (*A*
_{1} and *A*
_{2}) are much more intense than the signal and idler (*A*
_{3} and *A*
_{4}), the solutions of Eqs. (1) and (2) can be obtained:

Where *P*_{j}
=|*A*_{j}
(0)|^{2}. The Eqs. (5) and (6) show that the pumps are not depleted but experience a phase change due to SPM and XPM. By substituting Eqs. (5) and (6) in the Eqs. (3) and (4), one can show that:

Where *θ*=[Δ*β*-3*γ*(*P*
_{1}+*P*
_{2})]*z*.

By making the substitutions in the Eqs. (7) and (8):

One can rewrite Eqs. (7) and (8) in the form:

Where *k* is the total phase mismatch including the nonlinear phase shift:

The solutions of Eqs. (10) and (11) can be obtained:

Eqs. (13) and (14) can be expressed in the matrix form:

Where *B*(*z*)=[*B*
_{3}(*z*),*B**_{4}(*z*)]^{T}, *B*(0)=[*B*
_{3}(0),*B**_{4}(0)]^{T}.

The transfer matrix in the case of single-section HNLF is obtained by combining initial condition with Eqs. (13) and (14) [19]:

The parametric gain coefficient g is given [20]:

Where *u*=*γ*(*P*
_{1}+*P*
_{2})=*γP*
_{0} and $r=2\sqrt{{P}_{1}{P}_{2}}\u2044\sqrt{{P}_{1}^{2}\phantom{\rule{.2em}{0ex}}{P}_{2}^{2}}$.

The linear propagation-constant mismatch Δ*β* is shown [20]:

Define the parameters, the central frequency of the two pumps *ω*_{c}
=(*ω*
_{1}+*ω*
_{2})/2; the frequency detuning between the signal *ω*_{3}
and the central frequency Ω=*ω*
_{3}-*ω*_{c}
; the average difference of the pumps frequency Δ*ω*_{p}
=(*ω*
_{1}-*ω*
_{2})/2.

We expand *β*_{3}
around the central frequency of the two pumps *ω*_{c}
:

$$+\frac{1}{6}\frac{{d}^{3}\beta}{d{\omega}^{3}}\left({\omega}_{c}\right)\xb7{\left({\omega}_{3}-{\omega}_{c}\right)}^{3}+\dots $$

$$=\beta \left({\omega}_{c}\right)+\sum _{n=1}^{\infty}\frac{{\beta}^{\left(n\right)}}{n!}\left({\omega}_{c}\right)\xb7{\left({\omega}_{3}-{\omega}_{c}\right)}^{n}$$

Where *β*^{(n)}
denotes the nth derivative of *β* at *ω*_{c}
. Similarly, we can write *β*
_{1}, *β*
_{2} and *β*
_{4}, and utilize the substitutions, Ω=*ω*
_{3}-*ω*_{c}
=-(*ω*
_{4}-*ω*_{c}
), Δ*ω*_{p}
=*ω*
_{1}-*ω*_{c}
=-(*ω*
_{2}-*ω*_{c}
).

The linear propagation-constant mismatch Δ*β* can be rewritten:

From Reference [20], it is known that truncating Δ*β* after the fourth order should be a generally valid procedure, so Δ*β* can be expressed as following:

We replace the hyperbolic function cosh(*gz*)=(*e*^{gz}
+*e*
^{-gz})/2, sinh(*gz*)=(*e*^{gz}
-*e*
^{-gz})/2 in Eq. (16) and the transfer matrix becomes:

Where *ζ*=-(*κ*+2*ig*)/(2*ur*).

By iterating this procedure, the signal and idler amplitudes at the output of the *N*-th fiber segment can be written:

The small-signal power gain at the output can be expressed as the form:

Furthermore, we have also taken into account for the pump waves a splicing loss factor, and therefore the pump power at the input of each fiber segment becomes *P*_{k}
=*P*
_{0}
*α*
^{k-1}(*k*=1, 2⋯*N*), with *α*=0.87 [9].

From the preceding theory, it is known that the gain spectrum of two-pump FOPA mainly depends on the pump powers and wavelengths, the dispersion coefficient, nonlinear coefficient and length of HNLFs. In the case of the nonlinear coefficient of fiber, the pump powers and wavelengths being fixed, the gain spectrum is governed by the length and dispersion characteristics of each fiber segment. Actually the gain is determined by the even dispersion orders of fiber (*β*
^{(2)}, *β*
^{(4)}) depicted as in Eq. (21) [20]. However, the *β*
^{(2)} is related to the third- and fourth-order dispersion at the zero-dispersion wavelength (ZDW) of fiber as *β*
^{(2)}(*ω*_{c}
)=*β*
^{(3)}(*ω*
_{0})(*ω*_{c}
-*ω*
_{0})+*β*
^{(4)}(*ω*
_{0})(*ω*_{c}
-*ω*
_{0})^{2}/2 and tuned with the central frequency of the two pumps *ω*
_{c} and the zero-dispersion frequency of fiber *ω*
_{0} [21]. Generally, the third- and fourth-order dispersion of fiber is considered as constant [2,12,13,21], so a uniform gain spectrum can be obtained by choosing properly the length and ZDW of each fiber segment and the two pump wavelengths. Moreover, the ZDW can be chosen over a wide range by the means of selecting the appropriate core diameter.

In two-pump FOPA with multi-section fibers, the first fiber segment determines the approximate required bandwidth and gain and provides a broad non-uniform gain spectrum; the following fiber segments progressively smooth the gain spectrum by carefully tuning their parameters. Normally, parameters of the first fiber segment are determined in advance to provide a required broadband gain spectrum. And then the optimization algorithm is used to obtain the optimum parameters of following fiber segments. Supposing that *β*
^{(4)} of each fiber segment remains constant, for a FOPA composed of *N* fiber segments, there will be 2(*N*-1) parameters: *L*_{i}
and ${\beta}_{\mathrm{i}}^{\left(2\right)}$ (*i*=2…*N*), which are required to be optimized simultaneously. And ordinary algorithms are not effective. In contrast, GA is employed well to solve the problem of multivariate stochastic optimization and good results are obtained. In the following part, the operating process of GA to optimize multi-parameters will be described in detail. Fig. 1 shows the scheme of two-pump broadband FOPA with *N*-section HNLFs.

## 3. Genetic algorithm to optimize FOPA

Genetic algorithm (GA), a multivariate stochastic optimization algorithm, which is based on the mechanics of natural selection and natural genetics, is generally able to find good solutions in a reasonable amount of run time [22]. GA operates on a population of potential solutions applying the principle of survival of the fittest to produce better approximations to a solution. The use of a GA requires the determination of six fundamental issues: the chromosome representation, the creation of the initial population, the fitness function, the selection function, the genetic operators making up the reproduction function and the termination criteria.

*The chromosome representation*. For any GA, an “individual” is a feasible solution that is described by a coded datum called a “chromosome” with values within the variables upper and lower bounds. In the optimization for FOPA, the coded data contain amplifier information such as the length and second-order dispersion of HNLFs. Each individual (*popj*) is made of 2(*N*-1) “chromosomes” {*L*
_{2}, *L*
_{3}, … *L*_{N}
, *β*
^{(2)}2, *β*
^{(2)}3, …*β*
^{(2)}N}, which constitute the variables of the problem, where *L*_{i}
represents the length of each fiber segment and ${\beta}_{\mathrm{i}}^{\left(2\right)}$ is the second-order dispersion (*i*=2…*N*).

*The creation of the initial population*. At the first generation, a population of “individuals” is randomly created, each individual being a possible solution to the problem. It is very important to choose a proper number of individuals in a population (*popsize*). If there are no enough individuals, it is possible to be trapped in local optima; if there are too many individuals, it will spend lots of time to find the optimum individual. After many trails, the number of individuals in a population is set to 100 (*popsize*=100).

*The fitness function*. Fitness functions of many forms can be used in a GA, subject to the minimal requirement that the function can map the population into a partially ordered set. The individuals are ranked by fitness function. In the optimization for FOPA, a fitness function is given to represent the degree of satisfaction with the amplifier performance. Commonly, the optimized amplifier should have high gain and broad and flat gain profile, and therefore two fitness functions for this problem are reasonable: the bandwidth with preset gain ripple or the reciprocal of ripple with preset bandwidth. After many trails, the former or the combination of both shows better efficiency. In this work, we choose the bandwidth with 0.2-dB gain ripple as fitness function (*F*_{j}
=*fitness*(pop_{j}) *j*=1, 2 …, *popsize*).

*The selection function*. After generating an initial population that contains a certain number of individuals, we perform a series of processes (selection, crossover, and mutation) on the population to generate the next generation circularly. The selection of individuals to produce successive generations plays an extremely important role in GA. A probabilistic selection is performed based upon the individual’s fitness such that the better individuals have an increased chance of being selected. An individual in the population can be selected more than once with all individuals in the population having a chance of being selected to reproduce into the next generation. That is, the fittest individuals of any population tend to reproduce and survive into the next generation, thus improving successive generations. In this work, the selection method of roulette wheel, developed by Holland, is used. The probability (*P*_{i}
) for each individual is defined by the following equation, where *F*_{i}
equals the fitness of individual *i*.

*The genetic operators making up the reproduction function*. Genetic operators provide the basic search mechanism of the GA. The operators are used to create new solution based on existing solutions in the population. There are two basic types of operators: crossover and mutation, which are implemented at each generation. Crossover takes two individuals and produces two new individuals while mutation alters one individual to produce a single new solution. The application of these two basic types of operators and their derivatives depends on the chromosome representation used. Combine the selected *pop*_{j}
in pairs and then implement a single-point crossover on the counterpart of chromosomes in each pairs in order to create offspring, which inherits the traits of the parents. The crossover rate here is about 85%. In nature, mutation consists of modifying randomly a chromosome. The individual who has mutated is conserved in the case of a better adaptation to the environment. The mutation rate specifies the odds that a given gene in a chromosome will be mutated. In the case of bit representation, the gene will simply be flipped, that is, a one changed to a zero or a zero changed to a one. The mutation rate is changed during the evolution process and initially it is set to be about 8%.

*The termination criteria*. The GA moves from generation to generation selecting and reproducing parents until a termination criterion is satisfied. The most frequently used stopping criterion is a specified maximum number of generations. The trace of the optimization is shown in Fig. 2. It indicates that 50 generations of evolution are sufficient for obtaining the optimum solution.

## 4. Result and discussion

For two-pump FOPA, it is very important to choose the length and *β*
^{(2)} of the first fiber segment, which determine the approximately required bandwidth and gain and provide a broad non-uniform gain spectrum. And the effect of these parameters on the gain spectrum will be discussed in the following part. We consider the gain of a two-pump FOPA using a multi-section HNLFs configure at the output of the first-section fiber with nonlinear coefficient γ=20 km^{-1}W^{-1}, for which the ZDW is *λ*
_{0}=1500 nm with dispersion parameters *β*
^{(2)}=0 ps^{2}km^{-1} and *β*
^{(4)}=-2.85×10^{-4} ps^{4}km^{-1}. Here fiber parameters are same as those of Reference [9]. Supposing that the FOPA is pumped at 1490 nm and 1510 nm with a power of 0.25 W at each wavelength. The gain spectra of FOPA with different fiber length of 50, 100, 150 and 200 m are shown in Fig. 3(a). It is evident that the longer the fiber is, the higher the signal gain becomes, but the flat gain bandwidth goes by contraries.

As mentioned above, the gain spectrum is influenced by *β*
^{(2)} to a large extent, and therefore it is inevitable to find an appropriate value of *β*
^{(2)} to obtain a broad gain spectrum. Suppose that the fiber length is 200 m and the other parameters are same as those used in Fig. 3(a). Fig. 3(b) shows the gain spectra with different second-order dispersion of -0.1, 0 and 0.02 ps^{2}km^{-1}. The simulation indicates that different value of *β*
^{(2)} brings different shape of gain spectrum. A broad and uniform gain spectrum can be obtained when *β*
^{(2)}=0 ps^{2}km^{-1} along with a gain ripple of about 1 dB at the central wavelength of the two pumps. Moreover, when *β*
^{(2)} is positive value, there is a gain peak in the center of gain spectrum.

Furthermore, the nonlinear phase shift *γP*
_{0} has also a great effect on the gain spectrum. Suppose that the fiber length is 200 m, and the other parameters are same as those used in Fig. 3(a). It is evident that the increase of the nonlinear coefficient of fiber and the pump power can raise the gain of FOPA, but the flat gain bandwidth in the center of gain spectrum can’t be broaden to a large extent, as shown in Fig. 4(a) and Fig. 4(b).

To increase the gain bandwidth, the central frequency of the two pumps *ω*_{c}
must be chosen to be close to the zero-dispersion frequency of fiber *ω*
_{0}, in which case the effects of *β*
^{(4)} on the gain bandwidth can’t be ignored [21]. Fig. 5(a) shows the gain spectra for the same FOPA used in Fig. 3(a) with different *β*
^{(4)} of -2.85×10^{-4}, -2×10^{-6} and 2×10^{-6} ps^{4}km^{-1}. As shown in Fig. 5(a), the less the absolute value of *β*
^{(4)} is, the larger the gain bandwidth becomes. In common configurations, the two pump wavelengths are closely spaced, and the linear phase mismatch Δ*β* can compensate the nonlinear phase mismatch *γP*
_{0}. But when the wavelength separation between the two pumps Δ*λ* increases, Δ*β* becomes large and can’t be equalized by *γP*
_{0}, which makes it become impossible to maintain *k*=0 all the time. Hence, pump separation Δ*λ* is an important limiting factor and to a large extent influences the gain flatness. Fig. 5(b) shows the gain curves for the same FOPA used in Fig. 3(a) with different pump separation Δ*λ* of 20, 40 and 100 nm. When pump separation Δ*λ* is over 100 nm, the gain at the central wavelength of the two pumps approximately decreases to zero, and the gain ripple is over 10 dB, which accords with the results in Reference [12]. Therefore in order to obtain a broad and flat gain spectrum, the pump wavelengths must be chosen properly. In most recent experiments, pumps are indeed kept 40 to 50 nm apart [23,24]. Finally, the proper parameters of the first-section HNLF for two-pump FOPA are determined by preceding analyses, which are applied to the following two-pump FOPA with two-section HNLFs allocation.

For single-pump FOPA, four-section HNLFs are needed to obtain a broadband gain spectrum, as shown in Fig. 6(a), which has been discussed in Reference [10] in detail. However, for two-pump FOPA there is less gain ripple in the center of gain spectrum, and therefore it is possible to utilize less-section fibers to smooth gain spectrum. In the paper, a two-pump FOPA with two-section HNLFs is presented. Here, the two pump wavelengths are set to 1490 nm and 1510 nm separately, and both pump powers are 1 W with nonlinear coefficient of 60 km^{-1}W^{-1} and fourth-order dispersion of -2×10^{-6} ps^{4}km^{-1}. The optimum fiber parameters are obtained by using GA, L_{i} (m)=(25; 2.93) and ${\beta}_{\mathrm{i}}^{\left(2\right)}$(×10^{-2} ps^{2}km ^{-1})=(0; 4.29) (i=1,2). The zero-dispersion wavelength of the first section fiber is set to 1500 nm and the two pump wavelengths is so tuned that the central wavelength is kept closely to the zero-dispersion wavelength and phase matching is maintained over a broad spectral bandwidth. The numerical simulation shows the gain bandwidth may reach 346-nm with a gain of 20.3 dB, as shown in Fig. 6(b). In the FOPA, the long first-section HNLF provides a broad gain bandwidth, but gain flatness is very bad and has a gain ripple of over 3 dB in gain spectrum, as shown by solid curve G1 in Fig. 6(b); the second segment provides the higher gain in the center of the gain spectrum by using the positive second-order dispersion, which results in an average gain of more than 20.3 dB with less gain ripple, as shown by dotted curve G2 in Fig. 6(b).

For a two-pump FOPA using a three-section HNLFs configure, the gain spectrum can be driven up and broadened. The optimum fiber parameters are L_{i} (m)=(25; 2.0; 2.3) and ${\beta}_{\mathrm{i}}^{\left(2\right)}$ (×10^{-2} ps^{2}km^{-1})=(0; 3.98; -11.08), (i=1,2, 3) and the parametric gain of 21.04 dB is obtained with 0.2-dB uniformity over 438-nm bandwidth for a total amplifier length of about 30 m, but it will bring more splicing loss.

The above gain spectra are obtained in the case of phase-matching condition of FOPA being maintained, that is, the total phase mismatch *k* remains close to zero over a relatively broad spectral range. However, the core diameter of any fiber exhibits random variations related to manufacturing [25], which result in a randomly varying ZDW along the fiber. Random variations in the ZDW cause *β*
^{(2)} to vary randomly [21], which can modify the phase-matching condition between pump waves and signal waves and makes it become difficult to maintain *k*=0. As a result, the gain spectrum of FOPA becomes considerably no uniform, which has been discussed in Reference [9] for single-pump FOPA with four-section HNLFs and Reference [2] for two-pump FOPA with single-section HNLFs. For two-pump FOPA using a two-section HNLFs configure, the *β*
^{(2)} variations in the different random process are shown in Fig. 7(a), and the corresponding gain spectra in Fig. 7(b), which are similar to those of Reference [2]. From the curve G3 in Fig. 7(b), it is found that when the value of *β*
^{(2)} is maintained close to the constant optimum value, the variation of gain spectrum is very small.

In most research, the fourth-order dispersion is considered as a constant because its little variation is ignored [2,9,21], however, the impact of its fluctuation on gain performance of FOPA is also very important. Considering the fluctuation of *β*
^{(4)} as a random summation of sine functions:

Where *A*
_{1}=0.2×10^{-6} ps^{2}km^{-1} and *A*
_{2}=-0.02×10^{-6} ps^{2}km^{-1}, which account for the maximum amplitude of the fourth-order dispersion variation in each scale. *k*_{i,n}
=*k*
_{i,0}+*r*_{i,n}
Δ*k*_{i}
with *r*_{i,n}
randomly varying between -1 and +1, and *ψ*_{i,n}
is the phase randomly varying between 0 and 2π, *i*=1,2. The variations of the four-order dispersion are shown in Fig. 8(a), the corresponding gain spectra in Fig. 8(b). In numerical simulation, the parameters are same as those used in Fig. 6(b), and the results are in accordance with those shown in Fig. 5(a), smaller values of the fourth-order dispersion bring broader gain width, as shown by G1 in Fig. 8(b).

Moreover, for a given fiber set with a known dispersion profile, genetic algorithm can calculate the optimum pump wavelengths and powers of an ideal FOPA. For example, when the fiber parameters are same as those used in Fig. 6(b), the nearly same gain spectra can be obtained and the optimum pump wavelengths calculated by GA are 1489.9 nm and 1510.1 nm, which are close to the values in Fig. 6(b). The same method can be used to get the optimum pump powers of 0.71 W and 0.85 W, and the gain bandwidth of about 408 nm with a gain of 14.8 dB can be obtained.

## 5. Conclusion

We present a new two-pump fiber optical parametric amplifier, which is composed of two-section high nonlinear fibers. For many design parameters, such as pump powers and wavelengths, dispersion coefficient, nonlinear coefficient and length of HNLFs, govern the gain performance of FOPA, genetic algorithm is proposed to optimize these parameters, which is proved to be an effective method and a series of optimum results are obtained. With these optimum parameters, a two-pump FOPA using a two-section HNLFs configure can theoretically provide flat gain of 20.3 dB with 0.2-dB uniformity over 346-nm bandwidth, which can be potentially used in wideband WDM communication system to amplify WDM channels simultaneously.

## Acknowledgments

The research was supported in part by the National Natural Science Foundation of China (60377023 and 90304002).

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