## Accurate numerical simulation of short fiber optical parametric amplifiers

Optics Express, Vol. 16, Issue 6, pp. 3610-3622 (2008)

http://dx.doi.org/10.1364/OE.16.003610

Acrobat PDF (539 KB)

### Abstract

We improve the accuracy of numerical simulations for short fiber optical parametric amplifiers (OPAs). Instead of using the usual coarse-step method, we adopt a model for birefringence and dispersion which uses fine-step variations of the parameters. We also improve the split-step Fourier method by exactly treating the nonlinear ellipse rotation terms. We find that results obtained this way for two-pump OPAs can be significantly different from those obtained by using the usual coarse-step fiber model, and/or neglecting ellipse rotation terms.

© 2008 Optical Society of America

## 1. Introduction

1. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence,” J. Lightwave Technol. **15**, 1735–1746 (1997). [CrossRef]

*dz*, assumed to be uniform. This segment is much sorter than the fiber length, but larger than or of the order of the correlation length of the linear parameters. The birefringence and dispersion parameters in these segments are obtained by assuming that they are random variables with suitable probability density functions (pdf’s). Because

*dz*is larger than the correlation length of the linear parameters of the fiber, this approach is called the “coarse-step” method [1

1. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence,” J. Lightwave Technol. **15**, 1735–1746 (1997). [CrossRef]

2. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum. Electron . **QE-23**, 174–176 (1987). [CrossRef]

3. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. **15**, 751–765 (1997). [CrossRef]

4. M. E. Marhic, G. M. Williams, L. Goldberg, and J. M. P. Delavaux, “Tunable fiber optical parametric wavelength converter with 900 mW of CW output power at 1665 nm,” Proc. SPIE , **6103** 0W-1-0W-12 (2006). [CrossRef]

5. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **14**, 148–157 (1996). [CrossRef]

5. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **14**, 148–157 (1996). [CrossRef]

## 2. Modeling the linear properties of fibers with longitudinal variations.

6. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 1292–1294 (2004). [CrossRef]

7. F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 431–433 (2004). [CrossRef]

- Birefringence. One defines the birefringence correlation length
*L*_{b}as the maximum distance along*z*between two points after which it is no longer possible to predict, within any level of certainty, the birefringence strength and orientation at the second point when knowing those at the first one.*L*_{b}is determined by the longitudinal variations of the linear birefringence,*Δn*(*z*), and the orientation of the axes of linear birefringence, characterized by the angle*θ*(*z*) with respect to a fixed*x*axis. The random fluctuations of these parameters give rise to the well-known phenomenon of PMD, extensively studied because of its impact on optical-fiber communication systems. - Dispersion. Similarly, the dispersion is characterized by the variations of the ZDW,
*λ*_{0}(*z*). The autocorrelation of*λ*_{0}(*z*) has a width*L*_{d}, the dispersion correlation length, which is the scale length of the dispersion fluctuations.

*L*can be of the same order as, or even shorter than

*L*

_{d}and

*L*

_{b}. Then it is clear that the usual coarse-step model for describing birefringence and dispersion in numerical calculations is no longer applicable. The reason is that the standard coarse-step model considers uniform segments of fiber of length

*dz*≈

*L*

_{d},

*L*

_{b}≪

*L*, in which the dispersion and birefringence are chosen at random, according to some pdf’s (for instance a Gaussian pdf is used in [8

8. M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express. **12**, 136–142 (2004). [CrossRef] [PubMed]

1. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence,” J. Lightwave Technol. **15**, 1735–1746 (1997). [CrossRef]

*dz*≪

*L*,

_{d}*L*. For each segment of length

_{b}*dz*the values of

*Δn*(

*z*),

*θ*(

*z*) and

*λ*

_{0}(

*z*) are obtained by applying the second Wai-Menyuk model [5

5. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. **14**, 148–157 (1996). [CrossRef]

*L*

_{b}. This model is described in the Appendix.

*θ*(

*z*) and

*Δn*(

*z*) in two different situations: when they change abruptly at each

*L*, or when the Wai-Menyuk model is applied. In both cases we use

_{b}*L*= 50 m and an average birefringence

_{b}*Δn*= 10

^{-7}.

*Δn*(

*z*) and

*θ*(

*z*), as shown in Fig. 1, demonstrating that this more accurate modeling significantly changes the results given by the coarse-step method.

*λ*

_{0}(

*z*) Although not shown in this paper, we have performed preliminary simulations showing that the same happens if a fine-step Langevin process is applied for the modeling of ZDW fluctuations. We will not study ZDW fluctuations further in this paper; however the preceding reasoning already indicates that a fine-step model should also be applied to ZDW when modeling its fluctuations.

## 3. Keeping the ellipse rotation terms in the nonlinear propagation equations

*x*and

*y*components of the electric fields can be written as

*C*(

_{x}*z*,

*t*) and

*C*(

_{y}*z*,

*t*) are the slowly-varying envelopes (SVEs) of the electric field components, and

*C⃗*=

*C*+

_{x}x̂*C*, where

_{y}ŷ*x̂*and

*ŷ*are the unit vectors along the

*x*and

*y*axes.

*P*=

_{x}*C** and

_{x}C_{x}*P*=

_{y}*C**.

_{y}C_{y}*γ*is the fiber nonlinearity coefficient.

*β*

^{(n)}

_{x,y}represents the

*n*th derivative of

*β*at the center of the optical spectrum,

_{x,y}*ω*

_{c}; and Δ

*β*=

_{xy}*β*(

_{x}*ω*)-

_{c}*β*(

_{y}*ω*).

_{c}*N*of equal segments, with length

_{z}*d*=

_{z}*L*/

*N*. In each segment, the elementary contributions due to dispersion (including birefringence if present) and nonlinearity are calculated sequentially. For this reason, this is known as a split-step procedure. If birefringence and dispersion have random variations along the fiber, these can be introduced by varying the parameters in consecutive segments, as described in Section 2 We assume that

_{z}*C⃗*(

*z*,

*t*) is known at the end of the preceding section. Since linear birefringence and dispersion are best handled in the frequency domain, the Fourier transform of

*C⃗*(

*z*,

*t*),

*D⃗*(

*z*, Ω), is calculated by means of a fast Fourier transform (FFT). (ω is the angular frequency measured from the center frequency, i.e. Ω=

*ω*-

*ω*.) We then have

_{c}_{0}= Δ

*nω*

_{c}*dz*/2

*c*=

*π*Δ

*ndz*/

*λ*.

_{c}*β*and

*β*can be expanded as usual in terms of the desired orders of dispersion, generally up to the fourth order for fiber OPAs. (See the Appendix for more details.)

_{c}*C*)

_{y}^{2}

*C** and (

_{x}*C*)

_{x}^{2}

*C** can be neglected, because they lead to phase factors of the form exp[±2

_{y}*i*(

*β*-

_{x}*β*)

_{y}*z*]. These terms oscillate rapidly due to the difference in phase velocity between the two axes, and essentially average out to zero over several beat lengths. These terms are also associated with the phenomenon of ellipse rotation, and so neglecting them amounts to neglecting ellipse rotation. (These terms are also known as coherent-coupling terms). In such situations, we then keep only those terms on the right-hand sides that are synchronous. We then obtain the well-known equations

*P*and

_{x}*P*remain constant. Then Eq. (9) can easily be integrated, which yields

_{y}*dz*, and are easy to implement in numerical simulations. For these reasons they are widely used in code for simulating propagation in optical fibers.

*C*)

_{y}^{2}

*C** and (

_{x}*C*)

_{x}^{2}

*C** in Eq. (8) may not be justified, or may not be a very good approximation, as in the following cases. For example, if one wants the numerical method to work in the limit of an isotropic fiber, then these terms must be kept, because otherwise an incorrect limit is obtained when using circularly-polarized (CP) states of polarization (SOPs). Also, in a fiber with linear birefringence, if

_{y}*dz*is small, and/or the birefringence is weak, it may not be possible to argue that these terms will average out to zero. For OPAs in particular, if a high pump power and a fiber with very high

*γ*are used, only a short fiber is needed, and neglecting these terms is probably not justified. It is then necessary to keep them, and to use them in the numerical procedure.

*r̂*= (

*x̂*+

*iŷ*)/

**√2**and

*l̂*= (

*x̂*-

*iŷ*)/

**√2**. In that basis

*C⃗*has the components

*C*= (

_{r}*C*+

_{x}*iC*)/√2 and

_{y}*C*= (

_{l}*C*-

_{x}*iC*)/√2. Eq. (4) then leads to

_{y}*P*=

_{r}*C*

_{r}*C** and

_{r}*P*=

_{l}*C*

_{l}*C**. The terms

_{l}*P*and

_{r}*P*remain constant. Therefore Eq. (11) can be integrated, with the results

_{l}*θ*and

_{r}*θ*when they pass through the nonlinear section. When the phase difference between them, Δ

_{l}*θ*=

*θ*-

_{r}*θ*= 2(

_{l}*P*-

_{l}*P*)

_{r}*γdz*/3, is not equal to zero, this leads to the well-known effect of ellipse rotation, whereby the major axis of the ellipse of the input SOP is rotated by

*δθ*= Δ

*θ*/2.

*C*and

_{x}*C*are used in the step where linear birefringence and dispersion are dealt with. Then one switches to

_{y}*C*and

_{r}*C*. Finally one returns to

_{l}*C*and

_{x}*C*to work on the next section by means of

_{y}*C*= (

_{x}*C*+

_{r}*C*)/√2 and

_{l}*C*= (

_{y}*C*-

_{r}*C*)/(

_{l}*i*√2).

*C*and

_{x}*C*, without any approximation. Specifically, one can show that

_{y}*P*

_{0}=

*P*+

_{z}*P*=

_{y}*P*+

_{r}*P*is the total power. And

_{l}*dz*is obtained by rotating the input fields by

*δθ*, and by multiplying the result by a phase factor which is calculated as a self phase modulation (SPM) phase shift, which involves the total power regardless of how it is distributed among the SOPs.

*N*of points are used, and so either approach can be implemented without introducing significant computational overhead. For all the simulations we will present in this article we used 2

^{14}points, and the simulation time to obtain each gain spectrum was ~35 seconds, independently of the method used.

## 4. Numerical simulations

### 4.1. Effect of fine-step modeling of birefringence

10. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Amer. B **20**, 2425–2433 (2003). [CrossRef]

*θ*(

*z*) and Δ

*n*(

*z*) changing either smoothly or abruptly over each segment of length

*L*, corresponding to the dashed red and solid blue lines in Fig.1, respectively. We used 1 W of input power for each CW pump (at 1502.6 and 1600.6 nm),

_{b}*γ*= 10 W

^{-1}km

^{-1}and

*λ*

_{0x}(

*z*)=

*λ*

_{0y}(

*z*)=constant=1550 nm. We also used

*β*

^{(3)}=0.1 ps

^{3}/km,

*β*

^{(4)}=10

^{-4}ps

^{4}/km and

*β*

^{(n)}=0 for n>4. From

*Δn*, shown for example in Figure 1(b), it is also possible to calculate

*b*

^{(0)}and

*b*

^{(1)}from (A.7). From these relations, since the mean value of

*Δn*is 10

^{-7}, the mean value of

*b*

^{(0)}is ~0.2 m

^{-1}and the mean value of

*b*

^{(1)}is ~0.001 ps/m. From the relation

*D*= 2(2

_{p}*L*)

_{b}^{0.5}

*b*

^{(0)}(

*D*is the PMD coefficient), and since we used

_{p}*L*= 50 m, we obtain

_{b}*D*~0.6 ps/km

_{p}^{0.5}. This value is higher than in state of the art highly-nonlinear fiber (HNLF), with

*D*~0.2 ps/km

_{p}^{0.5}, because we purposely choose a high value of

*L*to show that our method is valid under any circumstance. The gain spectra are shown in Fig. 3 (they were obtained by applying the SSFM described in Section 3). Clearly, a more realistic modeling of

_{c}*θ*(

*z*) and

*Δn*(z) changes the final result.

*d*= 0.5 m for the generation of the birefringence parameters according to the WMM and for the integration of the coupled equations. When applying the coarse-step method, we also use

_{z}*dz*= 0.5 m for the integration, but the birefringence parameters vary as shown by the red curves in Fig. 1. This is because we use a low-order SSFM, and the numerical errors associated with it do not enable one to use

*dz*= 50 m. We could use

*dz*= 50 m if we had implemented a higher-order SSFM, or instead of the SSFM used a model considering only four waves in the time domain (as done in [6

6. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 1292–1294 (2004). [CrossRef]

7. F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 431–433 (2004). [CrossRef]

11. M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, “Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers,” J. Opt. Soc Amer. B **25**, 22–30 (2008). [CrossRef]

^{14}point in a spectral window of 25 THz (time increment of 0.04 ps) around the central frequency (in other words, increasing the number of sample points, or increasing the spectral window or shortening

*dz*do not change the results by an amount larger than 0.2 dB).

### 4.2. Effect of keeping the ellipse rotation terms

#### 4.2.1. Isotropic fiber

*b*

^{(0)}(

*z*)=

*b*

^{(1)}(

*z*)=

*b*

^{(2)}(

*z*)=0], where b is the birefringence parameter. As expected, we found that (away from the pumps) the SSFM with ER terms yielded exactly the same gain spectrum as the analytical solution using the four-wave model [10

10. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Amer. B **20**, 2425–2433 (2003). [CrossRef]

11. M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, “Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers,” J. Opt. Soc Amer. B **25**, 22–30 (2008). [CrossRef]

#### 4.2.2. Fiber with random birefringence

*θ*(

*z*) given by Fig. 1(a), and

*Δn*(

*z*) given by Fig. 1(b).

*L*and total pump powers

*P*

_{0}, namely:

*L*= 200 m and

*P*

_{0}= 2 W;

*L*= 20 m and

*P*

_{0}= 20 W;

*L*= 2 m and

*P*

_{0}= 200 W. Fig. 6 clearly shows again the importance of the ER terms for short fibers: in the case of the 2-m long fiber [Fig. 6(c)] the calculation without the ER terms underestimates the gain in decibels by about 7 dB, which is a very significant difference. Observe from Fig. 1 that in this case [Fig. 6(c)] the birefringence of the fiber is almost constant, since the changes of the birefringence parameters are small in the first 2 m of fiber. This result, together with the one shown in Fig. 5, shows the importance of ER terms in modeling the light propagation not just in isotropic fibers, but also in fibers with constant birefringence. As expected the difference becomes smaller as the length is increased, because of the averaging of ER due to random birefringence. As a result, when the length reaches 200 m [Fig. 6(a)] there is little difference between the results obtained with and without the ER terms.

## 5. Conclusion

**14**, 148–157 (1996). [CrossRef]

## 6. Appendix

*x*and

*y*denote the directions of the principal axes.

*β*and

_{x}*β*are the propagation constants for field components polarized along

_{y}*x*and

*y*, respectively; they are generally independent functions of the angular frequency

*ω*.

*β*is used to represent dispersion, just as in an isotropic fiber. It is generally approximated by a power-series expansion about

*ω*

_{c}. Letting ω=

*ω*-

*ω*,

_{c}*β*

^{(m)}is the

*m*th derivative of

*β*with respect to

*ω*, evaluated at

*ω*

_{c}.

*β*and

_{x}*β*as

_{y}*b*characterizes the birefringence, because when

*b*= 0 the propagation constants along the two axes are identical, and therefore there is no birefringence. More precisely, if we introduce the effective refractive indices for the two polarizations,

*n*=

_{x,y}*cβ*/

_{x,y}*ω*, we see that

*n*=

*n*-

_{x}*n*is by definition the fiber birefringence.

_{y}*b*in power series about the center frequency

*ω*

_{c}. This yields

*b*

^{(m)}is the

*m*th derivative of

*b*with respect to

*ω*, evaluated at

*ω*

_{c}.

*b*and its derivatives do not contribute to chromatic dispersion along the

*x*or

*y*axes, because from Eq. (A.2) the derivatives of

*β*and

_{x}*β*beyond the first order are not affected by

_{y}*b*at all. In particular this means that the fiber then has the same ZDW along both axes. To avoid ZDW fluctuations along

*z*one can also assume that the changes in

*n*(

_{x}*ω*) and

*n*(

_{y}*ω*) at each

*L*are given by the mere multiplication of

_{b}*n*(

_{x}*ω*) and

*n*(

_{y}*ω*) by some wavelength-independent factor, which changes the values of

*b*

^{(0)}and

*b*

^{(1)}, but does not change

*β*

^{(2)}, and thus does not produce ZDW fluctuations. This modeling enabled the study, in references [6

6. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 1292–1294 (2004). [CrossRef]

7. F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. **16**, 431–433 (2004). [CrossRef]

*Δn*is wavelength-dependent, or the spectral shapes

*n*(

_{x}*ω*) and

*n*(

_{y}*ω*) change along

*z*, both effects (PMD and ZDW fluctuations) are coupled, and proper simulations should consider the influence of PMD on ZDW fluctuations.]

## Acknowledgements

## References and links

1. | D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence,” J. Lightwave Technol. |

2. | C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum. Electron . |

3. | A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. |

4. | M. E. Marhic, G. M. Williams, L. Goldberg, and J. M. P. Delavaux, “Tunable fiber optical parametric wavelength converter with 900 mW of CW output power at 1665 nm,” Proc. SPIE , |

5. | P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. |

6. | F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, “Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. |

7. | F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. |

8. | M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express. |

9. | |

10. | M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifiers with linearly or circularly polarized waves,” J. Opt. Soc. Amer. B |

11. | M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, “Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers,” J. Opt. Soc Amer. B |

12. | M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: December 12, 2007

Revised Manuscript: January 29, 2008

Manuscript Accepted: February 16, 2008

Published: March 4, 2008

**Citation**

M. E. Marhic, A. A. Rieznik, G. Kalogerakis, C. Braimiotis, H. L. Fragnito, and L. G. Kazovsky, "Accurate numerical simulation
of short fiber optical parametric amplifiers," Opt. Express **16**, 3610-3622 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3610

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### References

- D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997). [CrossRef]
- C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. QE-23, 174-176 (1987). [CrossRef]
- A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997). [CrossRef]
- M. E. Marhic, G. M. Williams, L. Goldberg, and J. M. P. Delavaux, "Tunable fiber optical parametric wavelength converter with 900 mW of CW output power at 1665 nm," Proc. SPIE 6103, 165-176 (2006). [CrossRef]
- P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996). [CrossRef]
- F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (2004). [CrossRef]
- F. Yaman, Q. Lin, and G. P. Agrawal, "Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 431-433 (2004). [CrossRef]
- M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004). [CrossRef] [PubMed]
- www.photonics.incubadora.fapesp.br/portal/download.
- M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, "Fiber optical parametric amplifiers with linearly or circularly polarized waves," J. Opt. Soc. Am. B 20, 2425-2433 (2003). [CrossRef]
- M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008). [CrossRef]
- M. Karlsson and J. Brentel, "Autocorrelation function of the polarization-mode dispersion vector," Opt. Lett. 24, 939-941 (1999). [CrossRef]

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