Propagation of the Pearcey pulse with a linear chirp

Journal Pre-proofs Propagation of the Pearcey pulse with a linear chirp Yunqi Li, Yuanqiang Peng, Weiyi Hong PII: DOI: Reference:

S2211-3797(19)32870-0 https://doi.org/10.1016/j.rinp.2020.102932 RINP 102932

To appear in:

Results in Physics

Received Date: Revised Date: Accepted Date:

29 September 2019 30 December 2019 6 January 2020

Please cite this article as: Li, Y., Peng, Y., Hong, W., Propagation of the Pearcey pulse with a linear chirp, Results in Physics (2020), doi: https://doi.org/10.1016/j.rinp.2020.102932

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier B.V.

Propagation of the Pearcey pulse with a linear chirp YUNQI LI, YUANQIANG PENG, AND WEIYI HONG* Guangzhou Key Laboratory for Special Fiber Photonic Devices and Applications, South China Normal University, Guangzhou 510631, People’s Republic of China * [email protected]

Abstract: The propagation of the Pearcey pulse with a linear chirp is investigated in detail. We find that the chirp parameter has an impact on the focusing property of the Pearcey pulse. We further explain this phenomenon by the evolution of the phase distribution. In addition, we also discuss the propagation of Pearcey pulse with a linear chirp under the action of second-order dispersion and third-order dispersion, and an interesting phenomenon analogy to the refraction can be observed. 1. Introduction In recent years, novel beams have been intrigued extensive research due to its unique propagation properties and flexible generation methods in paraxial beam propagation or pulse propagation. Among them, Airy beams [1, 2, 3], Gaussian beams [4], and Bessel beams [5, 6] have been extensively studied and proposed various potential applications. Besides, the Pearcey beams also have been attracted much attention for its various generation method and unusual properties including auto-focusing, form-invariance, and self-healing [7]. In 2012, the Pearcey beams were theoretically introduced and experimentally verified by Ring et al. [7], and it was confirmed to be generated by using virtual source in 2014 [8]. A half Pearcey beam was produced by using the Fourier transform method in 2015 [9]. One-dimensional Pearcey beams were used to form accelerated hyper-geometric laser beams with auto-focus.in 2016 [10]. Recently, a number of extended researches about Pearcey beams have been proposed, such as the Pearcey solitons [11], the Pearcey Gaussian vortex beam [12, 13], and the properties of the circular Pearcey beam: focusing properties [14] and nonparaxial propagation [15]. Novel pulses, the pulsed version of novel wave packets, also arouse the interest of research. The dynamics of novel pulses exhibit extraordinary characteristics under different conditions [16-21], such as the third-order dispersion [16, 17], the Kerr nonlinearity [18], the high-order linear and nonlinear effects [19], etc. For instance, under the action of Kerr nonlinearity, the Airy pulse sheds solitons during propagation [18]. Under the action of the third order dispersions, the Airy pulse reaches the tight-focusing point, then undergoes an inversion (Rotationally symmetrical distribution with respect to the input pulse), finally continues to travel with an opposite acceleration [16]. Generally, the effect of the chirp imposed on the pulse during its propagation is widely investigated [16, 22-30], because the pulses emitted from laser sources are often chirped [16, 22]. It is found that the initial chirp plays an important role on the pulse, such as self-focusing of the pulse [22,23], pulse compression [24], and triggering filamentation [25]. Additionally, the chirp can be adopted to control the generation of supercontinuum [26, 27]. Recently, the dynamics of the Airy pulse with a linear chirp under the action of third-order dispersion also has been studied [31]. It is found that the Airy pulse is focused, and then forms a channel with several dispersion lengths during the propagation. It is worth noting that the larger the chirp parameter is, the shorter distance the Airy pulse focuses on. Therefore, we surmise that the Pearcey pulse also has self-focusing properties, unusual propagation characteristics under the influence of initial chirp. In this paper, we investigate the dynamics of the Pearcey pulse with a linear chirp and the effect of the chirp parameter on the Pearcey pulse. The potential physically mechanism for this phenomenon is also proposed by analyzing the evolution of the phase distribution of the

frequency components. Finally, we investigate the dynamics of the Pearcey pulse with a linear chirp under the action of second-order dispersion (SOD) and third-order dispersion (TOD). And we find a phenomenon similar to the refraction during propagation in the absence of any external media. The structure of this paper is as follows. In section 2 we investigate the dynamics of the Pearcey pulse with a linear chirp and the evolution of the phase distribution. In section 3 we discuss the propagation of the Pearcey pulse with a linear chirp under the action of SOD and TOD. And we conclude in section 4 with further discussion and outlook. 2. The Pearcey pulse with a linear chirp The Pearcey function is defined by an integral representation [32],

Pe  x, y   

+

-

ds exp i  s 4  s 2 x  sy   ,

(1)

where x and y are dimensionless variables transverse to propagation in the pulse propagation direction. Then we limit the Pearcey function by setting y = 0 in Eq. (1) to describe the temporal profile with the Pearcey distribution. The Pearcey pulse is defined as follows: 

Pe   t    ds exp i ( s 4  s 2t )  ,

(2)



where ± respectively represent the forward and backward Pearcey pulses. Therefore, the initial Pearcey pulse with a linear chirp (normalized by its total power) can be expressed in the form:

 2  U  z  0,    Pe    exp  iC  , 2 

where and

(3)

 is the temporal coordinate, z is propagation coordinate, C is the chirp parameter

  0.1937 is the normalized coefficient to satisfy







2

U  z  0,   d  1 .

We consider the propagation of an optical pulse in a fiber ignoring the high-order dispersion, which can be described by the nonlinear Schrodinger equation [33]:

i where

2

U  2  2U 2   P U U, 2 z 2 

is the second-order dispersion parameter of the fiber,

(4)



is a nonlinear coefficient

P is the total power of the pulse. By introducing the SOD length LD  T0 2 /  2 with the pulse width T0 (FWHM), the nonlinear length LNL  1 /   P  , t   / T0 and Z  z / LD , and

Eq. (4) can be normalized in the form:

i With

U sgn   2   2U 2   P0 U U , 2 2 Z t

(5)

P0  LD / LNL the normalized power of the pulse. In this paper, the split-step Fourier

method was used for solving the pulse propagation equations [33], and only the propagation of Pearcey pulse in the abnormal region is considered.

Fig. 1. Evolution of (a) the Pearcey pulse with C = 0, (b) the Pearcey pulse with C = -0.12, and (c) the Pearcey pulse with C = 0.12.

In the case of the normalized power of the pulse (assumed

P0  0 ), that is, in the linear

regime, Figure 1 displays the evolution of the Pearcey pulse with different chirp c. It is clearly seen that the Pearcey pulse with C = 0 reaches a focal point at Z = 2 and then undergoes an inversion. Interestingly, the dynamics of Pearcey pulse is different under the influence of negative or positive chirp as depicted in Figs. 1(b) and 1(c). For the negative chirp parameters, the Pearcey pulse focused at a longer distance than the chirp-free case C = 0, while for the positive chirp parameter, the Pearcey pulse focused at a shorter distance. Therefore, it is shown that the self-focusing process of the Pearcey pulse can be adjusted by changing the chirp parameter, which is similar to the case of the Gaussian pulse reported in Ref. [22]. The results in Figure 1 can also be understood optically. The linear chirp factor is like a “one-dimensional parabolic lens”. If C < 0, then the lens exp  iC t 2 / 2 is concave and





increases the distance to the autofocus of Pearcey pulse. If C > 0, then the lens is convex and reduces the distance to the focus.

Fig. 2. (a) Evolutions of the maximum amplitudes for the Pearcey pulses with different chirp. (b) The temporal distribution of |U| of the Pearcey pulse with C = -0.12 at different propagation distance. (c) The temporal distribution of |U| of the Pearcey pulse with C = 0.12 at different propagation distance.

In order to achieve a better visualization of the impact of the chirp on the pulse propagation, we show the maximum amplitude of the Pearcey pulses as a function of the propagation distance for the cases of different chirp parameters as depicted in Fig. 2(a). It is found that the larger the chirp parameter is, the shorter distance the Pearcey pulse focused on, as shown in Fig. 1. And the amplitude of the Pearcey pulse at the focal point increases with the

increasing chirp parameter. Further, we present the temporal distribution of |U| of the Pearcey pulses with a linear chirp at different propagation distance, as shown in Figs. 2(b) and 2(c) for the case of C= -0.12 and C = 0.12. It is clearly seen from the figure that, even though the signs of the chirp parameters are opposite, the evolution trends of |U| for these two cases are similar except for the focusing distances. For a short distance, the Pearcey pulse are basically unchanged, all the lobes of the Pearcey pulse merge together at the propagation distance Z = 4 in Fig. 2(b) or Z = 1.4 in Fig. 2(c). For a long distance, the Pearcey pulse undergoes an inversion and then mirrors its previous dynamics. And the evolution of the pulse shape is similar to that of Airy pulse in a fiber under the action of TOD [34].

Fig. 3. Spectral phase and spectra of (a) the Pearcey pulse with C = 0, (b) the Pearcey pulse with C = -0.12, and (c) the Pearcey pulse with C = 0.12 at different propagation distance.

For the focusing properties of the Pearcey pulse, similar to the case of the Airy pulse under the action of the third-order dispersion [31], it can also be understood by the phase change of the frequency component during the propagation of pulse. We show the distributions of the spectral intensity  2 and its corresponding phase  of the chirp-free Pearcey pulse at different propagation distances Z = 0, Z = 2 and Z = 4 as shown in Fig. 3(a). The distribution of



2

remains unchanged during the propagation since the propagation is in the linear regime. According to the Eq. (5), the group velocity dispersion  2 changes the phase of each spectrum of the pulse during the propagation process, and the phase change depends on the frequency component  and the propagation distance Z . Although this phase change does not affect the pulse spectrum, it can change the shape of the pulse. It is known that the initial phase distribution of the pulse with C = 0 is  2 , and the phase change  2 Z , proportional to the phase, is imposed on the spectrum during the propagation by the SOD. Consequently, almost all phases become the same when the pulse propagates to a certain distance (Z = 2) as shown in Fig. 3(a). In this case, all of the frequency components are synchronized, then the pulse shape becomes nearly the Gaussian [35]. Then, the Pearcey pulse undergoes an inversion and becomes defocusing during propagation, resulting in the phenomenon shown in the Fig.1. We further show the spectral phase and spectra of the Pearcey pulse with a linear negative/positive chirp at different propagation distance as depicted in Figs. 3(b) and 3(c). According to the discussion above, the pulse with linear negative/positive chirp would therefore experience longer/shorter propagation distance cancel the initial phase to evolve to a Gaussian shape. In order to understand this phenomenon clearly, we investigate the initial spectral phase and spectra of the Pearcey pulse with different chirp as shown in Fig. 4. Generally, the size of the chirp parameter affects the initial phase distribution that the larger the chirp parameter, the gentler the phase distribution. Therefore, the positive chirp parameters accelerate the Pearcey pulse focusing, and the negative chirp parameters delay the Pearcey pulse focusing.

Fig. 4. Initial spectral phase and spectra of the Pearcey pulses with different chirps.

In the nonlinear regime, the propagation of the Pearcey pulse with a linear chirp is affected by the nonlinear term in Eq. (5). Figure 5(a), (b) and (c) depicts the evolution of the Pearcey pulse in the nonlinear regime with the normalized power of the pulse P0 = 9. The focusing properties of the Pearcey pulse is affected under the action of the nonlinearity of Eq. (4), which is clearly seen that the soliton shedding from the Pearcey pulse exhibit periodical compression and stretching. The breathing soliton with a periodic evolution in amplitude is formed [36]. Compared with the breathing soliton shedding from the Sech pulse in the same situation [33], displayed in Fig.5. (e), (d) and (f), the soliton of Pearcey pulse has a drift on the time axis, which means that the soliton of Pearcey pulse has a shift in group velocity during the propagation process and the shift of group velocity is influenced by the chirp.

Fig. 5. Evolution of the Pearcey pulse and Sech pulse in the nonlinear regime with P0 = 9: the Pearcey pulse with (a) C = 0, (b) C = -0.12, (c) C = 0.12; the Sech pulse with (d) C = 0, (e) C = -0.12, (f) C = 0.12.

3. The Pearcey pulse under the action of SOD and TOD

To consider the influence of SOD and TOD on the Pearcey pulse, Eq. (5) is revised by the normalized nonlinear Schrodinger equation:

U sgn   2   2U i  2U 2 (6)    P0 U U , 2 2 2 6 t Z t where  3 is the third-order dispersion parameter of the fiber and  is a relative TOD strength i

parameter corresponding to  = 3 /   2 T0  .

Fig. 6. Evolution of the Pearcey pulses with C = 0 under the action of SOD and TOD: the positive Pearcey pulse with (a) ε = 1; (b) ε = -1; the negative Pearcey pulse with (c) ε = 1; (d) ε = -1.

Figure 6 depicts the evolution of the Pearcey pulses with C = 0 under the action of the SOD and TOD with different condition: the positive/negative pulse and the sign of SOD and TOD, respectively. It is found that under the influence of SOD, the dynamics of Pearcey pulse can be divided into two situations: The Pearcey pulses reaches a focal point at a short distance in Figs.6(a) and (d); the propagation of Pearcey pulses does not change overall at a long distances in Figs.6(b) and (c). Compared with Figure 1(a), the propagation distance of the Pierce pulse reaching the focus is shorter when ε> 0 under the action of the SOD and TOD. Interestingly, the autofocus of the Pierce pulse is greatly suppressed when the relative TOD strength parameter ε < 0, resulting in substantially no change in pulse propagation.

Fig. 7. Evolution of the Pearcey pulses with a linear chirp under the action of SOD and TOD: the positive Pearcey pulse with (a) C = -0.18, ε = 1; (b) C = -0.18, ε = -1; (c) C = 0.18, ε = 1; (d) C = 0.18, ε = -1; the negative Pearcey pulse with (e) C = -0.18, ε = -1; (f) C = -0.18, ε = 1; (g) C = 0.18, ε = -1; (h) C = 0.18, ε = 1.

Figure 7 displays the dynamics of the positive and negative Pearcey pulse under the action of the SOD and TOD with different condition: the positive/negative pulse, C = ± 0.18 and the sign of SOD and TOD, respectively. Due to the symmetry, there are four different dynamics for the parameters above. Figures. 7(a) and 7(e) depict that the Pearcey pulses maintain the original velocity for a certain distance and then undergo a divergence. The Pearcey pulses diverge gradually during its propagation under the action of SOD and TOD as shown in Figs. 7(b) and 7(f). And the Pearcey pulses reach the focal point at a fairly high speed and then mirror their previous dynamics as depicted in Figs. 7(c) and 7(g).

Fig. 8. (a) Evolution of the positive Pearcey pulse with C = 0.18 under the action of the SOD and TOD with comparable strengths ε = -1. (b) The temporal distribution of |U| of the positive Pearcey pulse with C = 0.18 under the action of the SOD and TOD with comparable strengths ε = -1.

An interesting phenomenon is observed in Figs. 7(d) and 7(h) that the pulse suddenly changes its velocity during propagation, which is analogous to the diffraction of a beam on the interface [37]. We take Fig. 7(d) as an example and show its details in Fig. 8(a) for the better visualization. It is shown that the main lobe of the pulse splits into two pulses marked by the arrows, and the weak one disperses quickly, which therefore results in the analogous “diffraction” phenomenon. We further show in Fig. 8(b) the temporal distribution of |U| at different propagation distances. The minor lobes merge together when the pulse propagates to Z = 0.9, and the main lobe starts to split into two lobes at Z = 1.2. The energy of these two lobes

redistribute during propagation under the actions of SOD and TOD: the former one quickly disperses while the energy of the later one increases.

Fig. 9. Evolution of the Pearcey pulses with a linear chirp under the action of SOD and TOD in the nonlinear regime with P0 = 25: the positive Pearcey pulses with (a) C = -0.18, ε = 1; (b) C = -0.18, ε = -1; (c) C = 0.18, ε = 1; (d) C = 0.18, ε = -1; the negative Pearcey pulses with (e) C = -0.18, ε = -1; (f) C = -0.18, ε = 1; (g) C = 0.18, ε = -1; (h) C = 0.18, ε = 1.

For the cases in the nonlinear regime, the dynamics of the Pearcey pulses with a linear chirp under the action of the SOD and TOD become complicated, as depicted in Fig. 9. Under the action of nonlinearity, the Pearcey pulse enters a soliton-shedding regime, similar to that reported in Ref. [18], and the energy of the solitons depend on the TOD strength. 4. Summary In conclusion, we investigate the dynamics of the Pearcey pulse with a linear chirp and find that the positive linear chirp accelerates the Pearcey pulse focusing, the negative linear chirp delays focusing, which can be understood by the evolution of the phase distribution of the frequency components. And the relationship between the focusing speed and the chirp parameter is investigated: the larger the chirp parameter is, the shorter propagation distance the Pearcey pulse focused on, which means that the focusing speed can be controlled by adjusting the chirp parameter of the Pearcey pulse. An interesting phenomenon is observed that the breathing soliton is forming under the action of the nonlinearity. In addition, the effects of SOD and TOD on the Pearcey pulse with different condition are discussed in detail: the positive/negative pulse, the positive/negative chirp, the sign of SOD and TOD. And we found that the Pearcey pulse can experience a phenomenon analogy to refraction during propagation under certain conditions: the positive Pearcey pulse with C = 0.18 and ε = -1, the negative Pearcey pulse with C = 0.18 and ε = 1. This may be a new candidate for fiber optic sensing since the dynamics of the Pierce pulse is sensitively dependent on the zero dispersion point, which is also sensitive to ambient temperature. Finally, the effect of nonlinearity is also discussed, the dynamics of the Pearcey pulse with a linear chirp become complicated and the soliton is shedding. Funding The research was supported by the National Natural Science Foundation of China (Grant No. 11874019). References 1.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

27. 28. 29. 30. 31. 32. 33. 34.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). A. E. Siegman, Lasers (University Science Books, 1986). V. V. Kotlyar and A. A. Kovalev, “Airy beam with a hyperbolic trajectory,” Opt. Commun. 313, 290-293 (2014). J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A. 4(4), 651–654 (1987). J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. M. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). D. M. Deng, C. D. Chen, X. Zhao, B. Chen, X. Peng, and Y. S. Zheng, “Virtual source of a Pearcey beam,” Opt. Lett. 39(9), 2703–2706 (2014). A. A. Kovalev, V. V. Kotlyar, S. G. Zaskanov, and A. P. Porfirev , “Half Pearcey laser beams,” J. Opt. 17, 035604 (2015). A.A.Kovalev, V. V. Kotlyar and A. P. Porfirev, “Auto-focusing accelerating hyper-geometric laser beams,” J. Opt. 18, 025610 (2016). A. Zannotti, M. Rüschenbaum, and C. Denz, “Pearcey solitons in curved nonlinear photonic caustic lattices,” J. Opt. 19, 094001 (2017). Y. L. Peng, C. D. Chen, B. Chen, X. Peng, M. L. Zhou, L. P. Zhang, D. D. Li, and D. M. Deng, “propagating of the Pearcey-Gaussian-vortex beam in free space and Kerr media,” Laser physics 26(12), 125401 (2016). X. Y. Chen, D. M. Deng, G. H. Wang, X. B. Yang, and H. Z. Liu, “Abruptly autofocused and rotated circular chirp Pearcey Gaussian vortex beams,” Opt. Lett. 44(4), 955-958 (2019). X. Y. Chen, D. M. Deng, J. L. Zhuang, X Peng, D. D. Li, L. P. Zhang, F. Zhao, X. B. Yang, H. Z. Liu, and G. H. Wang, “Focusing properties of circle Pearcey beams,” Opt. Lett. 43(15), 3626-3629 (2018). X. Y. Chen, D. M. Deng, J. L. Zhuang, X. B. Yang, H. Z. Liu, and G. H. Wang, “Nonparaxial propagation of abruptly autofocusing circular Pearcey Gaussian beams,” Appl. Opt. 57(28), 8418-8423 (2018). L. F. Zhang, K. Liu, H. Z. Zhong, J. G. Zhang, Y. Li, and D. Y. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23(3), 2566-2576 (2015). I. M. Besieris and A. M. Shaarawi, “Accelerating airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78(4), 046605 (2008) Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18) 17298-17307 (2011) L. F. Zhang, J.G. Zhang, Y. Chen, A. L. Liu, and G. C. Liu, “Dynamic propagation of finite-energy Airy pulses in the presence of higher-order effects,” J. Opt. Soc. Am. B 31(4), 889-897 (2014). W. Y. Hong, Q. Guo, and L. Li, “Dynamics of optical pulses in highly noninstantaneous Kerr media,” Phys. Rev. A 92(2), 023803 (2015). F. Deng, W. Y. Hong, and D. M. Deng, “Airy-type solitary wave in highly noninstantaneous Kerr media,” Opt. Express 24(14), 15997-16002 (2016). X. D. Cao, G. P. Agrawal, and C. J. McKinstrie, “Self-focusing of chirped optical pulses in nonlinear dispersive media,” Phys. Rev. A 49(5), 4085–4092 (1994). L. Bergé, J. J. Rasmussen, E. A. Kuznetsov, E. G. Shapiro, and S. K. Turitsyn, “Self-focusing of chirped optical pulses in media with normal dispersion,” J. Opt. Soc. Am. B 13(9), 1879–1891 (1996). J. Park, J. H. Lee and C. H. Nam, “Laser chirp effect on femtosecond laser filamentation generated for pulse compression,” Opt. Express 16(7), 4465–4470 (2008). R. Nuter, S. Skupin, and L. Bergé, “Chirp-induced dynamics of femtosecond filaments in air,” Opt. Lett. 30(8), 917–919 (2005). J. Kasparian, R. Sauerbrey, D. Mondelain, S. Niedermeier, J. Yu, J.-P. Wolf, Y.-B. André, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, M. Rodriguez, H. Wille, and L. Wöste, “Infrared extension of the super continuum generated by femtosecond terawatt laser pulses propagating in the atmosphere,” Opt. Lett. 25(18), 1397–1399 (2000). V. Kartazaev and R. R. Alfano, “Supercontinuum generated in calcite with chirped femtosecond pulses,” Opt. Lett. 32(22), 3293–3295 (2007). O. Varela, B. Alonso, I. J. Sola, J. San Román, A. Zaïr, C. Méndez, and L. Roso, “Self-compression controlled by the chirp of the input pulse,” Opt. Letters 35(21), 3649-3651 (2010). L. F. Zhang, X. Q. Fu, J. Q. Deng, J. Zhang, S. C. Wen, and D. Y. Fan, “Role of chirp in spatiotemporal instability ofbroadband pulsed laser,” J. Opt. 13(1), 015203 (2011). C. Milián, A. Jarnac, Y. Brelet, V. Jukna, A. Houard, A. Mysyrowicz, and A. Couairon, “Effect of input pulse chirp on nonlinear energy deposition and plasma excitation in water,” J. Opt. Soc. Am. B 31(11), 2829–2837 (2014). F. Deng and W. Y. Hong, “Chirp-Induced Channel of an Airy Pulse in an Optical Fiber Close to Its ZeroDispersion Point,” IEEE Photonics Journal 8(6), 7102807 (2016). T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,”, Phil. Mag. 37, 311–7(1946). G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001). R. Driben, Y. Hu, Z. G. Chen, B. A. Malomed, and R. Morandotti, “Inversion and tight focusing of Airy pulses under the action of third-order dispersion,” Opt. Lett. 38(14), 2499–2501(2013).

35. 36. 37.

M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion,” Appl. Opt. 18(5), 678–682(1979). Z. X. Deng, Y. Chen, J. Liu, C. J. Zhao and D. Y. Fan, “Graded-index breathing solitons from Airy pulses in multimode fibers,” Opt. Express 27(2), 483-493 (2019). Max Born and Emil Wolf, Principles of Optics 7th eidition (Cambridge, 1999).

Highlights     

The dynamics of the Pearcey pulse with a linear chirp have been investigated. We find that the chirp parameter has an impact on the focusing property of the Pearcey pulse, which can be understood by the evolution of the phase distribution of the frequency components. Under the action of the nonlinearity, the focusing property of the Pearcey pulse has been affected and the breathing soliton is forming. We discuss the propagation of Pearcey pulse with a linear chirp under the action of second-order dispersion and third-order dispersion. And we found that the Pearcey pulse can experience a phenomenon analogy to refraction. For the case in the nonlinear regime, the dynamics of the Pearcey pulse with a linear chirp under the action of the second-order dispersion and third-order dispersion become complicated and the soliton is shedding.