- Email: [email protected]
Propagation of a cos-Gaussian beam in a Kerr medium
Optics & Laser Technology 43 (2011) 483–487
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Propagation of a cos-Gaussian beam in a Kerr medium Ruipin Chen a,b,n, Yongzhou Ni a, XiuXiang Chu a a b
School of Sciences, Zhejiang A & F University, Lin’an 311300, Zhejiang Province, China Nanjing National Laboratory of Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e in fo
abstract
Article history: Received 18 December 2009 Received in revised form 21 June 2010 Accepted 15 July 2010 Available online 24 July 2010
The dynamic properties of cos-Gaussian beams in the presence of Kerr nonlinearity are investigated ¨ analytically and numerically using the nonlinear Schrodinger equation (NLS). Based on the moments method, evolution of a cos-Gaussian beam width in the root-mean-square (RMS) sense is obtained analytically. The beam propagation factors and the critical powers of the cos-Gaussian beams with a uniform wavefront are calculated. Using numerical simulation, it is found that although the RMS beam width broadens, the central parts of the beam give rise to an initial radial compression and a significant redistribution during propagation. The partial collapse of central parts of the beam is observed below the threshold for a global collapse. The cos-Gaussian beams eventually convert into cosh-Gaussian type beams in Kerr media with low initial power. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear optics Kerr effect Cos-Gaussian beam
1. Introduction Hermite-sinusoidal-Gaussian (HSG) beams have attracted particular interest since they were introduced in the rectangular coordinate system by Casperson and coworkers [1–3], and their production and propagation in free space [4], complex optical systems [5–7], and turbulence [8–10] have been reported. As a special case of HSG beams, cos-Gaussian beams have many interesting applications such as in optimizing the efficiency of laser amplifiers [1–3] because of their particular profile as a Gaussian beam modulated with a cos function. In the present paper, we investigate analytically and numerically the evolution of the cos-Gaussian beam in the Kerr medium ¨ using the nonlinear Schrodinger (NLS) equation. Based on the moments method [10–16], which provides a convenient and rigorous way to obtain the evolution of the relevant parameters despite the shapes of the beams varying during propagation, important information about the Kerr effect on the cos-Gaussian beam can be obtained. The critical power and the beam propagation factor M2Q as a function of the beam parameters Ox and Oy, that are associated with the cos part of the cos-Gaussian beam are analytically given. Numerical simulations for the nonlinear dynamics of the beams are carried out. It is found that although the RMS beam width broadens, the central parts of the beam give rise to radial compression and the beam profile becomes strongly deformed during the cos-Gaussian beam
propagation in a Kerr medium. The central parts of the beam partially collapse [17–19], while the RMS beam width still increases or remains constant. When the initial power is low and moderate, the cos-Gaussian beams will transform into cosh-Gaussian type beams during propagation in Kerr media.
2. The moments method analysis The propagation of an optical wave in a Kerr medium is governed by the NLS: @2 u @2 u @u 2n2 k2 2 juj u ¼ 0, þ þ 2ik @z n0 @x2 @y2
ð1Þ
where k is the linear wave number, n0 is the linear refractive index, x and y are the transverse coordinates, z is the longitudinal coordinate, and n2 is the third order nonlinear coefficient. In this section we use the moments method to reduce the NLS to a set of ordinary differential equations (ODEs). This method proceeds by analyzing the evolution of several integral quantities. A definition of these quantities is ZZ juj2 dx dy, I0 ðzÞ ¼ ð2aÞ s
I1 ðzÞ ¼
ZZ
ðx2 þy2 Þjuj2 dx dy,
ð2bÞ
s n Corresponding author at: School of Sciences, Zhejiang A & F University, Lin’an 311300, Zhejiang Province, China. E-mail address: [email protected] (R. Chen).
0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.07.005
I2 ðzÞ ¼
i k
ZZ @u* @u @u* @u þy u dx dy, u* u* x u @x @y @x @y s
ð2cÞ
484
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
2 I3 ðzÞ ¼ 2 k
ZZ s
! 2 2 2 @u þ @u k n2 juj4 dxdy: @x @y n0
ð2dÞ
The quantities are associated with the beam power (I0), beam width (I1), momentum (I2), and Hamiltonian (I3), and yield a closed set of coupled ODEs. An Ermakov–Pinney (EP) equation describing the dynamics of the scaled beam width can be obtained with the invariant Q¼ (I3I1/2) (I22/4) [12] 1=2
d2 I1 ðzÞ Q ¼ 3=2 : dz2 I1 ðzÞ
ð3Þ
The general solution of Eq. (3) can be given as I1 ðzÞ ¼ I1 ðz ¼ 0Þ þ
Q z2 : I1 ðz ¼ 0Þ
ð4Þ
For cos-Gaussian beams, the initial field is of the form [1,2] 2 x þ y2 ð5Þ cosðOx x=aÞcosðOy y=aÞ, uðx,yÞ ¼ A0 exp 2 2a where A0 is the amplitude, a is the beam width associated with the Gaussian part, and Ox and Oy are the parameters associated with the cos part. The invariant is Q¼
I02 k2
1
O2x 1 þ expðO2x Þ
O2y 1 þ expðO2y Þ
!"
#
1þ
2 2 O2x expðO2x Þ Oy expðOy Þ þ S , 2 1 þ expðOx Þ 1 þ expðO2y Þ
ð6Þ with 2
S¼
2
2
2
n2 k2 I0 ½1 þ 4 expð3Ox =2Þ þ 3expð2Ox Þ½1 þ 4 expð3Oy =2Þ þ 3expð2Oy Þ : 8pn0 ½1 þexpðO2x Þ2 ½1 þ expðO2y Þ2
ð7Þ For better demonstration of the dependence of nonlinearity, Pcr, which is defined as the critical power for the cos-Gaussian beam with a uniform wavefront [15,16], can be obtained from Eqs. (4)–(7): 4ce0 pn20 ð1 þ expðO2x ÞÞð1þ expðO2y ÞÞ n2 k2
Pcr ¼
1 þð1 þ O2x ÞexpðO2x Þ þ ð1 þ O2y ÞexpðO2y Þ þ ð1þ O2x þ O2y ÞexpðO2x þ O2y Þ ð1 þ 4 expð3O2x =2Þ þ 3 expð2O2x ÞÞð1 þ 4expð3O2y =2Þ þ3 expð2O2y ÞÞ
,
ð8Þ where c is velocity of light in vacuum and e0 is the permittivity of free space. The value of the critical power of the cos-Gaussian beam is dependent only on the beam profile of the transverse distribution and the nonlinear parameters of the medium. When the initial power exceeds the critical power, the cos-Gaussian beam RMS width goes to zero in a finite propagation distance as predicted by the moments method and a global collapse will occur. When Ox ¼ Oy ¼0, then Eq. (8) reduces to the Gaussian beam critical power: G Pcr ¼
pce0 n20 n2 k2
:
Fig. 1. (Color online) The critical powers of the cos-Gaussian beams for different parameters Ox and Oy.
ð9Þ
Fig. 1 shows that the values of the critical powers of the cosGaussian beams that are normalized with respect to the Gaussian beam critical power increase with the increasing beam parameters Ox and Oy.
3. Beam propagation factors of a cos-Gaussian beam in a Kerr medium The beam quality factor of a beam profile can be defined as the ratio of its far field divergence to the divergence of a Gaussian beam of same effective beam size [15]. The effective beam width
of the cos-Gaussian beam, W(z), can be obtained through the relation: W 2 ðzÞ ¼ I1 ðzÞ=I0 . It can be seen from Eq. (4) that the beam width increases linearly in the far field with a divergence y ¼ WðzÞ=z, which is given by y2 ¼ Q =½I1 ðz ¼ 0ÞI0 . We can get the 2 2 beam quality factor ðMQ2 Þ2 ¼ y =yGS ¼ k2 Q =I02 since the divergence of a Gaussian beam with a beamwaist, w0, is yGB ¼ l=pw0 , where the Gaussian beam size refers to an effective value corresponding to the second order moment of the intensity distribution is w20 ¼ 4W 2 ðz ¼ 0Þ. The values of M2Q of cos-Gaussian beams as a function of Ox and Oy with different initial powers in free space and Kerr media are presented in Fig. 2. It is found that the beam quality factors increase with increasing Ox and Oy but decrease with increasing initial powers in a focusing medium as recognized from Fig. 2. It shows a focusing nonlinearity can reduce the divergence of the cos-Gaussian beam and improve the beam propagation factor.
4. Numerical results and analysis In order to investigate further the evolution of the cosGaussian beam in a Kerr medium, we solve the NLS numerically using the alternating direction implicit (ADI) and Crank–Nicolson method. In the following calculation and simulation, unless specially noted, we take the following parameters: wavelength l ¼0.53pmffiffiffim, the beam width associated with Gaussian part a ¼ 10= 2 mm, the diffraction length of the beam Z0 ¼ ka2 ¼ 0:6 mm. The peak intensities as a function of the propagation distance with different initial powers are shown in Fig. 3(a) for Ox ¼ Oy ¼0.5 and Fig. 3(b) for Ox ¼ Oy ¼1.5. The plots have been normalized with respect to their initial peak intensities at x ¼y¼0. Although the moments method predicts beam broadening in the RMS sense, the peak intensities initially increase, which would suggest that the central parts of the beams initially compress as shown in Fig. 3(a) for Ox ¼ Oy ¼0.5 and Fig. 3(b) for Ox ¼ Oy ¼1.5 with focusing nonlinearity. Clearly, the RMS widths broaden because the departure of the outer parts of the beam from the center of the beam more than compensate the compression of the centre. Similar scenarios have been illustrated for nonlinear Bessel–Gaussian beam by Johannisson et al. [17]. The simulations show that when the initial power is
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
485
Fig. 2. (Color online) Beam quality factor M2Q of cos-Gaussian beam as a function of Ox and Oy: (a) Pin ¼ 0.5Pcr with defocusing nonlinearity, (b) Pin ¼0.5Pcr in free space, (c) Pin ¼ 0.5Pcr with focusing nonlinearity, and (d) Pin ¼0.9 Pcr with focusing nonlinearity.
Fig.3. (Color online) Peak intensity as a function of the propagation distance with different initial powers: (a) Ox ¼ Oy ¼1.5 and (b) Ox ¼ Oy ¼0.5.
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R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
Fig.4. (Color online) The contour plots of intensity distribution of cos-Gaussian beam with the beam parameter Ox ¼ Oy ¼ 1.5 at different propagation distances: (a) Pin ¼0.2 Pcr with defocusing nonlinearity, (b) Pin ¼ 0.2 Pcr with focusing nonlinearity, and (c) Pin ¼0.5 Pcr with focusing nonlinearity.
further increased, the intensity at the central parts of the beam will dominate and eventually lead to a partial collapse [17–19] as shown in Fig. 3(a) for Pin ¼0.6Pcr and in Fig. 3(b) for Pin ¼0.95Pcr, although the RMS beam width still increases. In order to further illustrate how the beam is evolved, the contour maps of intensity distribution of the cos-Gaussian beams with different initial powers Pin ¼ 0:2Pcr and 0:5Pcr at different propagation distances Z ¼0.5Z0, 8Z0, and 12Z0 are shown in Fig. 4. Again, the intensities are normalized with respect to their initial peak intensities. As demonstrated above, the moments method has shown that a non-diffracting situation in the RMS sense is possible to obtain by balancing linear diffraction and nonlinear focusing. However, using numerical simulations with the cosGaussian beams for the dynamic interaction between linear diffraction and nonlinear focusing with varying degrees of initial power, it has been found that the central parts of the beams become significantly distorted and the significant redistribution of the beam occurs even though the moments method predicts the RMS beam width remains constant or increases. Furthermore, it is of interest to consider the cos-Gaussian beam transverse intensity profile conversion. Numerical simulations show that the cos-Gaussian beam can eventually evolve into a cosh-Gaussian type beam as it propagates in a Kerr medium with low and moderate power as recognized from Fig. 4, just as the cosGaussian beams propagate in a turbulent atmosphere [8], complex optical systems [1], and the free space [3]. As shown in
Fig. 4, the propagation distances for the significant distortion and redistribution of the central parts of the beam are different with different initial powers. The propagation distance for the redistribution of the central parts of the beam increases with the increasing initial power during propagation in the focusing medium.
5. Conclusion Propagation of cos-Gaussian beams in a Kerr medium has been studied using the NLS. The evolution of a cos-Gaussian RMS beam width and the beam propagation factor as well as the critical powers of the beam with a uniform wavefront are demonstrated. The dynamic interaction between nonlinear focusing and linear diffraction has been analyzed using numerical simulations. It has been found that the central parts of the beam become significantly distorted and the significant redistribution of the beam occurs during propagation even though the RMS beam width remains constant or even increases. The partial collapse of central part of the beam appears while the RMS beam width still increases or remains constant. The cos-Gaussian beam eventually converts into a cosh-Gaussian type beam in a Kerr medium with low and moderate power. These results can be applied to many applications such as nonlinear interactions, optical communications, and optical limiting.
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
Acknowledgments This work is supported in part by the State Key Program for Basic Research of China under Grant no. 2006CB921805 and the Development Foundation of Zhejiang A & F University of Science and Research. References [1] Casperson LW, Hall DG, Tovar AA. Sinusoidal-Gaussian beams in complex optical systems. J Opt Soc Am A 1997;14:3341–8. [2] Casperson LW, Tovar AA. Hermite-sinusoidal-Gaussian beams in complex optical systems. J Opt Soc Am A 1998;15:954–61. [3] Tovar AA, Casperson LW. Production and propagation of Hermite-sinusoidalGaussian laser beams. J Opt Soc Am A 1998;15:2425–32. [4] Zhao D, Mao H, Liu H. Propagation of off-axial Hermite-cosh-Gaussian laser beams. J Opt A 2004;6:77–83. [5] Yu S, Guo H, Fu X, Hu W. Propagation properties of elegant Hermite-coshGaussian laser beams. Opt Commun 2002;204:59–66. [6] Lu¨ B, Zhang B, Ma H. Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams. Opt Lett 1999;24:640–2. [7] Chu X. Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere. Opt Express 2007;15:17613–8. [8] Eyyubo˘glu HT, Baykal Y. Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere. Opt Express 2004;12:4659–74.
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[9] Eyyubo˘glu HT, Baykal Y. Cosine-Gaussian laser beam intensity in turbulent atmosphere. Proc SPIE 2004;5743:131–41. [10] Eyyubog˘ lu HT. Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere. J Opt Soc Am A 2005;22: 1527–35. [11] Montesinos GD, Pe´rez–Garc´ia VM, Torres P. Stabilization of solitons of the ¨ multidimensional nonlinear Schrodinger equation: matter–wave breathers. Physica D 2004;191:193–210. [12] Pe´rez-Garc´ia VM, Torres P, Montesinos GD. The method of moments for ¨ nonlinear Schrodinger equations: theory and applications. SIAM J Appl Math 2007;67:990–1015. [13] Garc´ia–Ripoll JJ, Pe´rez–Garc´ia VM, Torres P. Extended parametric resonances ¨ in nonlinear schrodinger systems. Phys Rev Lett 1999;83:1715. [14] Centurion M, Porter MA, Kevrekidis PG, Psaltis D. Nonlinearity management in optics: experiment, theory, and simulation. Phys Rev Lett 2006; 97:033903. [15] Pare C, Be´langer PA. Beam propagation in a linear or nonlinear lens-like medium using ABCD ray matrices: the method of moments. Opt Quantum Electron 1992;24:S1051–70. [16] Vlasov SN, Petrishchev VA, Talanov VI. Averaged description of wave beams in linear and nonlinear media (the method of moments). Radiophys Quantum Electron 1971;14:1062–70. [17] Johannisson P, Anderson D, Lisak M, Marklund M. Nonlinear Bessel beams. Opt Commun 2003;222:107–15. [18] Fibich G, Ilan B. Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation. J Opt Soc Am B 2000;17:1749–58. [19] Fibich G, Gaeta AL. Critical power for self-focusing in bulk media and in hollow waveguides. Opt Lett 2000;25:335–7.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Propagation of a cos-Gaussian beam in a Kerr medium Ruipin Chen a,b,n, Yongzhou Ni a, XiuXiang Chu a a b
School of Sciences, Zhejiang A & F University, Lin’an 311300, Zhejiang Province, China Nanjing National Laboratory of Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e in fo
abstract
Article history: Received 18 December 2009 Received in revised form 21 June 2010 Accepted 15 July 2010 Available online 24 July 2010
The dynamic properties of cos-Gaussian beams in the presence of Kerr nonlinearity are investigated ¨ analytically and numerically using the nonlinear Schrodinger equation (NLS). Based on the moments method, evolution of a cos-Gaussian beam width in the root-mean-square (RMS) sense is obtained analytically. The beam propagation factors and the critical powers of the cos-Gaussian beams with a uniform wavefront are calculated. Using numerical simulation, it is found that although the RMS beam width broadens, the central parts of the beam give rise to an initial radial compression and a significant redistribution during propagation. The partial collapse of central parts of the beam is observed below the threshold for a global collapse. The cos-Gaussian beams eventually convert into cosh-Gaussian type beams in Kerr media with low initial power. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear optics Kerr effect Cos-Gaussian beam
1. Introduction Hermite-sinusoidal-Gaussian (HSG) beams have attracted particular interest since they were introduced in the rectangular coordinate system by Casperson and coworkers [1–3], and their production and propagation in free space [4], complex optical systems [5–7], and turbulence [8–10] have been reported. As a special case of HSG beams, cos-Gaussian beams have many interesting applications such as in optimizing the efficiency of laser amplifiers [1–3] because of their particular profile as a Gaussian beam modulated with a cos function. In the present paper, we investigate analytically and numerically the evolution of the cos-Gaussian beam in the Kerr medium ¨ using the nonlinear Schrodinger (NLS) equation. Based on the moments method [10–16], which provides a convenient and rigorous way to obtain the evolution of the relevant parameters despite the shapes of the beams varying during propagation, important information about the Kerr effect on the cos-Gaussian beam can be obtained. The critical power and the beam propagation factor M2Q as a function of the beam parameters Ox and Oy, that are associated with the cos part of the cos-Gaussian beam are analytically given. Numerical simulations for the nonlinear dynamics of the beams are carried out. It is found that although the RMS beam width broadens, the central parts of the beam give rise to radial compression and the beam profile becomes strongly deformed during the cos-Gaussian beam
propagation in a Kerr medium. The central parts of the beam partially collapse [17–19], while the RMS beam width still increases or remains constant. When the initial power is low and moderate, the cos-Gaussian beams will transform into cosh-Gaussian type beams during propagation in Kerr media.
2. The moments method analysis The propagation of an optical wave in a Kerr medium is governed by the NLS: @2 u @2 u @u 2n2 k2 2 juj u ¼ 0, þ þ 2ik @z n0 @x2 @y2
ð1Þ
where k is the linear wave number, n0 is the linear refractive index, x and y are the transverse coordinates, z is the longitudinal coordinate, and n2 is the third order nonlinear coefficient. In this section we use the moments method to reduce the NLS to a set of ordinary differential equations (ODEs). This method proceeds by analyzing the evolution of several integral quantities. A definition of these quantities is ZZ juj2 dx dy, I0 ðzÞ ¼ ð2aÞ s
I1 ðzÞ ¼
ZZ
ðx2 þy2 Þjuj2 dx dy,
ð2bÞ
s n Corresponding author at: School of Sciences, Zhejiang A & F University, Lin’an 311300, Zhejiang Province, China. E-mail address: [email protected] (R. Chen).
0030-3992/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2010.07.005
I2 ðzÞ ¼
i k
ZZ @u* @u @u* @u þy u dx dy, u* u* x u @x @y @x @y s
ð2cÞ
484
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
2 I3 ðzÞ ¼ 2 k
ZZ s
! 2 2 2 @u þ @u k n2 juj4 dxdy: @x @y n0
ð2dÞ
The quantities are associated with the beam power (I0), beam width (I1), momentum (I2), and Hamiltonian (I3), and yield a closed set of coupled ODEs. An Ermakov–Pinney (EP) equation describing the dynamics of the scaled beam width can be obtained with the invariant Q¼ (I3I1/2) (I22/4) [12] 1=2
d2 I1 ðzÞ Q ¼ 3=2 : dz2 I1 ðzÞ
ð3Þ
The general solution of Eq. (3) can be given as I1 ðzÞ ¼ I1 ðz ¼ 0Þ þ
Q z2 : I1 ðz ¼ 0Þ
ð4Þ
For cos-Gaussian beams, the initial field is of the form [1,2] 2 x þ y2 ð5Þ cosðOx x=aÞcosðOy y=aÞ, uðx,yÞ ¼ A0 exp 2 2a where A0 is the amplitude, a is the beam width associated with the Gaussian part, and Ox and Oy are the parameters associated with the cos part. The invariant is Q¼
I02 k2
1
O2x 1 þ expðO2x Þ
O2y 1 þ expðO2y Þ
!"
#
1þ
2 2 O2x expðO2x Þ Oy expðOy Þ þ S , 2 1 þ expðOx Þ 1 þ expðO2y Þ
ð6Þ with 2
S¼
2
2
2
n2 k2 I0 ½1 þ 4 expð3Ox =2Þ þ 3expð2Ox Þ½1 þ 4 expð3Oy =2Þ þ 3expð2Oy Þ : 8pn0 ½1 þexpðO2x Þ2 ½1 þ expðO2y Þ2
ð7Þ For better demonstration of the dependence of nonlinearity, Pcr, which is defined as the critical power for the cos-Gaussian beam with a uniform wavefront [15,16], can be obtained from Eqs. (4)–(7): 4ce0 pn20 ð1 þ expðO2x ÞÞð1þ expðO2y ÞÞ n2 k2
Pcr ¼
1 þð1 þ O2x ÞexpðO2x Þ þ ð1 þ O2y ÞexpðO2y Þ þ ð1þ O2x þ O2y ÞexpðO2x þ O2y Þ ð1 þ 4 expð3O2x =2Þ þ 3 expð2O2x ÞÞð1 þ 4expð3O2y =2Þ þ3 expð2O2y ÞÞ
,
ð8Þ where c is velocity of light in vacuum and e0 is the permittivity of free space. The value of the critical power of the cos-Gaussian beam is dependent only on the beam profile of the transverse distribution and the nonlinear parameters of the medium. When the initial power exceeds the critical power, the cos-Gaussian beam RMS width goes to zero in a finite propagation distance as predicted by the moments method and a global collapse will occur. When Ox ¼ Oy ¼0, then Eq. (8) reduces to the Gaussian beam critical power: G Pcr ¼
pce0 n20 n2 k2
:
Fig. 1. (Color online) The critical powers of the cos-Gaussian beams for different parameters Ox and Oy.
ð9Þ
Fig. 1 shows that the values of the critical powers of the cosGaussian beams that are normalized with respect to the Gaussian beam critical power increase with the increasing beam parameters Ox and Oy.
3. Beam propagation factors of a cos-Gaussian beam in a Kerr medium The beam quality factor of a beam profile can be defined as the ratio of its far field divergence to the divergence of a Gaussian beam of same effective beam size [15]. The effective beam width
of the cos-Gaussian beam, W(z), can be obtained through the relation: W 2 ðzÞ ¼ I1 ðzÞ=I0 . It can be seen from Eq. (4) that the beam width increases linearly in the far field with a divergence y ¼ WðzÞ=z, which is given by y2 ¼ Q =½I1 ðz ¼ 0ÞI0 . We can get the 2 2 beam quality factor ðMQ2 Þ2 ¼ y =yGS ¼ k2 Q =I02 since the divergence of a Gaussian beam with a beamwaist, w0, is yGB ¼ l=pw0 , where the Gaussian beam size refers to an effective value corresponding to the second order moment of the intensity distribution is w20 ¼ 4W 2 ðz ¼ 0Þ. The values of M2Q of cos-Gaussian beams as a function of Ox and Oy with different initial powers in free space and Kerr media are presented in Fig. 2. It is found that the beam quality factors increase with increasing Ox and Oy but decrease with increasing initial powers in a focusing medium as recognized from Fig. 2. It shows a focusing nonlinearity can reduce the divergence of the cos-Gaussian beam and improve the beam propagation factor.
4. Numerical results and analysis In order to investigate further the evolution of the cosGaussian beam in a Kerr medium, we solve the NLS numerically using the alternating direction implicit (ADI) and Crank–Nicolson method. In the following calculation and simulation, unless specially noted, we take the following parameters: wavelength l ¼0.53pmffiffiffim, the beam width associated with Gaussian part a ¼ 10= 2 mm, the diffraction length of the beam Z0 ¼ ka2 ¼ 0:6 mm. The peak intensities as a function of the propagation distance with different initial powers are shown in Fig. 3(a) for Ox ¼ Oy ¼0.5 and Fig. 3(b) for Ox ¼ Oy ¼1.5. The plots have been normalized with respect to their initial peak intensities at x ¼y¼0. Although the moments method predicts beam broadening in the RMS sense, the peak intensities initially increase, which would suggest that the central parts of the beams initially compress as shown in Fig. 3(a) for Ox ¼ Oy ¼0.5 and Fig. 3(b) for Ox ¼ Oy ¼1.5 with focusing nonlinearity. Clearly, the RMS widths broaden because the departure of the outer parts of the beam from the center of the beam more than compensate the compression of the centre. Similar scenarios have been illustrated for nonlinear Bessel–Gaussian beam by Johannisson et al. [17]. The simulations show that when the initial power is
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
485
Fig. 2. (Color online) Beam quality factor M2Q of cos-Gaussian beam as a function of Ox and Oy: (a) Pin ¼ 0.5Pcr with defocusing nonlinearity, (b) Pin ¼0.5Pcr in free space, (c) Pin ¼ 0.5Pcr with focusing nonlinearity, and (d) Pin ¼0.9 Pcr with focusing nonlinearity.
Fig.3. (Color online) Peak intensity as a function of the propagation distance with different initial powers: (a) Ox ¼ Oy ¼1.5 and (b) Ox ¼ Oy ¼0.5.
486
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
Fig.4. (Color online) The contour plots of intensity distribution of cos-Gaussian beam with the beam parameter Ox ¼ Oy ¼ 1.5 at different propagation distances: (a) Pin ¼0.2 Pcr with defocusing nonlinearity, (b) Pin ¼ 0.2 Pcr with focusing nonlinearity, and (c) Pin ¼0.5 Pcr with focusing nonlinearity.
further increased, the intensity at the central parts of the beam will dominate and eventually lead to a partial collapse [17–19] as shown in Fig. 3(a) for Pin ¼0.6Pcr and in Fig. 3(b) for Pin ¼0.95Pcr, although the RMS beam width still increases. In order to further illustrate how the beam is evolved, the contour maps of intensity distribution of the cos-Gaussian beams with different initial powers Pin ¼ 0:2Pcr and 0:5Pcr at different propagation distances Z ¼0.5Z0, 8Z0, and 12Z0 are shown in Fig. 4. Again, the intensities are normalized with respect to their initial peak intensities. As demonstrated above, the moments method has shown that a non-diffracting situation in the RMS sense is possible to obtain by balancing linear diffraction and nonlinear focusing. However, using numerical simulations with the cosGaussian beams for the dynamic interaction between linear diffraction and nonlinear focusing with varying degrees of initial power, it has been found that the central parts of the beams become significantly distorted and the significant redistribution of the beam occurs even though the moments method predicts the RMS beam width remains constant or increases. Furthermore, it is of interest to consider the cos-Gaussian beam transverse intensity profile conversion. Numerical simulations show that the cos-Gaussian beam can eventually evolve into a cosh-Gaussian type beam as it propagates in a Kerr medium with low and moderate power as recognized from Fig. 4, just as the cosGaussian beams propagate in a turbulent atmosphere [8], complex optical systems [1], and the free space [3]. As shown in
Fig. 4, the propagation distances for the significant distortion and redistribution of the central parts of the beam are different with different initial powers. The propagation distance for the redistribution of the central parts of the beam increases with the increasing initial power during propagation in the focusing medium.
5. Conclusion Propagation of cos-Gaussian beams in a Kerr medium has been studied using the NLS. The evolution of a cos-Gaussian RMS beam width and the beam propagation factor as well as the critical powers of the beam with a uniform wavefront are demonstrated. The dynamic interaction between nonlinear focusing and linear diffraction has been analyzed using numerical simulations. It has been found that the central parts of the beam become significantly distorted and the significant redistribution of the beam occurs during propagation even though the RMS beam width remains constant or even increases. The partial collapse of central part of the beam appears while the RMS beam width still increases or remains constant. The cos-Gaussian beam eventually converts into a cosh-Gaussian type beam in a Kerr medium with low and moderate power. These results can be applied to many applications such as nonlinear interactions, optical communications, and optical limiting.
R. Chen et al. / Optics & Laser Technology 43 (2011) 483–487
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