Bright spatial solitons in defocusing Kerr media supported by ... - ANU
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OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997
Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities Ole Bang and Yuri S. Kivshar Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Optical Sciences Centre, Australian National University, Canberra, ACT 0200, Australia
Alexander V. Buryak School of Mathematics and Statistics, University College, Australian Defence Force Academy, Canberra, ACT 2600, Australia, and Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received July 10, 1997 We show that resonant wave mixing that is due to quadratic nonlinearity can support stable bright spatial solitons, even in the most counterintuitive case of a bulk medium with defocusing Kerr nonlinearity. We analyze the structure and stability of such self-guided beams and demonstrate that they can be generated from a Gaussian input beam, provided that its power is above a certain threshold. 1997 Optical Society of America
It is well known that bright spatial solitons can exist only for focusing nonlinearity. In bulk Kerr [or x s3d ] media their existence further requires a saturating effect to arrest catastrophic self-focusing. Defocusing nonlinearity leads usually to an enhanced beam broadening and does not support any localized structures other than vortex and dark solitons, which require a background beam.1 However, theoretical and experimental results indicate that multidimensional bright solitons can exist in noncentrosymmetric materials with quadratic [or x s2d ] nonlinearity.2 In this Letter we demonstrate an important feature of the parametric self-trapping induced by resonant wave mixing in x s2d media. We show that even a weak quadratic nonlinearity can lead to self-focusing and stable bright solitons in a bulk medium with defocusing Kerr nonlinearity, provided that the fundamental and its second harmonic are nearly phase matched. This means that the parametric interaction in a x s2d medium can be strong enough not only to suppress the beam broadening that is due to diffraction but also to overcome the broadening effect of the defocusing Kerr nonlinearity. We consider beam propagation in noncentrosymmetric lossless bulk media with defocusing cubic nonlinearity, described by the dimensionless equations ≠w 1 =' 2 w 1 w p v 2 sjwj2 1 rjvj2 dw 0 , i ≠z (1) ≠v 1 2 2 2 2 2i 1 =' v 2 bv 1 w 2 shjvj 1 rjwj dv 0 , ≠z 2 (2) which are valid when spatial walk-off is negligible and the fundamental frequency v1 and its second harmonic v2 2v1 are far from resonance. The slowly varying complex envelope function of the fundamental w wsr, zd and of the second harmonic v vsr, zd are assumed to propagate with a constant polarization, e1 and e2 , along the z axis. The Laplacian =' 2 refers to the transverse coordinates r sx, yd. The physical electric field is EsR, Z, T d E0 fw expsiu1 dˆe1 1 0146-9592/97/221680-03$10.00/0
2v expsi2u1 dˆe2 g 1 c.c., where R r0 r, Z z0 z, and u1 k1 Z 2 v1 T . The real normalization parames2d s3d ters are3 E0 4xˆ 1 yf3jx˜ 1s jg, z0 2k1 r0 2 , and r0 2 s3d s2d 3jx˜ 1s jyh16m0 v1 2 f x˜ 1 g2 j, where m0 is the vacuum permeability and kp is the wave number at frequency vp . s3d s3d Furthermore, b 2z0 Dk, h 16x˜ 2s yx˜ 1s , and r s3d s3d 8x˜ 1c yx˜ 1s , where Dk 2k1 2 k2 ,, k1 is the wavesj d vector mismatch. The coeff icients x˜ p x˜ s j d svp d denote the Fourier components at vp of the jth-order s2d s2d susceptibility tensor. Thus x˜ 1 x˜ 2 represents the s3d s3d s3d quadratic nonlinearity, and x˜ ps and x˜ 1c x˜ 2c represent the parts of the cubic nonlinearity responsible for self- and cross-phase modulation, respectively. The system of Eqs. R (1) and (2) conserves the dimensionless power, P sjwj2 1 4jvj2 ddr, that pcorresponds to the physical power P0 P , where P0 0.5 e0 ym0 E0 2 r0 2 . Equations (1) and (2), but for focusing Kerr nonlinearity, were used in Ref. 4 for studies of collapse in arbitrary dimensions. After a simple transformation these equations correspond to the s1 1 1d-dimensional equations used in Ref. 5 and later derived rigorously in Ref. 3. Similar equations were recently shown to appear in the theory of self-focusing in quasi-phasematched x s2d media.6 In this Letter we are interested in s2 1 1ddimensional bright solitary waves, and therefore we look for spatially localized solutions to Eqs. (1) and (2) of the form vsr, zd v0 srdexps2ilzd , wsr, zd w0 srdexpsilzd, (3) where the real and radially symmetric functions p w0 srd and v0 srd decay monotonically to zero as r x2 1 y 2 increases. The real propagation constant l must be above cutoff, l . lc maxs0, 2by4d, for w0 and v0 to be exponentially localized. For a large class of materials and experimental settings we can neglect s3d s3d the dispersion of x s3d and set x˜ 1s x˜ 2s , and it is s3d s3d further reasonable to set x˜ 1s x˜ 1c . In this case we 1997 Optical Society of America
November 15, 1997 / Vol. 22, No. 22 / OPTICS LETTERS
get h 2r 16, which we use below. These values were also used in earlier papers on competing x s2d and x s3d nonlinearities.3,5 The existence of localized solutions to Eqs. (1) and (2) is a nontrivial issue. When w0 0, Eqs. (1) and (2) reduce to the stationary nonlinear Schr¨odinger equation for the second harmonic, =' 2 v0 2 s b 1 4ldv0 2 hv0 3 0, which does not permit spatially localized solutions for defocusing Kerr nonlinearity, h . 0. Similarly, when b .. 1 and b .. l, we obtain v0 ø w0 2 y2b from Eq. (2). Then Eq. (1) gives the stationary nonlinear Schr¨odinger equation for the fundamental, =' 2 w0 2 lw0 2 w0 3 0, and localized solutions are not possible either. Therefore, if bright spatial solitons should exist in defocusing Kerr media, both components should be nonzero, w0 fi 0 and v0 fi 0 (combined or C solutions5). Using a standard relaxation scheme, we numerically found the families of localized C solutions [Eqs. (3)] to Eqs. (1) and (2) for the allowed values of l and b. In Fig. 1 we show examples of the prof iles w0 srd and v0 srd for b 0.1 and different values of l. When l is small (i.e., low power), the profiles resemble the sechshaped solitons that exist in self-focusing Kerr media [Fig. 1(a)]. Increasing l also increases the beam amplitude [Fig. 1(b)]. However, for suff iciently large values of l (or of the power), the amplitudes saturate, and the beam broadens significantly [Fig. 1(c)] because of the defocusing effect of the cubic terms in Eqs. (1) and (2). Above l ø 0.00656, no localized solutions exist. Similarly, near the cutoff lc 0, the beam width increases rapidly and the amplitude decreases. Below the cutoff, no localized solution exist. In Fig. 2 we show the amplitude of the C solutions, w0 s0d and v0 s0d, and their power P as function of l for two particular values of b. Clearly the solutions exist in a limited region only, ranging from the cutoff lc to a certain upper limit at which the beam power tends to inf inity, even though the amplitude saturates, ref lecting the defocusing effect of the Kerr nonlinearity. For b 0.1 the power increases monotonically with l, whereas for b 20.02 it decreases in a narrow interval above the cutoff. In both cases there is a power threshold below which no solutions exist. The Vakhitov –Kolokolov stability criterion, ≠N y ≠l . 0, has been shown to apply to solitary waves supported by pure x s2d nonlinearity.7 Its derivation for Eqs. (1) and (2) is similar. According to this criterion the solutions are stable in the whole region of existence for b 0.1, whereas for b 20.02 they become unstable in a narrow region above cutoff. We made a series of calculations as shown in Fig. 2(b), found ≠Ny≠l, and identified the regions of existence and stability of the C solutions in the sl, bd plane. The results are summarized in Fig. 3(a). In regions I and II no localized solutions exist, in region I because l is below cutoff and in region II because the defocusing effect of the cubic nonlinearity becomes dominant. For b , 0 there is a narrow band [magnified f ive times in Fig. 3(a) to show it] where solutions exist but are unstable. This result is to be expected because the instability also exists for pure x s2d nonlin-
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earity.7 Stable solitons exist in the hatched region. We have confirmed their stability to propagation by simulation of Eqs. (1) and (2) for representative cases. In Fig. 3(b) we show the power of the stable solitons. As is also evident from Fig. 2(b), the stable solitons exist for all powers above a certain threshold. Two physical effects are apparent from Fig. 3 when we keep in mind that the effective mismatch b is proportional to both the wave-vector mismatch Dk and the s3d s2d ratio of the x s3d and x s2d coeff icients, b ~ Dk x˜ 1s yf x˜ 1 g2 . When jbj increases, the existence region becomes narrower, ref lecting that the defocusing cubic nonlinearity becomes progressively more dominant, either s3d s2d because jx˜ 1s j .. f x˜ 1 g2 or because the waves are not phase matched (see also Ref. 5). Similarly, for f ixed b, solitons exist only when the maximum intensity, or equivalently, l is sufficiently small. When the intensity, or l, becomes too high, the defocusing Kerr nonlinearity is again dominant, prohibiting the existence of stable bright solitons. This phenomenon is due to the
Fig. 1. Profiles w0 sr xd (solid curves) and v0 sr xd (dotted curves) of solutions (3) for h 2r 16, b 0.1, and (a) l 0.003, (b) l 0.006, (c) l 0.00655.
Fig. 2. Characteristics of the C solutions [Eqs. (3)] for h 2r 16 and b 20.02 (dotted curves) and b 0.1 (solid curves). (a) Amplitude versus l for the fundamental w0 s0d (upper curve) and the second harmonic v0 s0d (lower curve). ( b) Dimensionless power versus l.
Fig. 3. (a) Region of existence of stable (hatched ) and unstable (black) solitons in the sl, bd plane. (b) Power regime ( hatched ) of stable solitons versus b. Parameters: h 2r 16.
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OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997 s2d
Fig. 4. Evolution of a Gaussian input beam at the fundamental, wsr, 0d 0.1 expf2sry30d2 g, with vsr, 0d 0 and (a) b 0.29, ( b) b 0.1. Parameters: h 2r 16.
s3d
Fig. 5. Physical power threshold versus x˜ 1s for h 2r s2d 16, l1 1.064 mm, and values of x˜ 1 of 3 pmyV (solid curves) and 5.6 pmyV (dashed curves). In each case the upper curve is for k 21023 and the lower curve is for k 1023 .
competition between two kinds of nonlinearity and is qualitatively similar to, e.g., the effect of self-trapping in media with focusing cubic and defocusing quintic nonlinearity.8 We numerically analyzed the generation of bright solitons from a Gaussian input beam with power P 14.1 at the fundamental, without seeding of the second harmonic, corresponding to the filled circles in Fig. 3(b). The results, presented in Fig. 4, show two characteristic types of evolution. When the effective mismatch is sufficiently large, b 0.29, the input power is below threshold, and the evolution is strongly affected by the defocusing Kerr effect. Therefore the beam rapidly diffracts [Fig. 4(a)]. When the mismatch is so small s b 0.1d that the input power is sufficiently above threshold (part of the power is always lost to radiation), the parametric focusing is stronger, and we observe the formation of a localized self-trapped beam [Fig. 4(b)]. To get a feeling for the powers, nonlinear coefficients, and degree of phase matching required for generation of these bright solitons, we consider a fundamental wavelength of l1 1.064 mm, a phase mismatch of s2d s2d k 61023 , and x˜ 1 3 or x˜ 1 5.6 pmyV. Here 2 k ; n1 Dkyk1 , where n1 is the refractive index at the fundamental frequency. From the dimensionless threshold power shown in Fig. 3(b) and the parameter def initions given below Eqs. (1) and (2) we can then calculate the physical threshold power as function of s3d x˜ 1s . The result is shown in Fig. 5. For example, for
KTP with x˜ 1 5.6 pmyV and a phase mismatch of k 1023 the threshold is of the order of 10 kW for s3d a rather wide range of x˜ 1s values and is thus of the same order as the power in the solitons observed in s3d s2d KTP.9 When jx˜ 1s j increases or jx˜ 1 j decreases, the required power increases. For a negative wave-vector mismatch, Dk , 0, the threshold power is always s3d higher and the x˜ 1s region in which stable solitons exist is always narrower than for the corresponding positive mismatch. Consider k 1023 and the Gaussian initial condition used in Fig. 4, which generated a soliton for b 0.1. For a fixed effective mismatch of b 0.1, the ratio of the nonlinearity coefficients is f ixed at x3 yx2 2 24.2. Hence this input beam has a f ixed FWHM width of 42 mm, a power of x2 22 3 42.5 kW, and a peak intensity of x2 22 3 4.2 GWycm2 , where the normalized nonlinearity coefficients are def ined as s2d s3d x2 x˜ 1 ys5.6 pmyVd and x3 x˜ 1s ys103 pm2 yV2 d. In conclusion, we have analyzed the effect of quadratic nonlinearity on the existence and stability of bright spatial solitons in bulk media with defocusing Kerr nonlinearity. We have shown that stable bright spatial solitons can exist because of parametrically induced self-focusing and that they can be generated from Gaussian input beams with experimentally reasonable powers. From a physical point of view this phenomenon indicates an important feature of cascaded nonlinearities, permitting self-focusing effects even in a defocusing Kerr medium. The authors acknowledge useful discussions with S. Trillo and W. Torruellas. Part of this research has been supported by the Australian Department of Industry, Science and Tourism, through grant 74 under the International Science and Technology program. A. V. Burak acknowledges support of the Australian Research Council. References 1. Yu. S. Kivshar and B. Luther-Davies, ‘‘Optical dark solitons: physics and applications,’’ Phys. Rep. (to be published ). 2. For a general overview see G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum. Electron. 28, 1691 (1996). 3. O. Bang, J. Opt. Soc. Am. B 14, 51 (1997). 4. L. Berg´e, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, Phys. Rev. E 55, 3555 (1997). 5. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995). 6. C. B. Clausen, O. Bang, and Yu. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997). 7. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995). 8. E. M. Wright, B. L. Lawrence, W. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 2481 (1995). 9. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997
Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities Ole Bang and Yuri S. Kivshar Australian Photonics Cooperative Research Centre, Research School of Physical Sciences and Engineering, Optical Sciences Centre, Australian National University, Canberra, ACT 0200, Australia
Alexander V. Buryak School of Mathematics and Statistics, University College, Australian Defence Force Academy, Canberra, ACT 2600, Australia, and Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received July 10, 1997 We show that resonant wave mixing that is due to quadratic nonlinearity can support stable bright spatial solitons, even in the most counterintuitive case of a bulk medium with defocusing Kerr nonlinearity. We analyze the structure and stability of such self-guided beams and demonstrate that they can be generated from a Gaussian input beam, provided that its power is above a certain threshold. 1997 Optical Society of America
It is well known that bright spatial solitons can exist only for focusing nonlinearity. In bulk Kerr [or x s3d ] media their existence further requires a saturating effect to arrest catastrophic self-focusing. Defocusing nonlinearity leads usually to an enhanced beam broadening and does not support any localized structures other than vortex and dark solitons, which require a background beam.1 However, theoretical and experimental results indicate that multidimensional bright solitons can exist in noncentrosymmetric materials with quadratic [or x s2d ] nonlinearity.2 In this Letter we demonstrate an important feature of the parametric self-trapping induced by resonant wave mixing in x s2d media. We show that even a weak quadratic nonlinearity can lead to self-focusing and stable bright solitons in a bulk medium with defocusing Kerr nonlinearity, provided that the fundamental and its second harmonic are nearly phase matched. This means that the parametric interaction in a x s2d medium can be strong enough not only to suppress the beam broadening that is due to diffraction but also to overcome the broadening effect of the defocusing Kerr nonlinearity. We consider beam propagation in noncentrosymmetric lossless bulk media with defocusing cubic nonlinearity, described by the dimensionless equations ≠w 1 =' 2 w 1 w p v 2 sjwj2 1 rjvj2 dw 0 , i ≠z (1) ≠v 1 2 2 2 2 2i 1 =' v 2 bv 1 w 2 shjvj 1 rjwj dv 0 , ≠z 2 (2) which are valid when spatial walk-off is negligible and the fundamental frequency v1 and its second harmonic v2 2v1 are far from resonance. The slowly varying complex envelope function of the fundamental w wsr, zd and of the second harmonic v vsr, zd are assumed to propagate with a constant polarization, e1 and e2 , along the z axis. The Laplacian =' 2 refers to the transverse coordinates r sx, yd. The physical electric field is EsR, Z, T d E0 fw expsiu1 dˆe1 1 0146-9592/97/221680-03$10.00/0
2v expsi2u1 dˆe2 g 1 c.c., where R r0 r, Z z0 z, and u1 k1 Z 2 v1 T . The real normalization parames2d s3d ters are3 E0 4xˆ 1 yf3jx˜ 1s jg, z0 2k1 r0 2 , and r0 2 s3d s2d 3jx˜ 1s jyh16m0 v1 2 f x˜ 1 g2 j, where m0 is the vacuum permeability and kp is the wave number at frequency vp . s3d s3d Furthermore, b 2z0 Dk, h 16x˜ 2s yx˜ 1s , and r s3d s3d 8x˜ 1c yx˜ 1s , where Dk 2k1 2 k2 ,, k1 is the wavesj d vector mismatch. The coeff icients x˜ p x˜ s j d svp d denote the Fourier components at vp of the jth-order s2d s2d susceptibility tensor. Thus x˜ 1 x˜ 2 represents the s3d s3d s3d quadratic nonlinearity, and x˜ ps and x˜ 1c x˜ 2c represent the parts of the cubic nonlinearity responsible for self- and cross-phase modulation, respectively. The system of Eqs. R (1) and (2) conserves the dimensionless power, P sjwj2 1 4jvj2 ddr, that pcorresponds to the physical power P0 P , where P0 0.5 e0 ym0 E0 2 r0 2 . Equations (1) and (2), but for focusing Kerr nonlinearity, were used in Ref. 4 for studies of collapse in arbitrary dimensions. After a simple transformation these equations correspond to the s1 1 1d-dimensional equations used in Ref. 5 and later derived rigorously in Ref. 3. Similar equations were recently shown to appear in the theory of self-focusing in quasi-phasematched x s2d media.6 In this Letter we are interested in s2 1 1ddimensional bright solitary waves, and therefore we look for spatially localized solutions to Eqs. (1) and (2) of the form vsr, zd v0 srdexps2ilzd , wsr, zd w0 srdexpsilzd, (3) where the real and radially symmetric functions p w0 srd and v0 srd decay monotonically to zero as r x2 1 y 2 increases. The real propagation constant l must be above cutoff, l . lc maxs0, 2by4d, for w0 and v0 to be exponentially localized. For a large class of materials and experimental settings we can neglect s3d s3d the dispersion of x s3d and set x˜ 1s x˜ 2s , and it is s3d s3d further reasonable to set x˜ 1s x˜ 1c . In this case we 1997 Optical Society of America
November 15, 1997 / Vol. 22, No. 22 / OPTICS LETTERS
get h 2r 16, which we use below. These values were also used in earlier papers on competing x s2d and x s3d nonlinearities.3,5 The existence of localized solutions to Eqs. (1) and (2) is a nontrivial issue. When w0 0, Eqs. (1) and (2) reduce to the stationary nonlinear Schr¨odinger equation for the second harmonic, =' 2 v0 2 s b 1 4ldv0 2 hv0 3 0, which does not permit spatially localized solutions for defocusing Kerr nonlinearity, h . 0. Similarly, when b .. 1 and b .. l, we obtain v0 ø w0 2 y2b from Eq. (2). Then Eq. (1) gives the stationary nonlinear Schr¨odinger equation for the fundamental, =' 2 w0 2 lw0 2 w0 3 0, and localized solutions are not possible either. Therefore, if bright spatial solitons should exist in defocusing Kerr media, both components should be nonzero, w0 fi 0 and v0 fi 0 (combined or C solutions5). Using a standard relaxation scheme, we numerically found the families of localized C solutions [Eqs. (3)] to Eqs. (1) and (2) for the allowed values of l and b. In Fig. 1 we show examples of the prof iles w0 srd and v0 srd for b 0.1 and different values of l. When l is small (i.e., low power), the profiles resemble the sechshaped solitons that exist in self-focusing Kerr media [Fig. 1(a)]. Increasing l also increases the beam amplitude [Fig. 1(b)]. However, for suff iciently large values of l (or of the power), the amplitudes saturate, and the beam broadens significantly [Fig. 1(c)] because of the defocusing effect of the cubic terms in Eqs. (1) and (2). Above l ø 0.00656, no localized solutions exist. Similarly, near the cutoff lc 0, the beam width increases rapidly and the amplitude decreases. Below the cutoff, no localized solution exist. In Fig. 2 we show the amplitude of the C solutions, w0 s0d and v0 s0d, and their power P as function of l for two particular values of b. Clearly the solutions exist in a limited region only, ranging from the cutoff lc to a certain upper limit at which the beam power tends to inf inity, even though the amplitude saturates, ref lecting the defocusing effect of the Kerr nonlinearity. For b 0.1 the power increases monotonically with l, whereas for b 20.02 it decreases in a narrow interval above the cutoff. In both cases there is a power threshold below which no solutions exist. The Vakhitov –Kolokolov stability criterion, ≠N y ≠l . 0, has been shown to apply to solitary waves supported by pure x s2d nonlinearity.7 Its derivation for Eqs. (1) and (2) is similar. According to this criterion the solutions are stable in the whole region of existence for b 0.1, whereas for b 20.02 they become unstable in a narrow region above cutoff. We made a series of calculations as shown in Fig. 2(b), found ≠Ny≠l, and identified the regions of existence and stability of the C solutions in the sl, bd plane. The results are summarized in Fig. 3(a). In regions I and II no localized solutions exist, in region I because l is below cutoff and in region II because the defocusing effect of the cubic nonlinearity becomes dominant. For b , 0 there is a narrow band [magnified f ive times in Fig. 3(a) to show it] where solutions exist but are unstable. This result is to be expected because the instability also exists for pure x s2d nonlin-
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earity.7 Stable solitons exist in the hatched region. We have confirmed their stability to propagation by simulation of Eqs. (1) and (2) for representative cases. In Fig. 3(b) we show the power of the stable solitons. As is also evident from Fig. 2(b), the stable solitons exist for all powers above a certain threshold. Two physical effects are apparent from Fig. 3 when we keep in mind that the effective mismatch b is proportional to both the wave-vector mismatch Dk and the s3d s2d ratio of the x s3d and x s2d coeff icients, b ~ Dk x˜ 1s yf x˜ 1 g2 . When jbj increases, the existence region becomes narrower, ref lecting that the defocusing cubic nonlinearity becomes progressively more dominant, either s3d s2d because jx˜ 1s j .. f x˜ 1 g2 or because the waves are not phase matched (see also Ref. 5). Similarly, for f ixed b, solitons exist only when the maximum intensity, or equivalently, l is sufficiently small. When the intensity, or l, becomes too high, the defocusing Kerr nonlinearity is again dominant, prohibiting the existence of stable bright solitons. This phenomenon is due to the
Fig. 1. Profiles w0 sr xd (solid curves) and v0 sr xd (dotted curves) of solutions (3) for h 2r 16, b 0.1, and (a) l 0.003, (b) l 0.006, (c) l 0.00655.
Fig. 2. Characteristics of the C solutions [Eqs. (3)] for h 2r 16 and b 20.02 (dotted curves) and b 0.1 (solid curves). (a) Amplitude versus l for the fundamental w0 s0d (upper curve) and the second harmonic v0 s0d (lower curve). ( b) Dimensionless power versus l.
Fig. 3. (a) Region of existence of stable (hatched ) and unstable (black) solitons in the sl, bd plane. (b) Power regime ( hatched ) of stable solitons versus b. Parameters: h 2r 16.
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OPTICS LETTERS / Vol. 22, No. 22 / November 15, 1997 s2d
Fig. 4. Evolution of a Gaussian input beam at the fundamental, wsr, 0d 0.1 expf2sry30d2 g, with vsr, 0d 0 and (a) b 0.29, ( b) b 0.1. Parameters: h 2r 16.
s3d
Fig. 5. Physical power threshold versus x˜ 1s for h 2r s2d 16, l1 1.064 mm, and values of x˜ 1 of 3 pmyV (solid curves) and 5.6 pmyV (dashed curves). In each case the upper curve is for k 21023 and the lower curve is for k 1023 .
competition between two kinds of nonlinearity and is qualitatively similar to, e.g., the effect of self-trapping in media with focusing cubic and defocusing quintic nonlinearity.8 We numerically analyzed the generation of bright solitons from a Gaussian input beam with power P 14.1 at the fundamental, without seeding of the second harmonic, corresponding to the filled circles in Fig. 3(b). The results, presented in Fig. 4, show two characteristic types of evolution. When the effective mismatch is sufficiently large, b 0.29, the input power is below threshold, and the evolution is strongly affected by the defocusing Kerr effect. Therefore the beam rapidly diffracts [Fig. 4(a)]. When the mismatch is so small s b 0.1d that the input power is sufficiently above threshold (part of the power is always lost to radiation), the parametric focusing is stronger, and we observe the formation of a localized self-trapped beam [Fig. 4(b)]. To get a feeling for the powers, nonlinear coefficients, and degree of phase matching required for generation of these bright solitons, we consider a fundamental wavelength of l1 1.064 mm, a phase mismatch of s2d s2d k 61023 , and x˜ 1 3 or x˜ 1 5.6 pmyV. Here 2 k ; n1 Dkyk1 , where n1 is the refractive index at the fundamental frequency. From the dimensionless threshold power shown in Fig. 3(b) and the parameter def initions given below Eqs. (1) and (2) we can then calculate the physical threshold power as function of s3d x˜ 1s . The result is shown in Fig. 5. For example, for
KTP with x˜ 1 5.6 pmyV and a phase mismatch of k 1023 the threshold is of the order of 10 kW for s3d a rather wide range of x˜ 1s values and is thus of the same order as the power in the solitons observed in s3d s2d KTP.9 When jx˜ 1s j increases or jx˜ 1 j decreases, the required power increases. For a negative wave-vector mismatch, Dk , 0, the threshold power is always s3d higher and the x˜ 1s region in which stable solitons exist is always narrower than for the corresponding positive mismatch. Consider k 1023 and the Gaussian initial condition used in Fig. 4, which generated a soliton for b 0.1. For a fixed effective mismatch of b 0.1, the ratio of the nonlinearity coefficients is f ixed at x3 yx2 2 24.2. Hence this input beam has a f ixed FWHM width of 42 mm, a power of x2 22 3 42.5 kW, and a peak intensity of x2 22 3 4.2 GWycm2 , where the normalized nonlinearity coefficients are def ined as s2d s3d x2 x˜ 1 ys5.6 pmyVd and x3 x˜ 1s ys103 pm2 yV2 d. In conclusion, we have analyzed the effect of quadratic nonlinearity on the existence and stability of bright spatial solitons in bulk media with defocusing Kerr nonlinearity. We have shown that stable bright spatial solitons can exist because of parametrically induced self-focusing and that they can be generated from Gaussian input beams with experimentally reasonable powers. From a physical point of view this phenomenon indicates an important feature of cascaded nonlinearities, permitting self-focusing effects even in a defocusing Kerr medium. The authors acknowledge useful discussions with S. Trillo and W. Torruellas. Part of this research has been supported by the Australian Department of Industry, Science and Tourism, through grant 74 under the International Science and Technology program. A. V. Burak acknowledges support of the Australian Research Council. References 1. Yu. S. Kivshar and B. Luther-Davies, ‘‘Optical dark solitons: physics and applications,’’ Phys. Rep. (to be published ). 2. For a general overview see G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum. Electron. 28, 1691 (1996). 3. O. Bang, J. Opt. Soc. Am. B 14, 51 (1997). 4. L. Berg´e, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, Phys. Rev. E 55, 3555 (1997). 5. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995). 6. C. B. Clausen, O. Bang, and Yu. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997). 7. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995). 8. E. M. Wright, B. L. Lawrence, W. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 2481 (1995). 9. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
