## Control of the collapse of bimodal light beams by magnetically tunable birefringences

Optics Express, Vol. 18, Issue 9, pp. 8879-8895 (2010)

http://dx.doi.org/10.1364/OE.18.008879

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### Abstract

Using a system of coupled nonlinear Schrödinger equations (CNLSEs), we show that nonlinear light propagation in self-focusing Kerr media can be controlled via a suitable combination of linear and circular birefringences. In particular, magneto-optical effects are taken as a specific physical example, which enables the introduction of both types of birefringences simultaneously via the joint action of the Cotton-Mouton and the Faraday effect. We demonstrate the efficient management of the collapse of (2 + 1)D beams in magneto-optic dielectric media, which may result in either the acceleration or the suppression of the collapse. However, our study also shows that a complete stabilization of the bimodal beams (i.e., the propagation of two-dimensional solitary waves) is not possible under the proposed conditions. The analysis is performed by directly numerically solving the CNLSEs, as well as by using the variational approximation, both showing consistent results. The investigated method allows high-power beam propagation in Kerr media while avoiding collapse, thus offering a viable alternative to the techniques applied in non-instantaneous and/or non-local nonlinear media.

© 2010 OSA

## 1. Introduction

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*critical*) value of the beam power. These solutions are unstable, eventually leading to either a catastrophic collapse (singularity) or diffraction. Moreover, when the power exceeds the critical value, wave collapse always occurs [3,10

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*control*and eventually

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^{(2)}) and a cubic (χ

^{(3)}) nonlinearity [35

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**331**(4), 117–195 (2000). [CrossRef]

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46. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: equal propagation amplitudes,” Opt. Lett. **12**(8), 614–616 (1987). [CrossRef] [PubMed]

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33. O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. **83**(26), 5479–5482 (1999). [CrossRef]

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## 2. The theoretical model

### 2.1. Coupled nonlinear Schrödinger equations (CNLSEs)

*u*and

_{L}*u*of the left- (

_{R}*L*) and right- (

*R*) circularly polarized components, in the presence of the Kerr nonlinearity [5] and under the influence of the combined (here magnetically-induced) linear and circular birefringences [46

46. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: equal propagation amplitudes,” Opt. Lett. **12**(8), 614–616 (1987). [CrossRef] [PubMed]

49. Y. Barad and Y. Silberberg, “Polarization Evolution and Polarization Instability of Solitons in a Birefringent Optical Fiber,” Phys. Rev. Lett. **78**(17), 3290–3293 (1997). [CrossRef]

*w*

_{0}is the initial radius (waist) of the axisymmetric Gaussian beam,

*U*

_{L}_{,}

*are the complex amplitudes of the two components of the monochromatic field [i.e., the full components corresponding to the circular polarizations are defined as*

_{R}*k*

_{0}is the wave number in vacuum, λ

_{0}is the wavelength,

*n*

_{0}is the linear refractive index,

*n*

_{2}is the nonlinear Kerr coefficient, and η is the vacuum impedance. Equations (1) were derived by means of the slowly-varying envelope approximation (SVEA), and also neglecting the propagation losses. Indeed, in the materials typically used in the experimental conditions, such as YIG crystals, the latter postulation is acceptable, due to the intrinsic properties of the medium [61

61. Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. **103**(5), 053902 (2009). [CrossRef] [PubMed]

50. L. Bergé, O. Bang, and W. Królikowski, “Influence of four-wave mixing and walk-Off on the self-focusing of coupled waves,” Phys. Rev. Lett. **84**(15), 3302–3305 (2000). [CrossRef] [PubMed]

51. O. Bang, L. Bergé, and J. J. Rasmussen, “Fusion, collapse, and stationary bound states of incoherently coupled waves in bulk cubic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **59**(4), 4600–4613 (1999). [CrossRef]

*z*with the time coordinate

*t*in Eqs. (1), one can obtain the set of coupled Gross-Pitaevskii equations (GPEs) that is commonly used to describe BEC mixtures of two different spin states of the same atomic species [58

58. R. J. Ballagh, K. Burnett, and T. F. Scott, “Theory of an Output Coupler for Bose-Einstein Condensated Atoms,” Phys. Rev. Lett. **78**(9), 1607–1611 (1997). [CrossRef]

### 2.2. The variational approximation (VA)

59. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Progr. Opt. **43**, 71 (2002). [CrossRef]

*A*,

*φ*,

*θ*,

*w*

_{R}_{,}

*and*

_{L}*ξ*

_{R}_{,}

*are real functions of*

_{L}*z*. Specifically,

*A*and

*φ*are the common amplitude and phase of both polarization components,

*θ*determines the distribution of the power between them,

*ψ*is the relative phase, and

*w*

_{R}_{,}

*and*

_{L}*ξ*

_{R}_{,}

*are the widths and radial chirps, respectively.*

_{L}*ζ*is any variational parameter), are used to derive the evolution equations from the Lagrangian, Eq. (6). The first equation,

*ξ*and to derive a second-order ODE for the evolution of the beam width

*w*(

*z*):

15. L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. **303**(5-6), 259–370 (1998). [CrossRef]

33. O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. **83**(26), 5479–5482 (1999). [CrossRef]

34. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. **104**(5), 053902 (2010). [CrossRef] [PubMed]

50. L. Bergé, O. Bang, and W. Królikowski, “Influence of four-wave mixing and walk-Off on the self-focusing of coupled waves,” Phys. Rev. Lett. **84**(15), 3302–3305 (2000). [CrossRef] [PubMed]

51. O. Bang, L. Bergé, and J. J. Rasmussen, “Fusion, collapse, and stationary bound states of incoherently coupled waves in bulk cubic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **59**(4), 4600–4613 (1999). [CrossRef]

*c*= 0 (with no linear mixing between the circular polarization components), Eq. (7b) yields

*θ*equal to a constant. Concurrently, Eq. (7a), which governs the evolution of the beam width, decouples from the other equations (as expected; see [60

60. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of the optical pulses,” J. Opt. Soc. Am. B **8**(10), 2082 (1991). [CrossRef]

*θ*= 0 it can be demonstrated, using Eq. (7a), that, when

*z*) always takes place when

*Townes*solitons [60

60. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of the optical pulses,” J. Opt. Soc. Am. B **8**(10), 2082 (1991). [CrossRef]

*w*(

*z*), as governed by Eq. (7a), no longer decouples from the evolution of

*θ*. In fact, while the term sin

^{2}(2

*θ*) may take values in the range between 0 and 1,

*c*, Eq. (7a) takes the form of a non-autonomous ODE decoupled from Eqs. (7b) and (7c):where

*b*, is large, Eq. (7a) simplifies into the following form:

### 2.3. Magnetically-induced birefringences

*ε*, whose elements depend on the external magnetic field. Assuming that a magnetic field

**is applied in the (**

*H**x*,

*z*) plane at an angle

*α*with respect to the

*z*-axis, the dielectric permittivity tensor is written as [52

52. R. Kurzynowski and W. A. Woźniak, “Superposition rule for the magneto-optic effects in isotropic media,” Optik (Stuttg.) **115**(10), 473–475 (2004). [CrossRef]

*n*

_{0}is the refractive index in the absence of the magnetic field, and

*Q*,

*B*

_{1},

*B*

_{2}are material parameters depending on the magnetization of the medium, which vanish in the absence of the field (in particular,

*Q*~|

**| and**

*H**B*~|

_{i}**|**

*H*^{2}). The components of the dielectric tensor which are quadratic in |

**| account for the CM effect, while those linear in |**

*H***| represent the Faraday effect in transmission, or the MO Kerr effect in reflection (the latter one should not be confused with the commonly-known Kerr effect which gives rise to the electric nonlinearity of dielectric materials). In general, if the diagonal terms of the permittivity tensor are different, the material is linearly birefringent, whereas the off-diagonal terms, ~**

*H**iQ*, imply the presence of an optical activity (circular birefringence). Assuming that the beam propagates along the

*z*-axis, and defining the elements of the dielectric tensor as:

*A*are appropriate material constants, the following equation describes the resulting refractive index

_{i}*n*(square root of the dielectric permittivity) of the medium under the influence of the total magnetic field

**[52**

*H*52. R. Kurzynowski and W. A. Woźniak, “Superposition rule for the magneto-optic effects in isotropic media,” Optik (Stuttg.) **115**(10), 473–475 (2004). [CrossRef]

*α*two different solutions for

*n*

^{2}exist, indicating that the intrinsically isotropic medium becomes

*birefringent*under the action of the external magnetic field. In two particular cases, namely for

*α*= 0 (i.e., in the CM geometry), and for

*α*= π/2 (in the Faraday geometry) one obtains:and

**115**(10), 473–475 (2004). [CrossRef]

*n*and Δ

_{l}*n*are the linear and circular birefringences, i.e., the differences in the refractive index of the two wave components induced by the CM and the Faraday effect, respectively.

_{c}*b*~|

**|), while the linear birefringence caused by the CM effect (emerging when the direction of the light propagation is perpendicular to the magnetic field) is proportional to the square of the field,**

*H**c*~|

**|**

*H*^{2}[52

**115**(10), 473–475 (2004). [CrossRef]

*c*~

*b*

^{2}, which offers the possibility of controlling the combined birefringences strength. In the general case of an oblique orientation of the magnetic field with respect to the propagation direction (assuming that the orientation is confined to a plane), i.e., with components of the magnetic field

*H*_{||}= |

**| cos**

*H**α*and

*H*_{⊥}= |

**| sin**

*H**α*(which are parallel and perpendicular to the direction of the light propagation, respectively), the total birefringence can be written as:where

**, which may significantly affect the collapse scenario for the bimodal beam. Moreover, both birefringence coefficients**

*H**b*and

*c*, introduced in the theoretical model, are different from zero for an oblique direction of the magnetic field, whereas in the CM geometry

*b*= 0 and in the Faraday geometry

*c*= 0. However, in the vicinity of the collapse point, the paraxial approximation, which was assumed in the derivation of Eqs. (1), ceases to hold and the transversal components of the wave vector may appear even if they were previously absent. This means that in the CM geometry, a non-negligible contribution of the Faraday term (or equivalently, the CM term in the Faraday geometry) should be considered. As a result, more complex collapse dynamics may be expected in real conditions, as shown by our recent experimental data [61

61. Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. **103**(5), 053902 (2009). [CrossRef] [PubMed]

_{3}Fe

_{5}O

_{12}, where T is a trivalent metal. The best-known representative of this class of materials is yttrium iron garnet (Y

_{3}Fe

_{5}O

_{12}, abbreviated as YIG), which is highly transparent to near-infrared radiation, and is commonly used for the fabrication of optical isolators [62,63

63. O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. **119**(2), 345–348 (2005). [CrossRef]

*n*

_{0}= 2.22, ∆

*n*≈1.6·10

_{F}^{−4}and ∆

*n*≈1·10

_{CM}^{−4}, at the saturation field of ~0.2T [53–55

55. G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, “Magnetic circular dichroism and Faraday rotation spectra of Y_{3}Fe_{5}O_{12},” Phys. Rev. B **12**(7), 2562 (1975). [CrossRef]

*b*= 0.125,

*c*= 0.075 for a beam waist

*w*

_{0}= 5μm, in the Faraday and CM geometries, respectively. These values (and the ratio between them) can be tuned by adjusting the value and direction of the magnetic field, as well as by varying the spot-size and the wavelength of the injected beam (keeping also in mind that the circular magnetic birefringence strongly depends on the wavelength, while the linear magnetic birefringence is wavelength-independent for near-infrared radiation). Moreover, the range of the achievable birefringence parameters can be expanded using different magnetic materials. While in YIG only magnetic ions are ferric, magnetic rare-earth ions can be used in place of yttrium in order to improve the magnetic properties and thus increase the magneto-optic coefficients [53,63

63. O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. **119**(2), 345–348 (2005). [CrossRef]

64. M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. **94**(18), 181916 (2009). [CrossRef]

63. O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. **119**(2), 345–348 (2005). [CrossRef]

64. M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. **94**(18), 181916 (2009). [CrossRef]

## 3. Numerical results

### 3.1. The input conditions

*Townes profile*that remains axisymmetric, irrespective of the initial spatial profile of the beam [8

8. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett. **13**(15), 479–482 (1964). [CrossRef]

10. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. **25**(5), 335–337 (2000). [CrossRef] [PubMed]

13. T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. **4**(3), 218–222 (2008). [CrossRef]

**331**(4), 117–195 (2000). [CrossRef]

*ψ*

_{0}= 0,

*θ*

_{0}= π/4, and

*ξ*

_{0}= 0. Thus, the profiles of the components at the input are given by:which corresponds to an unchirped, linearly polarized Gaussian beam with width

*w*

_{0}and amplitude

*A*

_{0}.

### 3.2. Direct simulations (DS)

^{−3}over the transverse interval of

^{−3}, allowed us to keep the relative deviation of the total power below 10

^{−3}(with respect to the input value), over the considered distance (

*z*= 100

_{max}*L*). We have also assumed that, for such a high-density meshing, a change in the critical power for the collapse caused by the numerical discretization [31

_{D}31. O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Subcritical localization in the discrete nonlinear Schrödinger equation with arbitrary power nonlinearity,” Nonlinearity **7**(1), 205–218 (1994). [CrossRef]

*A*

_{0}, is varied, while the width

*w*

_{0}is kept constant), as well as for different combinations of the magnetically-induced birefringences. In addition to the beam-intensity distributions (which are displayed in Figs. 1(a)–1(e) and are proportional to

*R*is defined as:

*b*=

*c*= 0), the beam polarization components are coupled only via the XPM term (which becomes significant only close to the collapse point). In such a case, the diffraction of a low-input power beam is observed as shown in Fig. 1(a). The spatial divergence of the beam may be reduced by increasing the optical power. For a given initial width, if the input amplitude exceeds a critical value (

*A*≈1.124 for

_{cr}*b*=

*c*= 0), the collapse occurs after a finite propagation distance [in particular, at

*z*≈12

*L*in Fig. 1(b)]. The numerically found critical amplitude,

_{D}*A*= 1.124, if rescaled back to physical units, corresponds to an optical power of

_{cr}*P*≈9.8MW (taking

*n*

_{0}= 2.22, λ

_{0}= 1.2μm and

*n*

_{2}= 10

^{−20}m

^{2}/W). As shown in Fig. 1(d), the introduction of the circular birefringence alone (i.e., b ≠ 0 while c = 0) does not change the collapse distance. However, the collapse can be accelerated (i.e., it can occur after a shorter propagation distance) by the introduction of the linear birefringence, c ≠ 0, as shown in Fig. 1(c). Contrarily, when both birefringences are present, the collapse distance may be extended, as shown in Fig. 1(e). The latter outcome takes place only for a particular combination of the birefringence coefficients [see Fig. 2(c)], resulting from a specific interplay between the amplitude and phase mixing of the two polarization components. Thus, taking into account that the coefficients introduced in the model depend on the value of the external magnetic field, as described in section 2.3, the distance for the collapse may be magnetically tuned, allowing the possibility of controlling the self-focusing beams in bulk MO media.

*A*

_{0}= 1.135, which is above the critical value) is schematically presented in Fig. 2, where the total propagation length is limited to

*z*= 100

_{max}*L*. The figure demonstrates the tunability of the collapse in the (

_{D}*b*,

*c*)-plane. In particular, Fig. 2(a) shows the normalized final width of the beam (obtained by fitting the spatial profile of one of the beam components to a Gaussian function) at

*z*=

*z*. If the collapse occurs at

_{max}*z*≤

*z*, the final width of the beam is set to be

_{max}*w*= 0 [the grey area in Fig. 2(a)]. The corresponding normalized collapse distance,

*z*/

*L*, is shown in Fig. 2(b). The results are displayed for the

_{D}*u*component, while the dynamics of

_{R}*u*is essentially the same in the entire range of the considered parameters.

_{L}*w*/

*w*

_{0}≈1, thus defining the range of parameters where the propagating beams are potentially stable. As illustrated in Fig. 2(b), at a given value of the input power, the amplitude mixing between the polarization components dominates for

*c*>

*b*, which usually results in the acceleration of the collapse, while a prominent phase mixing (for

*b*>

*c*) may lead to the suppression of the collapse. Furthermore, distinctive regions in the chart can be singled out, as labeled in Fig. 2(a). Of particular importance are three characteristic (critical) values of the circular-birefringence coefficient

*b*(for a fixed value of

*c*): these are marked as

*b*,

_{A}*b*and

_{B}*b*in the figures. Two of them (

_{C}*b*and

_{A}*b*) correspond to the transition between the collapsing and diffracting solutions. The third critical point,

_{C}*b*, indicates parameters for which the beam divergence may be reduced, leading to a longer stable propagation length

_{B}*L*. Eventually, as one can see in Fig. 2(c), the length for which a nearly stable propagation takes place (before the beam eventually collapses or diffracts) may be significantly extended by precisely choosing the proper combination of the birefringence coefficients (

*b*,

*c*), and in particular, it can be increased to up to about 300 diffraction lengths in the specific case presented here.

*b*and

_{A}*b*) is independent of the linear birefringence coefficient

_{C}*c*. The obtained numerical results at these boundaries were determined to be well represented by power-functions to approximate the collapse length,

*L*:

*c*= 0.055 and

*c*= 0.07 [namely those for

*A*= 1.135 presented in Fig. (3)], the critical exponents

*α*were found to be around 0.05, indicating rapid changes (increasing with input optical power) of the collapse length

_{A,C}*L*close to the collapse-diffraction transition point. Thus, a strong delay of the collapse is possible for the parameters that are close enough to the critical values determined by

*b*and

_{A}*b*. For a given linear birefringence coefficient

_{C}*c*, a prolongation of the stable propagation of the beam is also possible if the value of the co-acting circular birefringence coefficient

*b*is approaching the above-mentioned third critical value,

*b*. In such cases, the range of suitable values for the propagation stabilization,

_{B}*∆b*, broadens as a function of the linear birefringence coefficient

_{B}*c*[see Figs. 3(a1) and 3(a2)]. The normalized beam waist radius as a function of the propagation distance for the birefringence coefficients approaching these critical values was also numerically determined, and is presented in Fig. 3(b1), 3(b2).

*b*feature a nearly linear dependence on

_{B}*c*, as shown in Fig. 4(a) . This plot allows one to determinate the required combination of the MO parameters for which the beam divergence may be reduced [examples are shown in Fig. 4(b1), 4(b2)], and, specifically, to identify situations [black squares in Fig. 4(a)] when the nearly stable propagation (before the eventual diffraction) can be extended. The quasi-stabilization line obtained for

*A*

_{0}= 1.135 is shown in Fig. 4(a), while the evident prolongation of the stable propagation [similar to that shown in Figs. 3(b1), 3(b2)] is observed only for a limited range of the combined birefringences, marked with the black squares in Fig. 4(a).

### 3.3. The variational approximation

^{−11}. In the case of collapse, the beam width varies so fast that even a drastic reduction of the step-size does not secure the conservation of the Hamiltonian. In particular, we have defined the collapse point as the propagation distance for which the numerical procedure fails to stay within the predetermined margins, fixed to 10

^{−3}and 10

^{−6}for the relative and absolute error tolerances, respectively, without reducing the step-size below the smallest value allowed by the numerical routine,

^{−14}.

59. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Progr. Opt. **43**, 71 (2002). [CrossRef]

*b*=

*c*= 0 and

*A*

_{0}= 1.135, used for the PDE simulations, corresponds to the initial condition of

*b*, which is easily observed in the case of the VA; see the black region in Fig. 2(d). Moreover, as it is presented in Fig. 5(a) , several sharp maxima of the propagation length within the stabilization region are obtained for critical values

_{B}*b*, in contrast to the DS results [Fig. 3(a1), 3(a2)]. The width of the stabilization range, ∆

_{Bi}*b*, as well as the number of peak values corresponding to the stable-propagation lengths, increases with the value of the linear birefringence coefficient

_{B}*c*(not shown here). Moreover, while the solutions found for values of

*b*taken in the vicinity of

*b*exhibit oscillations along the propagation distance with a constant period (as it was the case for the DS), irregular oscillations of the beam width for

_{A}*b*close to

*b*and

_{Bi}*b*are observed [see Fig. 5(b)]. Specifically, solutions around

_{C}*b*are prone to instabilities, and the transition between the diffraction and collapse is not as sharp as it was for the DS. In such a case collapse is observed for

_{C}*b*≥

*b*but the width of the beam exceed significantly its initial value before the collapse point [as seen on the bottom graph in Fig. 5(b) and as indicated by the solid dots in Fig. 5(a)]. However, it is still possible to estimate the critical values of the circular birefringence coefficient, and to fit it to the power-function approximation given by Eq. (20) (for instance,

_{C}*α*≈0.07 and

_{A}*α*≈0.12 in the present case).

_{C}## 4. Conclusions

61. Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. **103**(5), 053902 (2009). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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13. | T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. |

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24. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

25. | Y. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. |

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29. | N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. |

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32. | A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express |

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34. | A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. |

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40. | P. Pedri and L. Santos, “Two-dimensional bright solitons in dipolar Bose-Einstein condensates,” Phys. Rev. Lett. |

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42. | F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length,” Phys. Rev. A |

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60. | M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of the optical pulses,” J. Opt. Soc. Am. B |

61. | Y. Linzon, K. A. Rutkowska, B. A. Malomed, and R. Morandotti, “Magneto-optical control of light collapse in bulk Kerr media,” Phys. Rev. Lett. |

62. | J.-M. Liu, |

63. | O. Kamada, T. Nakaya, and S. Higuchi, “Magnetic field optical sensors using Ce:YIG single crystals as a Faraday element,” Sens. Actuators A Phys. |

64. | M. C. Sekhar, J.-Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce: and Bi: co-modified epitaxial iron garnet thin films,” Appl. Phys. Lett. |

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66. | J. C. Butcher, |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

(230.3810) Optical devices : Magneto-optic systems

(260.1440) Physical optics : Birefringence

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 16, 2010

Revised Manuscript: March 18, 2010

Manuscript Accepted: April 2, 2010

Published: April 13, 2010

**Citation**

Katarzyna A. Rutkowska, Boris A. Malomed, and Roberto Morandotti, "Control of the collapse of bimodal light beams by magnetically tunable
birefringences," Opt. Express **18**, 8879-8895 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-8879

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### References

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- O. Bang, D. Edmundson, and W. Królikowski, “Collapse of incoherent light beams in inertial bulk Kerr media,” Phys. Rev. Lett. 83(26), 5479–5482 (1999). [CrossRef]
- A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010). [CrossRef] [PubMed]
- L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3555–3570 (1997). [CrossRef]
- V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1080–1087 (1997). [CrossRef]
- S. Gatz and J. Herrmann, “Propagation of optical beams and the properties of two-dimensional spatial solitons in media with a local saturable nonlinear refractive index,” J. Opt. Soc. Am. B 14(7), 1795 (1997). [CrossRef]
- O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 046619 (2002). [CrossRef] [PubMed]
- M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27(16), 1460–1462 (2002). [CrossRef] [PubMed]
- P. Pedri and L. Santos, “Two-dimensional bright solitons in dipolar Bose-Einstein condensates,” Phys. Rev. Lett. 95(20), 200404 (2005). [CrossRef] [PubMed]
- I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19(3), 537 (2002). [CrossRef]
- F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003). [CrossRef]
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