Exact analytical Helmholtz bright and dark solitons

Transcription

1 Exact analytical Helmholtz bright and dark solitons P. CHAMORRO POSADA Dpto. Teoría de la Señal y Comunicaciones e Ingeniería Telemática ETSI Telecomunicación Universidad de Valladolid, Spain G. S. McDONALD Joule Physics Laboratory, School of Computing, Science and Engineering Institute of Materials Research University of Salford, UK

2 Introduction INDEX Introduction Exact bright Exact dark Propagation properties of Helmholtz soliton collisions Conclusions & Directions

3 Introduction Failure of the paraxial approximation Nonparaxial scenarios: Type I scenarios

4 Introduction Failure of the paraxial approximation Nonparaxial scenarios: Type I scenarios Intense self-focusing. Ultra-narrow beams (progressive miniaturisation of IT devices).

5 Introduction Failure of the paraxial approximation Nonparaxial scenarios: Type I scenarios Intense self-focusing. Ultra-narrow beams (progressive miniaturisation of IT devices). Type II scenarios (Helmholtz type of nonparaxiality).

6 Introduction Failure of the paraxial approximation Nonparaxial scenarios: Type I scenarios Intense self-focusing. Ultra-narrow beams (progressive miniaturisation of IT devices). Type II scenarios (Helmholtz type of nonparaxiality). Propagation of optical solitons at an arbitrary angle (rotation, steering or intrinsic). Simultaneous propagation of multiplexed soliton beams.

7 Introduction Failure of the paraxial approximation Nonparaxial scenarios: Type I scenarios Intense self-focusing. Ultra-narrow beams (progressive miniaturisation of IT devices). Type II scenarios (Helmholtz type of nonparaxiality). Propagation of optical solitons at an arbitrary angle (rotation, steering or intrinsic). Simultaneous propagation of multiplexed soliton beams. Scalar full Helmholtz theory (Kerr media)

8 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS)

9 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Considering a reference frame in the forward z direction, and a nonlinear Kerr medium, E(x, y, z) = 1 2 A(x, z) exp[ i(ω 0t kz)] + c.c.,

10 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Considering a reference frame in the forward z direction, E(x, y, z) = 1 2 A(x, z) exp[ i(ω 0t kz)] + c.c., and a nonlinear Kerr medium,we obtain the NNLS κ 2 u ζ 2 + i u ζ u 2 ξ 2 ± u 2 u = 0

11 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Considering a reference frame in the forward z direction, E(x, y, z) = 1 2 A(x, z) exp[ i(ω 0t kz)] + c.c., and a nonlinear Kerr medium,we obtain the NNLS ζ = z L D ξ = L D = kw2 0 2 κ 2 u ζ 2 + i u ζ u 2 ξ 2 ± u 2 u = 0 2x w 0 u(ξ, ζ) = (diffraction length) κ = 1 k 2 w 2 0 kn2 L D A(ξ, ζ) 2 (nonparaxiality parameter)

12 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS)

13 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE).

14 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected.

15 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0:

16 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0: ( ) 2 λ κ = 4π 2. w 0

17 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0: ( ) 2 λ κ = 4π 2. w 0 κ = 1 2 n 2 E 2 0 n 0.

18 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0: ( ) 2 λ κ = 4π 2. w 0 κ = 1 2 κ = 1 2 n 2 E0 2. ( n 0 ) 2 X0, Z 0

19 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0: ( ) 2 λ κ = 4π 2. w 0 κ = 1 2 κ = 1 2 n 2 E0 2. ( n 0 ) 2 X0,dimensionless angle tan φ = V Z 0

20 Introduction - the NNLS Non-Paraxial Non-Linear Schrödinger Equation (NNLS) Fully equivalent to the nonlinear Helmholtz equation (NHE). The NLS is recovered when the term κ ζζ can be neglected. κ 0: ( ) 2 λ κ = 4π 2. w 0 κ = 1 2 κ = 1 2 n 2 E0 2. ( n 0 ) 2 X0,dimensionless angle tan φ = V tan θ = 2κ tan φ. Z 0

21 Exact bright solitons Bright soliton solution The general bright soliton solution is given by the expression [ ] η(ξ + V ζ) 1 + 2κη u(ξ, ζ) = η sech exp i 2 ( 1 + 2κV κV 2 V ξ + ζ ) [ exp i ζ ] 2κ 2κ

22 Exact bright solitons Bright soliton solution The general bright soliton solution is given by the expression [ ] η(ξ + V ζ) 1 + 2κη u(ξ, ζ) = η sech exp i 2 ( 1 + 2κV κV 2 V ξ + ζ ) [ exp i ζ ] 2κ 2κ In the multiple limit κ 0, κv 2 0, κη 2 0, we recover the NLS fundamental soliton [ u(ξ, ζ) = η sech [η(ξ + V ζ)] exp iv ξ + i 1 ] 2 (η2 V 2 )ζ

23 Exact bright solitons Bright soliton solution The general bright soliton solution is given by the expression [ ] η(ξ + V ζ) 1 + 2κη u(ξ, ζ) = η sech exp i 2 ( 1 + 2κV κV 2 V ξ + ζ ) [ exp i ζ ] 2κ 2κ In the multiple limit κ 0, κv 2 0, κη 2 0, we recover the NLS fundamental soliton [ u(ξ, ζ) = η sech [η(ξ + V ζ)] exp iv ξ + i 1 ] 2 (η2 V 2 )ζ The general V 0 solutions can be obtained by transforming the V = 0 soliton.

24 Exact bright solitons Galilei invariance vs rotational invariance The NLS is invariant under the Galilei transformation ξ = ξ + V ζ ζ = ζ [ u (ξ ζ ) = exp i (V ξ + 12 )] V 2 ζ u(ξ, ζ)

25 Exact bright solitons Galilei invariance vs rotational invariance The NLS is invariant under the Galilei transformation ξ = ξ + V ζ ζ = ζ [ u (ξ ζ ) = exp i (V ξ + 12 )] V 2 ζ u(ξ, ζ) Whereas the NNLS is invariant under the transformation u (ξ, ζ ) = exp [ i ξ = ξ + V ζ 1 + 2κV 2 ( V ξ κV 2 2κ ( 1 ζ = 2κV ξ + ζ 1 + 2κV 2 ) )] 1 ζ u(ξ, ζ) 1 + 2κV 2

26 Exact bright solitons Galilei invariance vs rotational invariance The NLS is invariant under the Galilei transformation ξ = ξ + V ζ ζ = ζ [ u (ξ ζ ) = exp i (V ξ + 12 )] V 2 ζ u(ξ, ζ) Whereas the NNLS is invariant under the transformation u (ξ, ζ ) = exp [ i ξ = ξ + V ζ 1 + 2κV 2 ( V ξ κV 2 2κ ( 1 ζ = 2κV ξ + ζ 1 + 2κV 2 ) )] 1 ζ u(ξ, ζ) 1 + 2κV 2

27 Exact bright solitons Rotational invariance The previous transformation can be decomposed in three steps:

28 Exact bright solitons Rotational invariance The previous transformation can be decomposed in three steps: The nonparaxial solution is written in the original (unscaled) frame of reference. We introduce a rotation of angle θ, where tan θ = 2κV. The solution is transformed back to the scaled units.

29 Exact bright solitons Rotational invariance The previous transformation can be decomposed in three steps: The nonparaxial solution is written in the original (unscaled) frame of reference. We introduce a rotation of angle θ, where tan θ = 2κV. The solution is transformed back to the scaled units. The previous transformation is then written, in terms of θ, as [ ] [ ] 1 [ ] ξ cos θ = 2κ sin θ ξ ζ 2κ sin θ cos θ ζ.

30 Exact bright solitons Rotational invariance The previous transformation can be decomposed in three steps: The nonparaxial solution is written in the original (unscaled) frame of reference. We introduce a rotation of angle θ, where tan θ = 2κV. The solution is transformed back to the scaled units. The previous transformation is then written, in terms of θ, as [ ] [ ] 1 [ ] ξ cos θ = 2κ sin θ ξ ζ 2κ sin θ cos θ ζ. The additional phase term is introduced due to the phase reference used to obtain the normalised equation.

31 Exact bright solitons Properties of fundamental bright solitons Paraxial soliton Helmholtz soliton Soliton area A = ξ 0 η = 1 Soliton area A = ξ 0 η = 1 + 2κV 2 Conserved during propagation. Conserved during propagation. Independent of V V dependent ( 1 + 2κV 2 = 1/ cos(θ)). Soliton wave-vector Soliton wave-vector k = ( ( ( V, 1 2 η 2 V 2)) ( )) 1+2κη k = V 2, 1 1+2κη 2 1+2κV 2 2κ 1 1+2κV ( 2 ) V = 0 k z = 1 2 η2 V = 0 k z = 1 2κ 1 + 2κη2 1 ( ) 1+2κη V k = (, ) V k = 2 2κ, 1 2κ

32 Exact bright solitons Fundamental bright solitons u ξ κ = 10 3, θ 34 o V = 0 V = 15 y θ W y W W = W/cosθ x x

33 Dark solitons Dark soliton solution A general dark solution of the defocusing NNLS is found to be u(ξ, ζ) = u 0 (A tanh Θ + if ) exp [ i ( ) 1 4κu 2 1/2 ( κV 2 V ξ + ζ ) ] ( exp i ζ ) 2κ 2κ

34 Dark solitons Dark soliton solution A general dark solution of the defocusing NNLS is found to be u(ξ, ζ) = u 0 (A tanh Θ + if ) exp where Θ = u 0A (ξ + W ζ) (1 + 2κW 2 ) 1/2 [ i ( ) 1 4κu 2 1/2 ( κV 2 V ξ + ζ ) ] ( exp i ζ ) 2κ 2κ

35 Dark solitons Dark soliton solution A general dark solution of the defocusing NNLS is found to be u(ξ, ζ) = u 0 (A tanh Θ + if ) exp [ i ( ) 1 4κu 2 1/2 ( κV 2 V ξ + ζ ) ] ( exp i ζ ) 2κ 2κ where Θ = u 0A (ξ + W ζ) and W = V V 0 (1 + 2κW 2 ) 1/2 is a net 1 + 2κV V 0 transverse velocity involving V (choice of reference) and V 0 (intrinsic grey soliton velocity), given by V 0 = u 0 F [1 (2 + F 2 )2κu 2 0 ]1/2.

36 Dark solitons Dark solitons F and A are real constants, where F = ±(1 A 2 ). F = 0: black solitons. F > 0: grey solitons.

37 Dark solitons Dark solitons F and A are real constants, where F = ±(1 A 2 ). F = 0: black solitons. F > 0: grey solitons. In the paraxial limit, the NLS dark soliton is obtained: u(ξ, ζ) = u 0 (A tanh Θ + if ) exp where Θ = u 0 A [ξ + (V F u 0 )ζ]. ( iu 2 0ζ i 1 2 V 2 ξ + ζ ) 2κ

38 Dark solitons Properties of dark solitons Paraxial dark solitons exist for arbitrary values of background intensity, whereas Helmholtz dark solitons exist only for 4κu 2 0 < 1

39 Dark solitons Properties of dark solitons Paraxial dark solitons exist for arbitrary values of background intensity, whereas Helmholtz dark solitons exist only for 4κu 2 0 < 1 (2n 2 E 2 0 < n 0 ).

40 Dark solitons Properties of dark solitons Paraxial dark solitons exist for arbitrary values of background intensity, whereas Helmholtz dark solitons exist only for 4κu 2 0 < 1 (2n 2 E 2 0 < n 0 ). Paraxial grey solitons exist for any nonzero F < 1, while Helmholtz grey solitons exist only for 0 < F < F max = (1 4κu 2 0) 1/2 /(2κu 2 0) 1/2,

41 Dark solitons Properties of dark solitons Paraxial dark solitons exist for arbitrary values of background intensity, whereas Helmholtz dark solitons exist only for 4κu 2 0 < 1 (2n 2 E 2 0 < n 0 ). Paraxial grey solitons exist for any nonzero F < 1, while Helmholtz grey solitons exist only for 0 < F < F max = (1 4κu 2 0) 1/2 /(2κu 2 0) 1/2,where θ 0 = ±π/2 for F = F max.

42 Dark solitons Properties of dark solitons Paraxial dark solitons exist for arbitrary values of background intensity, whereas Helmholtz dark solitons exist only for 4κu 2 0 < 1 (2n 2 E 2 0 < n 0 ). Paraxial grey solitons exist for any nonzero F < 1, while Helmholtz grey solitons exist only for 0 < F < F max = (1 4κu 2 0) 1/2 /(2κu 2 0) 1/2,where θ 0 = ±π/2 for F = F max. The beam enlargement factor is 1/ 1 + 2κW 2 = sec(θ θ 0 ), where θ 0 = sec κV 2 0 and θ = sec κV 2.

43 Dark solitons Properties of dark solitons κ = Black solitons (top) with V = 25 and u 0 = 1. Grey solitons (bottom) with V = 10 and F = 0.8.

44 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods:

45 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods: Exact soliton solutions. Propagation invariants.

46 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods: Exact soliton solutions. Propagation invariants. Split-step FFT-BPM.

47 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods: Exact soliton solutions. Propagation invariants. Split-step FFT-BPM. Modified split-step method.

48 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods: Exact soliton solutions. Propagation invariants. Split-step FFT-BPM. Modified split-step method. An explicit highly accurate algorithm.

49 Numerical methods Numerical methods: Beam propagation schemes Analytical results of Helmholtz theory permit the testing of the numerical methods: Exact soliton solutions. Propagation invariants. Split-step FFT-BPM. Modified split-step method. An explicit highly accurate algorithm.

50 Numerical results - bright solitons Initial condition u(ξ, 0) = sech(ξ) exp( is 0 ξ) Exact paraxial soliton with V = S 0.

51 Numerical results - bright solitons Initial condition u(ξ, 0) = sech(ξ) exp( is 0 ξ) Exact paraxial soliton with V = S 0. When we observe the soliton propagation in the θ direction, the initial condition presents a perturbation of the soliton width of a factor 1 + 2κV 2. The asymptotic evolution can be described using IST techniques.

52 Numerical results - bright solitons Initial condition u(ξ, 0) = sech(ξ) exp( is 0 ξ) Exact paraxial soliton with V = S 0. When we observe the soliton propagation in the θ direction, the initial condition presents a perturbation of the soliton width of a factor 1 + 2κV 2. The asymptotic evolution can be described using IST techniques. When κη 2 << 1 S 0 = V 1 + 2κη κV 2 V = sin θ, 1 + 2κV 2 2κ the value of V is fixed by the initial condition and the asymptotic value of soliton area is sec θ = 1 + 2κV 2

53 Numerical results - bright solitons Initial condition u(ξ, 0) = sech(ξ) exp( is 0 ξ) S 0 = 5 S 0 = 10 S 0 = S 0 = 5 10 S 0 = S 0 = u m u m ζ ζ κ = κ =

54 Numerical results - bright solitons Initial condition u(ξ, 0) = sech(ξ) exp( is 0 ξ) A S 0 = 5 S 0 = 10 S 0 = ζ κ = 0.001

55 Numerical results - dark solitons Initial condition u(ξ, 0) = u 0 tanh(u 0 ξ) exp( is 0 ξ) Soliton width So=5 So=10 So= ζ Asymptotic values of the Helmholtz beam width are 1 + 2κV 2, where V = S 0 / 1 + 2κV 2. Fast convergence to the asymptotic solutions.

56 Numerical results - dark solitons Spontaneous generation of multiple dark solitons The initial condition u(ξ, 0) = u 0 tanh (u 0 aξ), 0 < a < 1, is used.

57 Numerical results - dark solitons Spontaneous generation of multiple dark solitons The initial condition u(ξ, 0) = u 0 tanh (u 0 aξ), 0 < a < 1, is used. The generation of 2N paraxial solitons is expected during evolution governed by the NLS, where N 0 is the largest integer that satisfies N 0 < 1/a.

58 Numerical results - dark solitons Spontaneous generation of multiple dark solitons The initial condition u(ξ, 0) = u 0 tanh (u 0 aξ), 0 < a < 1, is used. The generation of 2N paraxial solitons is expected during evolution governed by the NLS, where N 0 is the largest integer that satisfies N 0 < 1/a. The figure shows the paraxial (a) and Helmholtz (b) results for a = 0.26 and u 0 = 5.

59 Numerical results - dark solitons Spontaneous generation of multiple dark solitons V/(uF) Theory Non-paraxial Paraxial F Normalised transverse velocities of simulated grey solitons (symbols) and the corresponding analytical predictions (curves).

60 Soliton collisions Numerical results: conclusions are robust solutions that act as attractors of the nonlinear dynamics:

61 Soliton collisions Numerical results: conclusions are robust solutions that act as attractors of the nonlinear dynamics: Perturbed bright and dark solitons evolve asymptotically to exact Helmholtz solitons.

62 Soliton collisions Numerical results: conclusions are robust solutions that act as attractors of the nonlinear dynamics: Perturbed bright and dark solitons evolve asymptotically to exact Helmholtz solitons. The spontaneously generated fit the exact solutions presented.

63 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified:

64 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons.

65 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons. (Paraxial theory)

66 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons. (Paraxial theory) Almost exactly counterpropagating solitons.

67 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons. (Paraxial theory) Almost exactly counterpropagating solitons. (Paraxial theory)

68 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons. (Paraxial theory) Almost exactly counterpropagating solitons. (Paraxial theory) Collisions at intermediate angles.

69 Soliton collisions Helmholtz soliton collisions Three interaction geometries can be identified: Almost exactly copropagating solitons. (Paraxial theory) Almost exactly counterpropagating solitons. (Paraxial theory) Collisions at intermediate angles. (Helmholtz type of nonparaxiality)

70 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS.

71 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS. The simultaneous presence of the beams modulates the refractive index of the Kerr medium according to u 2 = u u u 1 u 2 + u 2 u 1 leading to three distinct effects:

72 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS. The simultaneous presence of the beams modulates the refractive index of the Kerr medium according to u 2 = u u u 1 u 2 + u 2 u 1 leading to three distinct effects: u j 2 u j (SPM).

73 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS. The simultaneous presence of the beams modulates the refractive index of the Kerr medium according to u 2 = u u u 1 u 2 + u 2 u 1 leading to three distinct effects: u j 2 u j (SPM). 2 u 3 j 2 u j (XPM).

74 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS. The simultaneous presence of the beams modulates the refractive index of the Kerr medium according to u 2 = u u u 1 u 2 + u 2 u 1 leading to three distinct effects: u j 2 u j (SPM). 2 u 3 j 2 u j (XPM). u 2 j u 3 j (Phase sensitive terms).

75 Soliton collisions Helmholtz soliton collisions Considering two beams u 1 (ξ, ζ) and u 2 (ξ, ζ) propagating at angles θ and θ to the ζ-axis and setting u(ξ, ζ) = u 1 (ξ, ζ) + u 2 (ξ, ζ) in the NNLS. The simultaneous presence of the beams modulates the refractive index of the Kerr medium according to u 2 = u u u 1 u 2 + u 2 u 1 leading to three distinct effects: u j 2 u j (SPM). 2 u 3 j 2 u j (XPM). u 2 j u 3 j (Phase sensitive terms). κ 2 u j ζ + i u j ζ u j 2 ξ 2 + [ u j 2 + (1 + h) u 3 j 2] u j = 0

76 Soliton collisions Helmholtz soliton collisions: geometry Geometry of Helmholtz soliton collisions in which interactions are dominated by the individual beam intensities. Top panel: copropagating solitons; bottom panel: counterpropagating solitons

77 Soliton collisions Helmholtz soliton collisions: numerical results Intensity profiles of two interacting solitons with equal amplitudes. Left panel: two co-propagating solitons; right panel: two counterpropagating solitons.

78 +++ Soliton collisions Helmholtz soliton collisions: numerical results e κ 2 u ζ θ Copropagation Counterpropagation + i u ζ u 2 ξ 2 + u 2 u = 0 + κ 2 uj ζ o + o o o 0.01 o o o o o o o oooooooooooo + ooo oooo o ooooo o o ++ + o o o o Copropagation (i) ocounterpropagation (i) o Copropagation (c) Counterpropagation (c) + 1e θ + o + + i u j ζ u j [ 2 ξ 2 + u j 2 + (1 + h) u 3 j 2] u j = 0 Magnitude of the trajectory phase shift as a function of the interaction angle θ for both copropagation and counterpropagation configurations (κ = 10 3 ).

79 Soliton collisions Helmholtz soliton collisions: numerical results Small amount of radiation is found in the numerical simulations of soliton collisions.

80 Soliton collisions Helmholtz soliton collisions: numerical results Small amount of radiation is found in the numerical simulations of soliton collisions. Small reshaping effect is found after collisions.

81 Soliton collisions Helmholtz soliton collisions: numerical results Small amount of radiation is found in the numerical simulations of soliton collisions. Small reshaping effect is found after collisions. Failure to obtain multisoliton solutions.

82 Soliton collisions Helmholtz soliton collisions: numerical results Small amount of radiation is found in the numerical simulations of soliton collisions. Small reshaping effect is found after collisions. Failure to obtain multisoliton solutions. Non-integrability of the NHE.

83 Soliton collisions Helmholtz soliton collisions: numerical results Small amount of radiation is found in the numerical simulations of soliton collisions. Small reshaping effect is found after collisions. Failure to obtain multisoliton solutions. Non-integrability of the NHE. are robust modes (individually and during interactions).

84 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality.

85 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles.

86 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions:

87 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions: Nonparaxial numerical beam propagation techniques.

88 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions: Nonparaxial numerical beam propagation techniques. Robustness of soliton solutions.

89 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions: Nonparaxial numerical beam propagation techniques. Robustness of soliton solutions. Soliton collisions at arbitrary angles.

90 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions: Nonparaxial numerical beam propagation techniques. Robustness of soliton solutions. Soliton collisions at arbitrary angles. Usefulness of the exact solutions to describe nonlinear beam propagation.

91 Conclusions & directions CONCLUSIONS Helmholtz type of nonparaxiality. Kerr soliton theory generalised to non-vanishing propagation angles. Exact bright and dark soliton solutions: Nonparaxial numerical beam propagation techniques. Robustness of soliton solutions. Soliton collisions at arbitrary angles. Usefulness of the exact solutions to describe nonlinear beam propagation.

92 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons.

93 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons. Helmholtz-Manakov soliton families: solutions, properties and interactions.

94 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons. Helmholtz-Manakov soliton families: solutions, properties and interactions. Higher dimensional Helmholtz modes (stripes,rings, vortices, etc.).

95 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons. Helmholtz-Manakov soliton families: solutions, properties and interactions. Higher dimensional Helmholtz modes (stripes,rings, vortices, etc.). in alternative media.

96 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons. Helmholtz-Manakov soliton families: solutions, properties and interactions. Higher dimensional Helmholtz modes (stripes,rings, vortices, etc.). in alternative media. Medium inhomogeneities: interfaces and devices.

97 Conclusions & directions CURRENT AND FUTURE WORK Interaction of Helmholtz dark solitons. Helmholtz-Manakov soliton families: solutions, properties and interactions. Higher dimensional Helmholtz modes (stripes,rings, vortices, etc.). in alternative media. Medium inhomogeneities: interfaces and devices.