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Literature Summary: Instabilities, solitons, and rogue
waves in PT-coupled nonlinear waveguides
J. Schoenfeld
Southern Met...
Goals
Study rogue waves in PT-symmetric optical models based on
dual-core couplers
Examine how the presence of balanced di...
Outline
Introduction to PT-symmetry
The model
CW solutions
Modulational instability
J. Schoenfeld Southern Methodist Unive...
Introduction: PT-Symmetry in quantum mechanics
ˆP: ˆx → −ˆx, ˆp → −ˆp, ˆT: ˆx → ˆx, ˆp → −ˆp, ˆi → −ˆi
ˆH = ˆp2
2 + V (ˆx)...
The model
Consider the following system of linearly coupled NLSEs for ψ1 and ψ2:
i
∂ψ1
∂z
= −
∂2ψ1
∂x2
+ χ1 |ψ1|2
+ χ |ψ2|...
The model continued
i
∂ψ1
∂z
= −
∂2ψ1
∂x2
+ χ1 |ψ1|2
+ χ |ψ2|2
ψ1 + iγψ1 − ψ2,
i
∂ψ2
∂z
= −
∂2ψ2
∂x2
+ χ |ψ1|2
+ χ1 |ψ2|2
...
Model: Parametrization
In optics, this situation can be achieved with two lossy waveguides
coupled in parallel, where one ...
Model: Parametrization
We introduce the following convenient parameterization,
γ = sin(δ), 0 < δ <
π
2
. (3)
We seek PT-sy...
Model: Parameterization
To ensure conventional symmetry is not broken, we do the following:
If ψ1(x, z), ψ2(x, z), is a so...
CW Solutions
We are looking for a solution of the form
ψ1 = ρeikx−iΩz
. (5)
Thus,
∂ψ1
∂z
= iΩρeikx−iΩz
,
∂ψ1
∂x
= ikρeikx−...
Modulational instability
Ansatz:
ψj = ρ ei(−1)j δ/2
+ ηj e−i(βz−κx)
+ νj ei(βz−κz)
eikx−ibz
, (8)
for j = 1, 2 and |ηj | /...
Modulational instability
Observe, there are three sources of MI:
χ1 + χ < 0
Stems from β1(k) due to long-wavelengths excit...
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  1. 1. Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear waveguides J. Schoenfeld Southern Methodist University, Department of Mathematics December 10, 2013 J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 1 / 12
  2. 2. Goals Study rogue waves in PT-symmetric optical models based on dual-core couplers Examine how the presence of balanced dissipation and gain affects the modulational instability of the background and possibly the creation of waves localized in space and time J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 2 / 12
  3. 3. Outline Introduction to PT-symmetry The model CW solutions Modulational instability J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 3 / 12
  4. 4. Introduction: PT-Symmetry in quantum mechanics ˆP: ˆx → −ˆx, ˆp → −ˆp, ˆT: ˆx → ˆx, ˆp → −ˆp, ˆi → −ˆi ˆH = ˆp2 2 + V (ˆx) is PT-symmetric if [ˆH, ˆP ˆT] = 0, which implies that VR(−ˆx) = VR(ˆx), and VI (−ˆx) = −VI (ˆx). Proof. Assume (ˆH ˆP ˆT)f = (ˆP ˆT ˆH)f , for any f. Then observe ˆTf (x) = f ∗ (x), ˆHf (x) = [ ˆp2 2 + V (ˆx)]f (x), ˆP ˆTf (x) = f ∗ (−x), ˆT ˆHf (x) = [ ˆp2 2 + V ∗ (ˆx)]f ∗ (x), ˆH ˆP ˆTf (x) = [ ˆp2 2 + V (ˆx)]f ∗ (−x). ˆP ˆT ˆHf (x) = [ ˆp2 2 + V ∗ ( ˆ−x)]f ∗ (−x). Therefore, V (ˆx) = V ∗ (−ˆx). Thus, ˆH and ˆP ˆT share eigenfunctions. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 4 / 12
  5. 5. The model Consider the following system of linearly coupled NLSEs for ψ1 and ψ2: i ∂ψ1 ∂z = − ∂2ψ1 ∂x2 + χ1 |ψ1|2 + χ |ψ2|2 ψ1 + iγψ1 − ψ2, (1) i ∂ψ2 ∂z = − ∂2ψ2 ∂x2 + χ |ψ1|2 + χ1 |ψ2|2 ψ2 − iγψ2 − ψ1. (2) Describes a set of two parallel planar waveguides, where z and x are the dimensionless propagation and transverse coordinates. Initial condition: optical beam shone into waveguides input at z = zi . Also, describes a dual-core fiber coupler, where here x represents the temporal variable. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 5 / 12
  6. 6. The model continued i ∂ψ1 ∂z = − ∂2ψ1 ∂x2 + χ1 |ψ1|2 + χ |ψ2|2 ψ1 + iγψ1 − ψ2, i ∂ψ2 ∂z = − ∂2ψ2 ∂x2 + χ |ψ1|2 + χ1 |ψ2|2 ψ2 − iγψ2 − ψ1. The two equations are coupled nonlinearly by the cross-phase modulation (XPM) ≈ χ, linearly by the last term. (Here, the coupling constant is scaled to be equal to 1). γ > 0 describes the mutual balance in gain in Eq. (1) and dissipation in Eq. (2) , i.e. the PT symmetry. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 6 / 12
  7. 7. Model: Parametrization In optics, this situation can be achieved with two lossy waveguides coupled in parallel, where one is being pumped by the external source of gain-providing atoms. Though the first core carries the gain, the linear coupling between this core and its lossy partner cause the zero state to be neutrally stable, resulting in the propagation of linear waves. In this case, modes do not arise spontaneously, but can be excited by input beams. This situation occurs when the gain/loss term is small compared to the linear coupling through which the core with gain transfers energy to the lossy one. Here, this occurs when γ ≤ 1. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 7 / 12
  8. 8. Model: Parametrization We introduce the following convenient parameterization, γ = sin(δ), 0 < δ < π 2 . (3) We seek PT-symmetric and antisymmtric solutions to Eqs. (1) and (2) such that ψ2(x, z) = ±e±iδ ψ1(x, z). (4) Then ψ1 is such that i ∂ψ1 ∂z = − ∂2ψ1 ∂x2 + (χ1 + χ) |ψ1|2 ψ1 + iγψ1 − ψ2 = − ∂2ψ1 ∂x2 + (χ1 + χ) |ψ1|2 ψ1 + isin(δ) e±iδ ψ1 = − ∂2ψ1 ∂x2 + (χ1 + χ) |ψ1|2 ψ1 cos(δ)ψ1 J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 8 / 12
  9. 9. Model: Parameterization To ensure conventional symmetry is not broken, we do the following: If ψ1(x, z), ψ2(x, z), is a solution to Eq. (1) and (2), then so is ψ2(x, −z), ψ1(x, −z) . This is equivalent to δ → π − δ. So now, 0 < δ < π. The values δ and π − δ correspond to the two different solutions with the same gain and dissipation. PT-symmetric solutions : 0 ≤ δ ≤ π 2 PT-antisymmetric solutions : π 2 ≤ δ ≤ π J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 9 / 12
  10. 10. CW Solutions We are looking for a solution of the form ψ1 = ρeikx−iΩz . (5) Thus, ∂ψ1 ∂z = iΩρeikx−iΩz , ∂ψ1 ∂x = ikρeikx−iΩz , ∂2ψ1 ∂x2 = (ik)2 ρeikx−iΩz , ... → Ω = k2 + (χ1 + χ)ρ2 cos(δ). (6) So generalizing this, the CW solutions (up to a trivial phase shift) to Eqs. (1) and (2) are ψ (cw) j = ρeikx−ibz+i(−1)j δ/2 , (7) where k is the background current and b = k2 + (χ1 + χ)ρ2 − cos(δ). Note, both cores have equal amplitudes of the fields. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 10 / 12
  11. 11. Modulational instability Ansatz: ψj = ρ ei(−1)j δ/2 + ηj e−i(βz−κx) + νj ei(βz−κz) eikx−ibz , (8) for j = 1, 2 and |ηj | / |νj | 1. The branches, β = β1,2(k), of the dispersion relation for the stability eigenvalues are as follows: β1(k) = 2kκ ± κ κ2 + 2ρ2(χ1 + χ), (9) β2(k) = 2kκ ± (κ2 + 2cosδ)(κ2 + 2cosδ + 2ρ2(χ1 + χ)) (10) Note: Due to the Galilean invariance of Eqs. (1) and (2), the instability is not affected by boost k. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 11 / 12
  12. 12. Modulational instability Observe, there are three sources of MI: χ1 + χ < 0 Stems from β1(k) due to long-wavelengths excitations. Not influenced by gain and dissipation. cos(δ) < max[0, ρ2(χ − χ1)] Stems from β2(k) due to the linear coupling between NLSEs. Gain and dissipation (δ = 0, π) is very different than a conservative system (δ = 0orδ = π). Occurs only due to imbalance of gain and loss Results in nearly homogeneous grow/ decay of the field in the waveguide with gain/ dissipation. J. Schoenfeld Southern Methodist University, Department of Mathematics ()Literature Summary: Instabilities, solitons, and rogue waves in PT-coupled nonlinear wavegDecember 10, 2013 12 / 12

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