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Optics Letters

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  • Editor: Alan E. Willner
  • Vol. 36, Iss. 4 — Feb. 15, 2011
  • pp: 588–590
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Nonlinear polarization bistability in optical nanowires

Wen Qi Zhang, M. A. Lohe, Tanya M. Monro, and Shahraam Afshar V.  »View Author Affiliations


Optics Letters, Vol. 36, Issue 4, pp. 588-590 (2011)
http://dx.doi.org/10.1364/OL.36.000588


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Abstract

Using the full vectorial nonlinear Schrödinger equations that describe nonlinear processes in isotropic optical nanowires, we show that there exist structural anisotropic nonlinearities that lead to unstable polarization states that exhibit periodic bistable behavior. We analyze and solve the nonlinear equations for continuous waves by means of a Lagrangian formulation and show that the system has bistable states and also kink solitons that are limiting forms of the bistable states.

© 2011 Optical Society of America

Here, we analyze an aspect of the nonlinear interactions of the two polarizations of a mode in optical nanowires that occurs within the VNLS model, which has not been previously explored. We reveal the existence of anisotropic nonlinear behavior with respect to the two polarizations of a mode that is structural in origin. This anisotropy originates from the structure of the waveguide in the subwavelength regime, not from the anisotropy of the waveguide materials and so differs from that reported in [5

5. B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B 27, 956 (2010). [CrossRef]

]. Furthermore, this anisotropy leads to periodic bistable polarization states (defined below), properties of which we describe here.

For waveguides with isotropic materials such as glass, the nonlinear interactions of the two polarizations are usually described by the coupled NLS equations [1

1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

]:
Ajz+n=1in1n!βjnnAjtn=i(γj|Aj|2+γc|Ak|2)Aj+iγcAj*Ak2exp(2izΔβjk),
(1)
where j, k=1, 2(jk) are the two polarization modes, A1, A2 are the amplitudes of the corresponding fields, βjn are the nth order propagation constants, Δβjk=Δβkj is the linear birefringence, γj, γc, and γc are the effective nonlinear coefficients representing self phase modulation, cross phase modulation, and coherent coupling of the two polarization modes, respectively. The weak guidance approximation assumes that the effective mode areas of the two polarization modes are equal [1

1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

], leading to
γ1=γ2=3γc/2=3γc.
(2)

Figure 1 shows the γ coefficients calculated using Eq. (3), for elliptical waveguides surrounded by air, with chalcogenide glass (n=2.8, n2=1.1×1017m2/W at λ=1.55μm) as the host material. Evidently the equalities (2) do not generally hold for these γ values. Figure 1 also shows that γ1, γ2 are asymmetric with respect to the diagonal line where the fiber is circular. This indicates that elliptical shapes have higher γ values than circular shapes, and that in elliptical waveguides the γ values of the modes polarized along the major/minor axes are different, similar to the γ values in waveguides with anisotropic materials. As a consequence, the nonlinear behavior in birefringent waveguides includes anisotropic properties, which we refer to as structurally induced anisotropic nonlinearity.

We now solve Eq. (1) in the static case for general values of the γ coefficients, where Eq. (2) are not necessarily satisfied, in particular for γ1+γ22(γc+γc). We substitute Aj=Pjeiϕj for j=1,2 into Eq. (1), where Pj is the power of the field Aj with phase ϕj. For continuous waves we find that P1+P2=P0 is constant in z. Define the following dimensionless variables:
v=P1P0,θ=2Δϕ,τ=2γcP0z,a=Δβ12γcP0γcγ2γc,b=γ1+γ22γc2γc,
(4)
where Δϕ=ϕ1ϕ2+zΔβ12 is the phase difference between the two fields. Evidently b depends only on the nanowire parameters, whereas a also depends on the total power P0. From Eq. (1) we obtain
v˙dvdτ=v(1v)sinθ,
(5)
θ˙dθdτ=a+2bv+(12v)cosθ.
(6)
We choose initial values θ0=θ(0), v0=v(0) with 0<v0<1, where we regard τ as a “time” variable, then it can be shown from Eq. (5) that 0<v(τ)<1 for all τ>0, i.e. v always remains within the physical region. We solve Eqs. (5, 6), in terms of periodic elliptic functions by observing that Γ=av+bv2+v(1v)cosθ is a constant of the motion, enabling us to write v˙2=Q(v) where Q is the fourth degree polynomial Q(v)=v2(1v)2(Γ+avbv2)2. The minimum and maximum values of v, denoted vmin, vmax respectively, occur when v˙=0, i.e. at zeroes of Q. Since Q(0), Q(1)<0 and Q(v0)=v02(1v0)2sin2θ00 we deduce that Q generally has at least two real zeroes in the interval (0,1). We integrate v˙=Q(v) over the half-period in which v increases, in order to find τ as a function of v, and also the period T:
vminvduQ(u)=ττ0,T=2vminvmaxduQ(u),
(7)
where vmin=v(τ0). These integrals may be evaluated in terms of elliptic integrals of the first kind, see for example [6

6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

] (Sections 3.145, 3.147). In particular, T is expressible in terms of the complete elliptic integral K, and so can be written as an explicit function of a, b, v0, θ0, i.e. as a function of the waveguide parameters and the initial power and phase of the input fields. The precise formulas depend on the relative location of the roots of Q.

In addition to periodic solutions, Eqs. (5, 6) also have soliton solutions. These static kink solitons may be regarded as limits of the periodic bistable solutions, but with infinite period, and are found by solving the differential equation for θ with the boundary condition cosθ1 as |τ|. An explicit exact solution is
cosθ=1+2κ1(κ+1)cosh2κ(τc),
(14)
where κ=(a1)(a+2b1)/2(b1) and c is any constant. We have κ>0 for the values of a, b under consideration, namely, 1<a<2b1; however, at b=1 the solutions (14) do not exist. Given θ from Eq. (14), the power v is obtained from Eq. (6), and (5) is then also satisfied. The soliton (14) can propagate in time as a pulse over the length of the waveguide according to the evolution Eq. (1) and maintains its identity as a soliton provided that the boundary conditions remain intact.

As an example of bistable solutions, consider an elliptical waveguide made of chalcogenide glass with major/minor diameters equal to 640 and 620nm, with γ values calculated from Eq. (3) giving b=2.4. Consider initial values for the system that correspond to an unstable steady state, by setting P1(0)=p1+ΔP, P2(0)=p2 where p1=150.5W, p2=793.3W and ΔP (in units of watts) is a perturbation on p1, together with θ0=0. We plot the period T as a function of ΔP in Fig. 2. At ΔP=0 we have a=1.3 (hence 1<a<2b1 is satisfied) and v0=p1/(p1+p2)=(a1)/2(b1) corresponds to the unstable steady state solution (8), for which T is infinite. The insets in Fig. 2 show P1, P2, and cosΔϕ as functions of z, where the periodicity of the power functions and the bistability of cosΔϕ are evident. The polarization vector flips through an angle 20° over each period.

In conclusion, we have shown that within a full vectorial model of nonlinear processes in optical nanowires, we obtain γ coefficients that do not necessarily satisfy the relations (2). This results in structurally induced anisotropic nonlinearities for isotropic material-based linearly birefringent waveguides. The model allows continuous wave solutions of three types: steady state solutions, periodic (including bistable) solutions, and kink soliton solutions. The bistable states and the soliton solutions (14) exist only for the extended range of γ coefficients. Properties of the bistable states can in principle be utilized to construct photonic devices such as optical logical gates.

T. M. Monro acknowledges the support of an ARC Federation Fellowship, and M. Lohe acknowledges the support of the Larry Biedenharn Fellowship.

Fig. 1 Contour plots of γ1, γ2, 3γc/2, 3γc in units of (W·m)1 as functions of the major/minor diameters for elliptical waveguides.
Fig. 2 The period T as a function of ΔP. The insets show the periodic variation of the two polarization powers P1, P2, and cosΔϕ, as the pulse (for ΔP=100W) propagates along the fiber.
1.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

2.

S. Afshar V. and T. M. Monro, Opt. Express 17, 2298 (2009). [CrossRef]

3.

S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, Opt. Lett. 34, 3577 (2009). [CrossRef]

4.

F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. St. J. Russell, Phys. Rev. Lett. 105, 093904 (2010). [CrossRef] [PubMed]

5.

B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B 27, 956 (2010). [CrossRef]

6.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

OCIS Codes
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(130.4310) Integrated optics : Nonlinear
(190.1450) Nonlinear optics : Bistability
(190.3270) Nonlinear optics : Kerr effect
(190.4360) Nonlinear optics : Nonlinear optics, devices
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 6, 2010
Manuscript Accepted: January 7, 2011
Published: February 15, 2011

Citation
Wen Qi Zhang, M. A. Lohe, Tanya M. Monro, and Shahraam Afshar V., "Nonlinear polarization bistability in optical nanowires," Opt. Lett. 36, 588-590 (2011)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-588


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References

  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  2. S. Afshar V. and T. M. Monro, Opt. Express 17, 2298 (2009). [CrossRef]
  3. S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, Opt. Lett. 34, 3577 (2009). [CrossRef]
  4. F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. St. J. Russell, Phys. Rev. Lett. 105, 093904 (2010). [CrossRef] [PubMed]
  5. B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B 27, 956 (2010). [CrossRef]
  6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

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