Analysis of bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg gratings

Xin-Hong Jia; Zheng-Mao Wu; Guang-Qiong Xia

doi:10.1364/OPEX.12.002945

1. Introduction

Nonlinear Bragg grating (NLBG) exhibits particular properties near the edge of “photonic band gap” (PBG). Outside PBG, slow Bragg soliton related to great anomalous group velocity dispersion (GVD) has been used for ultra-short optical pulse compression successfully [1,2,3]; inside PBG, optical bistability occurs [4] when a positive feedback loop (among inner optical intensity, nonlinear refractive index, and Bragg resonance) causes the Bragg wavelength shift to longer or shorter wavelength, which depends on the increasing or decreasing directions of incident radiation. The optical bistability has wide applications in optical signal processing, optical memory, optical limiting, optical switching and optical gate operations, etc [5–7]. Many efforts have been made to offer additional feasibility of NLBG, such as the switching-on threshold, the switching time, the on-off switching ratio and dynamic stability, etc. The used technologies mainly include spatial taper, phase shift, chirp and nonlinear refractive index axial varying, etc [8–14]. For the nonuniform gratings, earlier studies mostly were carried out under the quasi-continuous-wave (CW) hypothesis [8–12], but it is essential to study its dynamic properties since the self-pulsing and chaos may happen in the upper branch of hysteresis due to modulation instability (MI) [15]. Liu et al have analyzed the dynamic properties of chirped NLBG [13], comparisons of dynamic properties between uniform and phase shifted NLBG have also been performed by H. Lee et al [14]. However, report on dynamic stability of linearly tapered nonlinear Bragg grating (LT-NLBG) has not yet been found, and the physical mechanisms of LT-NLBG steady characteristics also need to be studied further. In this paper, the bistable steady characteristics and dynamic stability of LT-NLBG have been analyzed based on the nonlinearly coupled mode equations (NLCME). The numerical simulations show that higher on-off switching ratio and broader stability regime can be achieved for negative-tapered grating compared with the positive-tapered grating and free-tapered grating, which may provide an instructive insight from a practical viewpoint.

2. Theoretical model

2.1 Nonlinearly coupled mode equations

The axial distribution of refractive index n can be described by

n (z) = n_{0} + n_{1} (z) \cos [\frac{2 π}{Λ} z + ϕ (z)] + n_{2} {∣ E (z) ∣}^{2}

where E is the inner electric field of grating, Λ is the grating period, φ is the spatial phase shift, n ₀, n ₁ and n ₂ denote the effective mode refractive index, linear refractive index modulation amplitude, and nonlinear refractive index coefficient, respectively.

The inner electric field E can be expressed by the sum of two terms

E = A_{f} \exp [i (β_{0} z - ϖ t)] + A_{b} \exp [- i (β_{0} z + ϖ t)]

where ω is the carrier angular frequency, t is the time, β ₀=π/Λ is the Bragg wave number, A _f and A _b represent the slowly varying amplitude of forward and backward wave, respectively.

Substituting Eqs. (1) and (2) into the wave equations, and assuming that the loss and material dispersion can be neglected (in this paper, the nonlinear medium of the NLBG is assumed to be Erbium-doped fiber without pump. Even though its loss and material dispersion coefficients near 1.55µm are large, the total loss and material dispersion are negligible due to very short length selected in calculations), the response time of material is fast enough, as well as the carrier wavelength is close to Bragg wavelength, one can obtain the following nonlinearly coupled mode equations [2]

\frac{\partial A_{f}}{\partial z} + \frac{1}{v_{g}} \frac{\partial A_{f}}{\partial t} = i [δ A_{f} + Γ ({∣ A_{f} ∣}^{2} + 2 {∣ A_{b} ∣}^{2}) A_{f} + κ A_{b}]

\frac{\partial A_{b}}{\partial z} - \frac{1}{v_{g}} \frac{\partial A_{b}}{\partial t} = - i [δ A_{b} + Γ ({∣ A_{b} ∣}^{2} + 2 {∣ A_{f} ∣}^{2}) A_{b} + κ^{*} A_{f}]

where v _g is the light group velocity in the grating medium, δ, Γ and κ account for the detuning, nonlinear coefficient, and coupling coefficient, respectively, which can be expressed by

δ = β - β_{0} = n_{0} \frac{ϖ}{c} - β_{0}, Γ = \frac{2 π n_{2}}{λ_{0}}, κ (z) = \frac{π n_{1} (z) η}{λ_{0}} \exp [i ϕ (z)]

where c is the light velocity in vacuum, λ ₀=2 n ₀ Λ is the Bragg wavelength, η is the confinement factor. For LT-NLBG with no spatial chirp and phase-shift (φ(z)=0), κ can be written as [11]

κ (z) = κ_{0} [1 + Δ κ (z - L ⁄ 2) ⁄ L]

where L is the total length of grating, κ ₀ is the coupling coefficient of the grating center, and Δκ characterizes the variation slope of coupling coefficient.

The boundary conditions are given by

z = 0 : A_{f} (0, t) = A_{i} (0, t), A_{r} (0, t) = A_{b} (0, t)

z = L : A_{b} (L, t) = 0, A_{r} (L, t) = A_{f} (L, t)

where A _i, A _r and A _t are the slowly varying amplitudes of the incident, reflected and transmitted wave, respectively.

Setting the partial derivatives with respect to t in Eqs. (3a) and (3b) equal to zeros, the axial evolving equations of slowly varying amplitude under steady-state can be deduced. As a result, the bistable steady characteristics of LT-NLBG can be analyzed numerically by means of the fourth-order Runge-Kutta method together with boundary conditions.

2.2 Linear stability analysis

In practice, noise fluctuation exists inevitably. Even though the outer fluctuation is not concerned, the inner fluctuation may be amplified and eventually yields the unintentional self-pulsing and chaos associated with modulation instability. The dynamic stability of NLBG can be examined by the standard linear stability analysis as follows.

The slowly varying amplitude can be decomposed into real and imaginary parts as

A_{f} = u + iw, A_{b} = v + iy

If A _f0=u ₀+iw ₀ and A _b0=v ₀+iy ₀ are used to denote the steady solutions of NLCME, the small fluctuations can be added by

A_{f} (z, t) = [u_{0} (z) + g_{u} (z, t)] + i [w_{0} (z) + g_{w} (z, t)]

A_{b} (z, t) = [v_{0} (z) + g_{v} (z, t)] + i [y_{0} (z) + g_{y} (z, t)]

where g _u and g _w are the fluctuation real and imaginary parts of forward wave, g _v and g _y are the fluctuation real and imaginary parts of backward wave.

Substituting Eqs. (8a) and (8b) into Eqs. (3a) and (3b), and neglecting the contributions of second and higher-order terms of fluctuation, after some simplifications, the temporal evolving equations of fluctuation are found to satisfy

\frac{\partial g_{u}}{\partial z} + \frac{1}{v_{g}} \frac{\partial g_{u}}{\partial t} = A_{1} g_{u} + B_{1} g_{w} + C_{1} g_{v} + D_{1} g_{y}

\frac{\partial g_{w}}{\partial z} + \frac{1}{v_{g}} \frac{\partial g_{w}}{\partial t} = A_{2} g_{u} + B_{2} g_{w} + C_{2} g_{v} + D_{2} g_{y}

\frac{\partial g_{v}}{\partial z} - \frac{1}{v_{g}} \frac{\partial g_{v}}{\partial t} = A_{3} g_{u} + B_{3} g_{w} + C_{3} g_{v} + D_{3} g_{y}

\frac{\partial g_{y}}{\partial z} - \frac{1}{v_{g}} \frac{\partial g_{y}}{\partial t} = A_{4} g_{u} + B_{4} g_{w} + C_{4} g_{v} + D_{4} g_{y}

where the coefficients A _j, B _j, C _j, D _j (j=1, 2, 3, 4) are related to the steady solutions and structure parameters of NLBG, defined by

A_{1} = - f_{1} (u_{0}, w_{0}), B_{1} = - f_{2} (w_{0}, u_{0}, v_{0}, y_{0}),

A_{2} = f_{2} (u_{0}, w_{0}, v_{0}, y_{0}), B_{2} = - A_{1},

A_{3} = D_{2}, B_{3} = 2 f_{1} (w_{0}, y_{0}) + κ^{*},

A_{4} = - 2 f_{1} (u_{0}, v_{0}) - κ^{*}, B_{4} = C_{1},

with

f_{1} (ξ, η) = 2 Γ ξ η

f_{2} (ξ, η, ζ, χ) = δ + Γ (3 ξ^{2} + η^{2} + 2 ζ^{2} + 2 χ^{2})

The special solutions of Eqs. (9a)–(9d) take the form of

g_{u} = u_{1} (z) \exp (st), g_{w} = w_{1} (z) \exp (st),

where s=α+iσ is the eigenvalue.

Substituting Eq. (12) into Eqs. (9a)–(9d), the resulting ordinary differential equations are

\frac{d u_{1}}{dz} - (A_{1} u_{1} + B_{1} w_{1} + C_{1} v_{1} + D_{1} y_{1}) = - \frac{s}{v_{g}} u_{1}

\frac{d w_{1}}{dz} - (A_{2} u_{1} + B_{2} w_{1} + C_{2} v_{1} + D_{2} y_{1}) = - \frac{s}{v_{g}} w_{1}

- \frac{d v_{1}}{dz} + (A_{3} u_{1} + B_{3} w_{1} + C_{3} v_{1} + D_{3} y_{1}) = - \frac{s}{v_{g}} v_{1}

- \frac{d y_{1}}{dz} + (A_{4} u_{1} + B_{4} w_{1} + C_{4} v_{1} + D_{4} y_{1}) = - \frac{s}{v_{g}} v_{1}

The boundary conditions of inner fluctuation can be written as

g_{u} (0, t) = g_{w} (0, t) = g_{v} (L, t) = g_{y} (L, t) = 0

From Eqs. (12) and (14), the above boundary conditions become

u_{1} (0) = w_{1} (0) = v_{1} (L) = y_{1} (L) = 0

The numerical solutions of eigenvalue s can be obtained by applying finite-difference method from Eqs. (13a)–(13d) and (15). The fundamental algorism is introduced briefly below: Splitting the grating length L into N+1 equal sections, then the spatial point z can be approximated to z=z _n=nh (n=0,1, … N, N+1). Here, h is the step size, n=0 and n=N+1 correspond to the boundary mesh points, respectively. Using the central and three point differences at boundary and interior mesh point, respectively, thus the discretization of NLCME can be performed and the problem is converted into the set of linear homogenous equations in terms of u _1n, w _1n, v _1n and y _1n. Setting the determinant of coefficient matrix equal to zero for finding non-trivial solution, then all of the eigenvalues can be obtained. It should be stressed, however, that owing to the local truncation error has the order of O (h ²), the deviation between the calculated value and exact value is larger for points which are far from the origin at curve in α/κ ₀ v _g□σ/κ ₀ v _g plane [16]. Even so, the precision is high enough in the vicinity of the origin. In fact, such an approach can provide sufficient stability information of system because the chief factor affecting system stability is the eigenvalue with largest real part.

Assuming that s _m=α_m+iσ_m is the eigenvalue with largest real part, then the system is unstable for α_m>0, and is asymptotically stable for α_m<0, while α_m=0 corresponds to the stability boundary. Furthermore, the value of α_m represents the decay (α_m<0) or growth (α_m>0) rate of fluctuation, and can be used to measure the high-amplitude relaxation oscillation (α_m<0) or self-pulsing (α_m>0) angular frequency of fluctuation.

3. Results and discussions

To facilitate description, the light intensity I, the detuning δ, and the eigenvalue s are normalized as I/I_c, δL, and s/κ ₀ v _g, respectively in following discussions, where I _c=4λ₀/3πn ₂ L is the critical input intensity [11]. The medium of LT-NLBG is supposed to be Erbium-doped fiber (self-focusing material) as mentioned before, and the used data in calculations are: n ₀=1.46, η=0.8, n ₂=6.9×10^-15m²/W [17], L=1cm, κ ₀=3 cm ^-1, and λ ₀=1.55µm. The following discussions will concentrate on how the linearly tapered parameter Δκ affects the bistable steady characteristics and dynamic stability of the system for different detuning.

Fig. 1. Steady-state transmission spectrum of LT-NLBG for different Δκ in linear case.

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Fig. 2. Steady-state input-output characteristics of LT-NLBG for three different Δκ with δL=2.5.

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To visualize the effect of linear taper on PBG (-κ ₀<δ<κ ₀ when Δκ=0), Fig. 1 shows the transmission spectrum of LT-NLBG for different Δκ in linear case under steady-state. It can be seen that the range of PBG is extended with the increasing magnitude of Δκ. Noting that the PBG is overlapped for the Δκ with the same magnitude but opposite sign, because no spatial chirp and phase shift is introduced, and the average coupling strength is identical (see Eq. (5)).

Fig. 3. Axial distribution of forward wave intensity with δL=2.5, where figures (a) and (b) are for Δκ=-1 and Δκ=1, respectively.

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Figure 2 shows the steady-state input-output characteristics of LT-NLBG for three different Δκ values. To be understood easily, the axial distribution of forward wave intensity for Δκ=-1 and Δκ=1 are also indicated in Fig. 3(a) and Fig. 3(b). The detuning is chosen to be δL=2.5, which is near an edge of PBG. From Fig. 1, it can be seen that, the bistable steady characteristics of LT-NLBG exhibits two interesting features: (I) Compared with the free-tapered grating, the negative-tapered grating increases the switching-on threshold, and the opposite occurs for the positive-tapered grating; (II) the negative-tapered grating decreases the transmittance of lower branch, and increases the transmittance of upper branch, i. e., increases the on-off switching ratio, while the positive-tapered grating is the opposite. The above results are consistent with the preceding work of Ref. [11], however, in which the interpretation has not been given. Here we present the reason of these features as follows: When the input intensity is low, then the nonlinear effect is weak, and the coupling term (the third term on right side) in Eqs. (3a) and (3b) takes the lead. Under the case, for the negative-tapered grating, the coupling coefficient decreases gradually along the axial direction, which makes the peak value of forward wave intensity also decreases (see Fig. 3(a)), then the transmittance is small. By contrast, for the positive-tapered grating, the reverse coupling coefficient varying (see Fig. 3(b)) causes a relatively large transmittance. When the input intensity exceeds the switching-on threshold to form bistability, the nonlinear effect is strong, and the distribution feedback term (the second term on right side) in Eqs. (3a) and (3b) takes the lead. Meantime, for the negative-tapered grating, with the increase of input intensity, the varying rate of peak value of forward wave intensity is more and more slow (see Fig. 3(a)), which makes the inner energy of grating diffuse gradually. As a result, a larger switching-on threshold is required to excite bistability, and the allocated energy at right facet (transmittance) is also larger. This explanation can be generalized to the positive-tapered grating by noting that the inner energy of grating is more convergent (see Fig. 3(b)), which leads to a smaller switching-on threshold and transmittance.

Fig. 4. Dependence of α_m on transmitted intensity for three values of taper parameter Δκ with δL=2.5.

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Figure 4 shows the dependence of α_m on the transmitted intensity, and the stable and unstable regime of bistability loop has been marked in Fig. 2 as well. In most cases, the lower branch is stable, and the middle-branch with negative slope is unstable, so we focus on the stability in upper branch. Under the system parameters given above, this diagram indicates that, for Δκ=0, the stable regime of upper branch is 0.14≤I _t/I _c≤0.21 (section AB), and for Δκ=-1, the stable regime is 0.14≤I _t/I _c≤0.31 (section AC), while the whole upper branch is unstable for Δκ=1. This means that the negative-tapered grating widens the stable region efficiently, but the positive-tapered grating is the opposite. In essence, modulation instability stems from the interplay of GVD and nonlinear effects (including SPM and XPM) [18]. It has been proven that NLBG has greater GVD near the edge of PBG. During this process, new lower and higher frequency components are generated for both forward and backward waves, which correspond to Stokes and anti-Stokes band, respectively, so it can be treated as four-wave mixing (FWM). Compared with the free-tapered grating, for the negative-tapered grating, the coupling strength between every frequency component is weaker due to its inner diffusing energy, so the fluctuation has slower growth rate (for α_m>0) or faster decay rate (for α_m<0) on the whole. Similar discussions can be adopted to analyze the positive-tapered grating. The above discussions imply that, the negative-tapered grating not only enlarges the stable regime, but also enhances the stability degree in a sense. In practice, if the grating is operated at pulse state, the negative-tapered grating can effectively weaken the amplitude of relaxation oscillation and timing-jitter.

Fig. 5. Steady-state input-output characteristics of LT-NLBG for three different Δκ with δL=0.

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When the detuning is tuned close to the centre of PBG, new features of the bistable steady characteristics and dynamic stability will be revealed. In Fig. 5, the steady-state input-output characteristics of LT-NLBG for three different Δκ has been plotted at δL=0. From the comparison of Fig. 2 and Fig. 5, it can be seen that, the on-off switching ratio and switching-on threshold are higher for smaller detuning once Δκ is fixed, and the whole upper branch is unstable. This tendency is difficult to change even in the presence of the tapered. The reason is that, in the case of smaller detuning, the frequency of incident light lies in the centre of reflection spectrum, and the transmitted light is “stopped” when the input intensity is lower, therefore, the required switching-on threshold to excite bistability is larger, and the on-off switching ratio is also larger. Meantime, if the grating operates at upper branch, the incident light is so strong that the inner energy changes drastically, which amplifies the fluctuation rapidly to result in severe instability.

Fig. 6. Mapping of the stability boundary for three different Δκ.

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Finally, to give an overall insight for the effects of the tapered on dynamic stability of LT-NLBG, for the first time to the authors’ knowledge, the stability boundary for three different Δκ has been mapped out in Fig. 6, where the upper and lower region of every curve represent the stable and unstable areas, respectively, and the horizontal dashed line corresponds to an edge of PBG with Δκ=0. From this diagram, it can be found that, with the increase of the tapered parameter Δκ from negative value, the stability boundary shifts upward, in other words, the stability regime is more and more narrow. Additionally, it is wider for the detuning near an edge inside the PBG. The above features are in good agreement with the previous subsections.

4. Conclusions

By using the NLCME, this paper has demonstrated the effect of the linearly tapered on bistable steady characteristics and dynamic stability of LT-NLBG. Numerical simulations show that when the incident frequency is tuned near an edge of PBG, the negative-tapered grating increases the switching-on threshold, enlarges the stability region, and enhances the stability degree compared with the free-tapered grating and the positive-tapered grating. Despite this paper deals only with linear stability analysis for NLBG with cosine refractive index modulation, further study shows that, for arbitrary nonlinear periodic structures, the influence of the linearly tapered on its dynamic stability exhibits similar behaviors.

Acknowledgments

The authors acknowledge the support from the Key Project of the Ministry of National Education of the People’s Republic of China (Grant No. 03140) and the Commission of Science and Technology of Chongqing City of the People’s Republic of China.

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Analysis of bistable steady characteristics and dynamic stability of linearly tapered nonlinear Bragg gratings

Abstract

1. Introduction

2. Theoretical model

2.1 Nonlinearly coupled mode equations

2.2 Linear stability analysis

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (33)

Optics Express