Optical Bistability and Symmetry Breaking at Resonance Scattering of Light by a Finite Photonic Crystal with a Nonlinear Resonant Cavity

Abstract

Optical bistability and symmetry breaking under the conditions of plane wave incidence on a system of coupled photonic crystal microcavities containing an inclusion with Kerr susceptibility have been studied. The T-matrix modal method has been generalized for the case of nonlinear microcavities supporting one monopole mode. It has been shown that both phenomena considerably depend not only on external field strength but also on angle of incidence and photonic crystal size.

APPENDIX

The solution of the problem of plane wave scattering by a set of dielectric cylinders with permittivity ε will be described with the T-matrix model method [50].

We will follow the approach used in [49]. In this method, the electric field is expanded in cylindrical harmonics both inside and outside of the jth cylinder in a local coordinate system (see Fig. 14):

$$\begin{gathered} {{E}_{z}} = \sum\limits_{m = - \infty }^{ + \infty } {{{c}_{{jm}}}{{J}_{m}}(\sqrt \varepsilon {{k}_{0}}{{r}_{j}}){{e}^{{im{{\varphi }_{j}}}}},\quad {{r}_{j}} < {{R}_{j}},} \\ {{E}_{z}} = \sum\limits_{m = - \infty }^{ + \infty } {{{a}_{{jm}}}{{J}_{m}}({{k}_{0}}{{r}_{j}}){{e}^{{im{{\varphi }_{j}}}}}} \\ + \sum\limits_{m = - \infty }^{ + \infty } {{{b}_{{jm}}}H_{m}^{{(1)}}({{k}_{0}}{{r}_{j}}){{e}^{{im{{\varphi }_{j}}}}},\quad {{r}_{j}} > {{R}_{j}}.} \\ \end{gathered} $$

(4)

Fig. 14.

Global (xy) and local (x_j  y_j, x_l  y_l) coordinate systems.

Here, k₀ = ω/c, coefficients a_jm are the harmonic amplitudes of the field incident on the jth cylinder, and b_jm are the amplitudes of the scattered field. Amplitudes a_jm are presented in the form

$${{a}_{{j,m}}} = a_{{j,m}}^{{{\text{inc}}}} + a_{{j,m}}^{{{\text{rods}}}},$$

(5)

where

$$a_{{j,m}}^{{{\text{inc}}}} = {{( - 1)}^{m}}{{e}^{{i{{k}_{0}}{{R}_{j}}\sin ({{\theta }_{{{\text{inc}}}}} - {{\theta }_{j}}) - im{{\theta }_{{{\text{inc}}}}}}}}$$

(6)

is the amplitude of the plane wave incident on the system and (R_j, θ_j) are the coordinates of the jth cylinder relative to the global coordinate system. Amplitudes $a_{{j,m}}^{{{\text{rods}}}}$ represent the field scattered from the rest of cylinders (l ≠ j). Using the Graph formula, one can relate amplitudes $a_{{j,m}}^{{{\text{rods}}}}$ to scatt1ered wave amplitudes b_l, m (l ≠ j):

$$a_{{j,m}}^{{{\text{rods}}}} = \sum\limits_{q = - \infty }^{ + \infty } {\sum\limits_{l \ne j}^{} {{{b}_{{l,q}}}{{e}^{{i(m - q){{\theta }_{{j,i}}}}}}H_{{m - q}}^{{(1)}}(k{{r}_{{j,l}}}).} } $$

(7)

From (5)–(7), we obtain

$${{{\mathbf{a}}}_{j}} = {{{\mathbf{Q}}}_{j}} + \sum\limits_{l \ne j}^{} {{{{\hat {T}}}_{{j,l}}}{{{\mathbf{b}}}_{l}},} $$

(8)

where

$${{a}_{{j,m}}} = {{({{{\mathbf{a}}}_{j}})}_{m}},$$

(9)

$${{b}_{{j,m}}} = {{({{{\mathbf{b}}}_{j}})}_{m}},$$

(10)

$${{({{{\mathbf{Q}}}_{j}})}_{m}} = {{( - 1)}^{m}}{{e}^{{i{{k}_{0}}{{R}_{j}}\sin ({{\theta }_{{{\text{inc}}}}} - {{\theta }_{j}}) - im{{\theta }_{{{\text{inc}}}}}}}},$$

(11)

$${{({{\hat {T}}_{{j,l}}})}_{{m,q}}} = {{e}^{{i(q - m){{\theta }_{{j,l}}}}}}H_{{m - q}}^{{(1)}}({{k}_{0}}{{r}_{{j,l}}}).$$

(12)

To close system (8), we use coupling between amplitudes (b_j) and (a_j) through the diagonal t-matrix:

$${{{\mathbf{b}}}_{j}} = {{\hat {t}}_{j}}{{{\mathbf{a}}}_{j}},$$

(13)

where

$$\begin{gathered} {{{\hat {t}}}_{{m,m}}}\, = \, - (\sqrt \varepsilon {{k}_{0}}{{J}_{m}}({{k}_{0}}R)J_{m}^{'}({{k}_{0}}\sqrt \varepsilon R)\, - \,{{k}_{0}}{{J}_{m}}({{k}_{0}}\sqrt \varepsilon R)J_{m}^{'}({{k}_{0}}R)) \\ \times \,{{(\sqrt \varepsilon {{k}_{0}}H_{m}^{{(1)}}({{k}_{0}}R)J_{m}^{'}({{k}_{0}}\sqrt \varepsilon R)\, - \,{{k}_{0}}{{J}_{m}}({{k}_{0}}\sqrt \varepsilon R)H_{m}^{{(1)'}}({{k}_{0}}R))}^{{ - 1}}}. \\ \end{gathered} $$

(14)

Thus, we arrived at a set of linear equations for amplitudes b_j:

$${{{\mathbf{b}}}_{j}} - \sum\limits_{l \ne j}^{} {{{{\hat {t}}}_{j}}{{{\hat {T}}}_{{j,l}}}{{{\mathbf{b}}}_{l}} = {{{\hat {t}}}_{j}}{{{\mathbf{Q}}}_{j}}.} $$

(15)

If the permittivity of the cylinders contains a Kerr nonlinear correction, ε = ε₀ + λ|E_z(r)|², formula (15) remains valid if coupling between amplitudes ${\mathbf{b}}_{j}^{{({\text{nonlin}})}}$ and ${\mathbf{a}}_{j}^{{({\text{nonlin}})}}$ through the nonlinear t-matrix is known:

$${\mathbf{b}}_{j}^{{({\text{nonlin}})}} = \hat {t}_{j}^{{({\text{nonlin}})}}{\mathbf{a}}_{j}^{{({\text{nonlin}})}}.$$

(16)

The existence of such coupling in the form of formula (6) is not obvious. Below, an explicit expression for $\hat {t}_{j}^{{({\text{nonlin}})}}$ will be derived under two conditions: (i) nonlinear cylinders are placed only in resonant cavities and (ii) a monopole mode is excited in each cavity in the case of light resonance scattering, when nonlinearity becomes a critical factor; that is, s-scattering by nonlinear cylinders dominates.

The equation for the E_z component of the field at the nonlinear cylinder in the cylindrical coordinate system has the form

$$\begin{gathered} \left[ {\frac{{{{\partial }^{2}}}}{{\partial {{r}^{2}}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{1}{{{{r}^{2}}}}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}\, + k_{0}^{2}({{\varepsilon }_{0}} + \lambda {\text{|}}{{E}_{z}}(r,\varphi ){{{\text{|}}}^{2}})} \right]{{E}_{z}}(r,\varphi ) = 0. \\ \end{gathered} $$

(17)

As before, the wave function is sought in the form of a series:

$${{E}_{z}}(r,\varphi ) = \sum\limits_{m \in Z}^{} {{{\psi }_{m}}(r){{e}^{{im\varphi }}}.} $$

(18)

After (18) is substituted into (17), the latter can approximately be represented as a set of uncoupled equations:

$$\psi _{m}^{{''}} + \frac{1}{r}\psi _{m}^{'} + k_{0}^{2}\left( {{{\varepsilon }_{0}} - \frac{{{{m}^{2}}}}{{{{r}^{2}}}}} \right){{\psi }_{m}} = 0,\quad m \ne 0,$$

(19a)

$$\psi _{0}^{{''}} + \frac{1}{r}\psi _{0}^{'} + k_{0}^{2}({{\varepsilon }_{0}} + \lambda {\text{|}}{{\psi }_{0}}{{{\text{|}}}^{2}}){{\psi }_{m}} = 0,\quad m = 0.$$

(19b)

The nonlinear term should be taken into account only in the equation for ψ₀, since |ψ_m| ≪ |ψ₀| at resonance s-scattering. In the absence of resonance, the nonlinearity term can be ignored. Thus, the solution of the problem reduces to the solution of Eq. (19b). This equation can be solved approximately using the perturbation theory with the consideration of the fact that λ|ψ₀|² ≪ ε₀ even at resonance scattering.

To this end, let us cast an auxiliary boundary-value problem:

$$\psi (r'') + \frac{1}{r}\psi (r)'\, + k_{0}^{2}({{\varepsilon }_{0}} + \lambda {\text{|}}\psi (r{\text{)}}{{{\text{|}}}^{2}})\psi (r) = 0,$$

(20)

with

$$\psi (r = 0) < \infty ,\quad \psi (r = R) = A.$$

In what follows, we put ψ ≡ ψ₀. The problem can be solved using Green’s function g(r):

$$rg{\kern 1pt} ''(r) + g{\kern 1pt} '\, + \left( {k_{0}^{2}r - \frac{{{{m}^{2}}}}{r}} \right)g(r) = - \delta (r - \rho ),$$

(21)

with

$$\begin{gathered} g(r = 0) < \infty , \\ g(r = R) = 0. \\ \end{gathered} $$

Equation (21) has an explicit solution:

$$\begin{gathered} g(r,\rho ) = \frac{{\pi {{J}_{m}}({{k}_{0}}{{r}_{ < }})}}{{2{{J}_{m}}({{k}_{0}}R)}}({{J}_{m}}({{k}_{0}}{{r}_{ < }}){{Y}_{m}}({{k}_{0}}R) \\ \, - {{J}_{m}}({{k}_{0}}R){{Y}_{m}}({{k}_{0}}{{r}_{ > }})), \\ \end{gathered} $$

(22)

where r_> = max(r, ρ) and r_< = min(r, ρ). Then, Eq. (20) has the formal solution

$$\begin{gathered} \tilde {\psi }(r) = \int\limits_0^R {g(r,\rho )[k_{0}^{2}{{\varepsilon }_{0}}\rho A + \lambda k_{0}^{2}\rho {\text{|}}\psi {{{\text{|}}}^{2}}\psi ]d\rho ,} \\ \tilde {\psi }(r) = \psi - A. \\ \end{gathered} $$

(23)

It is easy to check that

$$A\int\limits_0^R {g(r,\rho )k_{0}^{2}\rho d\rho = A\frac{{{{J}_{0}}(kr)}}{{{{J}_{0}}(kR)}} - A,} $$

(24)

where

$$k = \sqrt {{{\varepsilon }_{0}}} {{k}_{0}}.$$

Then, at λ = 0, we have

$$\tilde {\psi }(r) = A\left( {\frac{{{{J}_{0}}(kr)}}{{{{J}_{0}}(kR)}} - 1} \right).$$

(25)

Now, integral equation (23) can be written as

$$\psi (r) = \psi {{(r)}_{g}} + \lambda k_{0}^{2}\int\limits_0^R {\rho {\text{|}}\psi (\rho {\text{)}}{{{\text{|}}}^{2}}\psi (\rho )g(r,\rho )d\rho ,} $$

(26)

where

$$\psi {{(r)}_{g}} = A\frac{{{{J}_{0}}(kr)}}{{{{J}_{0}}(kR)}}.$$

In the limit λ → 0, Eq. (26) can be approximately solved by iteration. In the first Born approximation, we have

$$\begin{gathered} \psi (r) \approx \psi {{(r)}_{g}} + \lambda k_{0}^{2}\int\limits_0^R {\rho {\text{|}}{{\psi }_{g}}(\rho ){{{\text{|}}}^{2}}{{\psi }_{g}}(\rho )g(r,\rho )d\rho } \\ = \psi {{(r)}_{g}} + {{\psi }_{{{\text{corr}}}}}(r). \\ \end{gathered} $$

(27)

Correction term ψ_corr(r) can be reduced to the form

$$\begin{gathered} {{\psi }_{{{\text{corr}}}}}(r) = \lambda k_{0}^{2}\int\limits_0^R {\rho {\text{|}}{{\psi }_{g}}{{{\text{|}}}^{2}}{{\psi }_{g}}g(r,\rho )d\rho } \\ = \frac{{\pi \gamma }}{{2{{k}^{2}}}}{{J}_{0}}(kr)\left[ {\frac{{{{Y}_{0}}(kr)}}{{{{J}_{0}}(kR)}}\int\limits_0^{kR} {xJ_{0}^{4}(x)dx} } \right. \\ \left. {\, - \int\limits_0^R {x{{Y}_{0}}(x)J_{0}^{3}(x)dx} } \right] - \lambda k_{0}^{2}{{Y}_{0}}(kr)\int\limits_0^{kr} {xJ_{0}^{4}(x)dx} \\ \, + \lambda k_{0}^{2}{{J}_{0}}(kr)\int\limits_0^{kr} {x{{Y}_{0}}(x)J_{0}^{3}(x)dx,} \\ \end{gathered} $$

(28)

where

$$\gamma = \frac{{\lambda k_{0}^{2}{\text{|}}A{\text{|}}A}}{{J_{0}^{3}(kR)}}.$$

Note that ψ_corr(r = R) = 0. To sew together solution (27) inside the cylinder and the external solution, it is necessary to know $\psi _{{{\text{corr}}}}^{'}$(r = R). This derivative is found from Eq. (28):

$$\psi _{{{\text{corr}}}}^{'}(r = R) = - \frac{\lambda }{{{{\varepsilon }_{0}}}}\frac{{{\text{|}}A{{{\text{|}}}^{2}}A}}{{RJ_{0}^{4}(kR)}}\int\limits_0^{kR} {xJ_{0}^{4}(x)dx.} $$

(29)

Then, the approximate solution inside the cylinder takes the form

$$\psi (r) = {{\psi }_{g}}(r) + {{\psi }_{{{\text{corr}}}}}(r),$$

$$\psi (r = R) = A,$$

$$\psi '(r = R) = k\frac{{J_{0}^{'}(kR)}}{{{{J}_{0}}(kR)}}A - \varkappa {\text{|}}A{{{\text{|}}}^{2}}A,$$

(30)

where

$$\varkappa = \frac{\lambda }{{{{\varepsilon }_{0}}}}\frac{1}{{RJ_{0}^{4}(kR)}}\int\limits_0^{kR} {xJ_{0}^{4}(x)dx.} $$

Now we sew together the internal solution and external solution

$$\psi (r) = {{a}_{0}}{{J}_{0}}({{k}_{0}}r) + {{b}_{0}}H_{0}^{{(1)}}({{k}_{0}}r),$$

to obtain

$${{b}_{0}} = {{t}_{0}}{{a}_{0}} - \frac{{\varkappa {\text{|}}A{{{\text{|}}}^{2}}A}}{{{{k}_{0}}H_{0}^{{(1)'}}({{k}_{0}}R) - k\frac{{J_{0}^{'}(kR)}}{{{{J}_{0}}(kR)}}H_{0}^{{(1)}}({{k}_{0}}R)}},$$

(31)

where t₀ is the element of diagonal t-matrix (14) at resonance scattering.

To eliminate parameter A from expression (31), one can take its value from the linear problem (λ = 0):

$$A \approx {{b}_{0}}\left[ {\frac{{{{J}_{0}}({{k}_{0}}R)}}{{{{t}_{0}}}} + H_{0}^{{(1)}}({{k}_{0}}R)} \right] + O(\Lambda ).$$

(32)

Then,

$${{t}_{0}}{{a}_{0}} \approx {{b}_{0}}\left( {1 + \Lambda {\kern 1pt} |{\kern 1pt} {{b}_{0}}{\kern 1pt} {{|}^{2}}} \right),$$

(33)

where

$$\Lambda = \frac{{\varkappa {{{\left| {\frac{{{{J}_{0}}({{k}_{0}}R)}}{{{{t}_{0}}}} + H_{0}^{{(1)}}({{k}_{0}}R)} \right|}}^{2}}\left( {\frac{{{{J}_{0}}({{k}_{0}}R)}}{{{{t}_{0}}}} + H_{0}^{{(1)}}({{k}_{0}}R)} \right)}}{{{{k}_{0}}H_{0}^{{(1)'}}({{k}_{0}}R) - k\frac{{J_{0}^{'}(kR)}}{{{{J}_{0}}(kR)}}H_{0}^{{(1)}}({{k}_{0}}R)}}.$$

Thus, formula (33) sets necessary coupling between amplitudes a₀ and b₀. As a result, formula (15) in the nonlinear case takes the form

$$\left( {1 + \Lambda \sum\limits_{k \in {{N}_{k}}}^{} {{\text{|}}{{b}_{{k0}}}{{{\text{|}}}^{2}}{{{\hat {P}}}_{k}}} } \right){{{\mathbf{b}}}_{j}} - \sum\limits_{l \ne j}^{} {{{{\hat {t}}}_{j}}{{{\hat {T}}}_{{jl}}}{{{\mathbf{b}}}_{l}} = {{{\hat {t}}}_{j}}{{{\mathbf{Q}}}_{j}},} $$

(34)

where

$${{({{\hat {P}}_{k}}{{b}_{j}})}_{m}} = {{\delta }_{{jk}}}{{\delta }_{{0m}}}{{b}_{{j0}}},$$

and N_k is a set of nonlinear cylinders. In essence, Eq. (34) represents a closed set of equations for amplitudes b_j. In computer calculations, azimuthal number m takes a finite series of values: m ∈ [–M, –M + 1, …, +M].