Plasmon Particle Array Lasers

Abstract

Diffractive arrays of strongly scattering noble metal particles coupled to a high-index slab of gain material can form the basis for plasmonic distributed feedback lasers. In this chapter, we discuss recent theoretical and experimental results describing the electromagnetic properties of these structures. Particularly, we investigate bandgap topology versus detuning between the plasmonic and Bragg resonances. We examine the complex dispersion relation, accounting for the fact that the particles are electrodynamic scatterers with radiation loss, that couple via a stratified medium system supporting guided modes. From the complex dispersion of this array we can deduce loss and outcoupling properties of the various Bloch modes, giving a handle on its lasing properties. From the experimental side, we show how to measure the dispersion relation using fluorescence microscopy, and systematically examine the array dispersion for realized plasmonic lasers as function of detuning between particle and lattice resonance. We conclude the chapter with a vision towards employing disordered, quasiperiodic and random plasmonic arrays to induce different optical responses, and experimentally demonstrate the exceptional robustness of lasing to disorder in these systems.

Appendices

Appendix A: 1D Green’s Function

First we define the normalized longitudinal wavenumbers $\zeta_{i}^{X} = \sqrt {\epsilon_{ri} - \xi_{X}^{2} }$ , with $X = TE/TM$ and subject to the radiation condition ${\text{Im}}\left\{ {\zeta_{i}^{X} } \right\}\, \ge \,0$. Then, the 1D Green’s function used in (8.8) is given by

$$g(\omega ,\,z,\,z^{{\prime }} )\, = \,\frac{1}{2}\frac{{Z_{2}^{X} }}{{D_{X} }}\left( {e^{{ik_{z}^{X} |z - z^{{\prime }} |}} \, + \,R_{1}^{X} e^{{ik_{z}^{X} (2h - (z + z^{{\prime }} ))}} \, + \,R_{3}^{X} e^{{ik_{z}^{X} (z + z^{{\prime }} )}} \, + \,R_{1}^{X} R_{3}^{X} e^{{ik_{z}^{X} (2h - |z - z^{{\prime }} |)}} } \right)$$

(8.9)

where h is the SU8 layer thickness and $k_{z}^{X} \, = \,k_{0} \zeta_{2}^{X}$, and

$$R_{i}^{X} \, = \,\frac{{Z_{i}^{X} - Z_{2}^{X} }}{{Z_{i}^{X} + Z_{2}^{X} }},\;\;\;{\kern 1pt} i = 1,\,3$$

(8.10)

$$Z_{i}^{TM} = \eta_{0} \frac{{\zeta_{1}^{TM} }}{{\epsilon_{ri} }},\quad Z_{i}^{TE} = \frac{{\eta_{0}}}{{\zeta_{i}^{TE}}}\quad i = 1,2,3$$

(8.11)

and

$$\left. {D_{X} \, = \,\frac{d}{d\xi }(1\, - \,R_{1}^{X} R_{3}^{X} e^{{2ik_{0} \zeta_{2}^{X} h}} )} \right|_{{\xi_{X} = k_{X} /k_{0} }}$$

(8.12)

Appendix B: Ewald Summation

The convergence of the infinite summation in (8.5) can be significantly accelerated by using the Ewald summation technique [50–52]. First, we write

$$C(\omega ,\,k_{x} ,\,k_{y} )\, = \,2A_{TE} \left( {S(k_{TE} )\, + \,\frac{{S_{xx} (k_{TE} )}}{{k_{TE}^{2} }}} \right)\, - \,2A_{TM} \frac{{S_{xx} (k_{TM} )}}{{k_{TM}^{2} }}$$

(8.13)

with $k_{TE} \, = \,k_{0} \xi_{TE}$, and $k_{TM} \, = \,k_{0} \xi_{TM}$, and

$$S(k)\, = \,\mathop {\lim }\limits_{{x^{{\prime }} y^{{\prime }} \to 0}} \mathop{\sum}\nolimits^{{\prime }} H_{0}^{(1)} (kR_{mn} )e^{{id(mk_{x} + n_{k} y)}} ,$$

(8.14)

$$S_{xx} (k) = \partial_{{x^{{\prime }} x^{{\prime }} }} S(k)$$

(8.15)

Where $R_{mn} = \sqrt {(x^{{\prime }} - md)^{2} + (y^{{\prime }} - nd)^{2} }$. The primed summation sign in (8.14) is used to exclude the $(m,\,n)\, = \,(0,\,0)$ term from the infinite two dimensional summation. The summation can also be written as

$$S(k)\, = \,\mathop {\lim }\limits_{{x^{\prime}y^{\prime} \to 0}} \sum {H_{0}^{(1)} } (kR_{mn} )e^{{id(mk_{x} + n_{k} y)}} \, - \,H_{0}^{(1)} (k\rho^{{\prime }} ),$$

(8.16)

where $\rho^{{\prime }} = \sqrt {x^{{{\prime }2}} + y^{{{\prime }2}} }$. The unprimed summation is used for infinite summation $(m,\,n)\, \in \,( - \infty ,\,\infty )\, \times \,( - \infty ,\,\infty )$. Next we replace the Hankel function by one of its integral representations

$$H_{0}^{(1)} (kR_{mn} ) = - \frac{2i}{\pi }\int_{0}^{\infty } {\frac{du}{u}} e^{{\left( {k^{2} /4u^{2} - R_{mn}^{2} u^{2} } \right)}}$$

(8.17)

Note that since $R_{mn}^{2} \, > \,0$, and assuming that $k^{2} \, > \,0$, to formally guarantee convergence of the integral representation in (8.17), we have to require that u goes to infinity along the line $argu\, = \, - \pi /4$. However, once we use this representation and derive an alternative, rapidly converging representation for the summation, we may apply Cauchy theorem and calculate the required integrals along a more convenient path. The semi-infinite integration path above is decomposed into two intervals, $0\, \to \,E$, and $E\, \to \,\infty$, where E is an arbitrarily chosen constant picked as a trade-off between fast convergence of $S_{1}$ and $S_{2}$. We define

$$S_{1} \, = \,\sum - \frac{2i}{\pi }\int\limits_{0}^{E} {\frac{du}{u}} e^{{\left( {k^{2} /4u^{2} - R_{mn}^{2} u^{2} } \right)}} e^{{id(mk_{x} + nk_{y} )}}$$

(8.18)

$$S_{2} \, = \,\mathop{\sum}\nolimits^{{\prime }} \frac{2i}{\pi }\int\limits_{E}^{\infty } {\frac{du}{u}} e^{{\left( {k^{2} /4u^{2} - R_{mn}^{2} u^{2} } \right)}} e^{{id(mk_{x} + nk_{y} )}}$$

(8.19)

$$C\, = \,\frac{2i}{\pi }\int\limits_{0}^{E} {\frac{du}{u}} e^{{\left( {k^{2} /4u^{2} - \rho^{{{\prime }2}} u^{2} } \right)}}$$

(8.20)

such that $S = S_{1} + S_{2} + C$. Note that as long as $E\, \gg \,k/2$, the integration in the summands of $S_{2}$ yields a Gaussian decay of the summands with respect to the summation indexes hence the summation over this part of the integral convergence rapidly. Similarly, the integration required to calculate C converges rapidly. The only issue left is the slow convergence of $S_{1}$ which is similar to the poor convergence of the original series. In this case, however, we are able to apply Poisson summation to accelerate the convergence. We obtain

$$S_{1} \, = \,\frac{4i}{{d^{2} }}\sum\limits_{p,q} {\frac{{e^{{k_{zpq}^{2} /4E^{2} }} }}{{k_{zpq}^{2} }}}$$

(8.21)

where ${\mathbf{k}}_{\rho pq} \, = \,(k_{x} ,\,k_{y} ) - 2\pi /d(p,\,q)$, and $k_{zpq}^{2} \, = \,k^{2} - {\mathbf{k}}_{\rho pq} \cdot {\mathbf{k}}_{\rho pq}$, $p,\,q\, \in \,( - \infty ,\,\infty )\, \times \,( - \infty ,\,\infty )$. The convergence of the summation $S_{1}$ in its new representation is Gaussian, therefore, practically only a few terms are required.

Finally, we have $S_{xx} = S_{1xx} + S_{2xx} + C_{xx}$ where

$$S_{1xx} = - \frac{4i}{{d^{2} }}\sum\limits_{p,q} {\frac{{e^{{k_{zpq}^{2} /4E^{2} }} }}{{k_{zpq}^{2} }}} \left( {k_{x} - \frac{2\pi }{d}p} \right)^{2}$$

(8.22)

$$S_{2xx} \, = \,\mathop{\sum}\nolimits^{{\prime }} \frac{4i}{\pi }\int\limits_{E}^{\infty } {du(1\, - \,2m^{2} d^{2} u^{2} )u\, \times \,e^{{\left( {k^{2} /4u^{2} - R_{mn}^{2} u^{2} } \right)}} e^{{id(mk_{x} + nk_{y} )}} }$$

(8.23)

$$C_{xx} = - \frac{4i}{\pi }\int_{0}^{E} d uue^{{\left( {k^{2} /4u^{2} - \rho^{{{\prime }2}} u^{2} } \right)}}$$

(8.24)