in the past few decades , there has been a growing interest in the development of artificial nano - materials .
one class of material such as this is photonic crystals ( pcs ) , which are periodic dielectric or semiconductor structures that display a photonic band gap ( pbg ) in their electromagnetic wave transmission characteristics [ 1 - 4 ] . the existence of the pbg has inspired the design of various nano - optical and opto - electronic devices . since the formation of the pbg is due to the multiple bragg reflections within a pc , ordinary pcs are a combination of lossless materials for which the stop band has a small width , almost less than 25% of the central frequency .
this is due to the fact that a significant percentage of incident radiation passes through the periodic structure cores so that the reflected radiation is weaker than the transmitted radiation . because of this , bragg scattering comes into play when there are periodicities along the direction of propagation . to exhibit a noticeable pbg with fewer periodicities
metals would be one of the best options since they are more reflective than dielectric or semiconductor materials over a broad range of frequencies , due to their imaginary dielectric component that exists in even the optical and near infrared regions , where metals are dispersive and absorptive in these frequency regions [ 6 - 8 ] .
recently , considerable progress has been made in constructing these periodic arrangements which are called metallic or metallodielectric photonic crystals ( mpc / mdpc ) in one- , two- , and three - dimensional systems .
for instance , one - dimensional ( 1d ) mdpcs with metal thicknesses on the order of hundreds of nanometres has been proved transparent to visible light while blocking ultraviolet and infrared .
this characteristic can be used for example in laser safety glasses , uv protective films and flat panel displays .
two - dimensional ( 2d ) mpc structures are usually made of metallic rods or nanodisks that are periodically arranged on a waveguide layer .
waveguided mpcs have exhibited unique optical properties because of the strong coupling that exists between the particle plasmon resonance and waveguide mode .
such structures have a number of potential applications in biosensors   and all - optical switching , among others . in the case of three - dimensional ( 3d ) structures ,
there are also several applications such as high - efficiency light sources   and thermal photovoltaic power generation . besides these advantages
, pcs can be used for radiation suppression and emission enhancement below the electronic band gap and near a photonic band edge , respectively , where a pc functions as a reservoir for an exited light emitter or active medium such as an atom , a molecule or a quantum dot ( qd ) . according to fermi s  golden rule ,
the decay rate is proportional to the local density of states ( ldos ) that counts the number of electromagnetic modes available to the photons for emission into the environment .
thus , any modification in the ldos would lead to manipulation of the decay rate .
recently , the inhibition , enhancement and quantum interference ( qi ) effects of spontaneous emission ( se ) from the qds doped in 3d  pcs have been widely studied , both experimentally and theoretically [ 15 - 19 ] . controlling spontaneous emission by using quantum optics would lead to several interesting effects , such as optical gain enhancement   and photoluminescence enhancement , optical switching , quantum information processing   and electromagnetically induced transparency .
qi in a three- or multi - level atomic system can arise from the superposition of ses when electron transitions take place between the upper and lower levels . under certain circumstances
the initially excited atomic system may not decay to its ground level due to a cancellation of se by qi between atomic transition levels . due to this , dark states with zero absorption amplitude would appear causing the multi - level atomic system to act like a transparent medium , which has potential applications for optical switches and photonic devices . in this paper , the effects of electronic qi on the absorption coefficient of qds have been investigated for small and large concentration of the dopants ( qds ) .
we consider that the qds are four  energy level systems where the two upper levels are very close , coupled to a lower one via the same and single field continuum and damped by the mpc interaction .
recently , theoretical and experimental studies have shown that it is possible to make 3d mpcs that contain nano - sized metallic spheres which are transparent to visible and near infrared light . in this paper , we have used an ideal 3d isotropic mpc model made from metallic spheres of radius a with a frequency dependent refractive index n1( ) , which are arranged periodically in a background dielectric material with a constant refractive index ( n2 ) .
the dispersion relation and photonic band structure of this idealized theoretical 3d model was developed by s. john in the following references .
although many simplifying approximations have been used in this model , it is sufficient for our purpose as it leads to qualitatively correct physics and exhibits many of the observed and computed characteristics of 3d mpc which opens a band gap in the rage of visible frequencies . in many photonic band structure calculations related to mpcs
, the refractive index function for metallic materials is derived using the drude model , which offers an excellent fit to measured data over a wide frequency range . using this model , the refractive index for a metallic material
is expressed as where p and  are the frequency of the incident laser beam , the plasma frequency and the damping factor of the conduction - band electrons , respectively .
the plasma frequency is defined by , where n0 is the electron density , meff is the effective mass of the electron , e is the electron charge and 0 is the permittivity of the free space
. the damping rate ( which is also called the electron collision rate ) is the inverse of the mean electron collision time .
the parameter  is frequency independent and therefore absorption can be neglected at optical frequencies , since / = 1 .
( 1 ) , for frequencies below p , the local wave vector is imaginary and the metal behaves as a dispersive and absorptive environment .
nonetheless , if the diameter of the metallic spheres in our crystal structures can be chosen close to or smaller than the relevant skin depth of the corresponding metal , the em wave can be transmitted by tunnelling through the structure .
the position of the pbg can be determined by selection of proper thicknesses and refractive indices .
since we are interested in optical frequencies , the radius of the metallic spheres and lattice constant of the pc have been chosen in reduced units as a = 0.25c/p and l = 10.5c/p , respectively . in these parameters
,  is the reduced planck constant and p is the plasmon energy , i.e. p = p . a plot of the band structure of the pc consisting of spherical metal nanoparticles in a dielectric background ( n2 = 1.5 ) has been illustrated in fig .
plot of the bloch wave vector k as a function of the normalized photon energy for a metallic pbg .
the vertical dashed lines show v/p and c/p which are the maximum normalized energy of the lower energy band and the minimum normalized energy of the upper energy band , respectively .
the nano - sized active medium in the 3d mpc considered here is assumed to be four - level qds , with diameters ranging from 2 to 10 nanometres , and two upper levels |c and |b which are close to one another .
the remaining two lower levels are denoted as |a and |d. the two upper levels are dipole coupled to the |a state via the same single field continuum .
we also assume that se is allowed from the excited states ( upper levels ) to the |a state and from |a state to |d state , whereas the transition
|b  |d , |c  |d and |c  |b are inhibited in the electric dipole approximation .
there are many potential qds suitable for our theoretical model such as cdse / zns   or inas / gaas   core  shell qds .
four - level qd with the two upper levels |c and |b which are near one another , and two lower levels |a and |d. here ,  is the probe field frequency while ca and ba are the transition frequencies.c and b are the decay rates from the exited states to the |a state . the decay rate from |a to |d
is given as a , whereas c = ca   and b =   ba are the detunings of the atomic transition energies.k is the detuning of the probe field , which has a central frequency at the middle point of the two upper levels
in this section , the interaction between qds doped within the pbg reservoir and a probe field with slowly varying amplitude is investigated . the total semiclassical hamiltonian of the system can be written as here , hq , hqf , hr , hqr and hqq are the hamiltonians of the four - level qd , the qd 
( 2 ) , the equation of motion of the density matrix elements can be written as follows : in eqs .
( 3 - 9 ) , ij ( ij = a , b , c or d ) are density matrix elements ( coherences ) , p is the strength of quantum interference and is defined by p = acab/acab . in this paper , the maximum quantum interface has been considered , which corresponds to a dipole transition moment ac that is parallel to ab .
this gives p = 1 . here , ac and ab are the electric dipole moments induced by the transitions
since the two upper energy levels are very close ( bc = 0.03 ev ) , it is reasonable to consider ab = ab = .  is the rabi frequency of the probe field , defined as  = e/2 , where the dipolar transition moments and external field e are parallel .
the parameters  and  are related to the interaction of qds when the mpc is densely doped .
this interaction is called dipole  dipole interaction ( ddi ) , and its effect was calculated using mean - field theory .
the dependency of all decay rates to energy and local density of states can be written as here , the function z( ) is called the form factor , which contains the information about the electron  photon interaction and is obtained in reference [ 39 - 41 ] . for simplicity ,
all parameters have been normalized with respect to ( bc)/2 , which gives a constant value for the resonant energies ab and ac . here , 0 is the decay rate ( line - width ) for an excited electron in a qd when it is located in a vacuum .
the expression of the absorption coefficient is written in terms of density matrix coherence as : here , n is the concentration of quantum dots and  is the energy of the incident laser beam .
in order to study the linear response of the system , we have calculated the normalized absorption coefficient given in eq . ( 11 ) ( /0 ) by using a very low driving field (  = 0.01 ) .
we consider that the metallic spheres are made of silver with p = 9 ev .
the two upper resonant energies ( ab = 2.78 ev and ac= 2.783 ev ) are considered to be far away from the upper edge band gap in the first brillouin zone where b  c= 1.570 . the value of a can be set by changing the resonant energy ad .
this decay rate ( a ) can be totally suppressed if the resonant energy lies within the band gap . the system of equations in ( 38 ) has been solved numerically for cases where ddi was neglected and taken into account while the system approaches a steady state configuration .
the results have been shown in figs . 3 and 4 where the normalized absorption coefficient versus the detuning parameter has been drawn for different values of lower decay rate ( a = 0.000 , 0.005 and 0.100 ) . as one can see , in both cases
this means that when the lower resonance state ( i.e. |a-|d ) lies within the band gap , the normalized absorption coefficient is zero , and when it goes further from the band gap it is nonzero .
this behaviour demonstrates the switching between absorption and nonabsorption states that can be used to make optical switches .
numerical plots of the normalized absorption coefficient ( /0 ) versus dimensionless detuning parameter ( k ) for different values of lower decay rate ( a = 0.0 , 0.005 and 0.1 ) when the ddi is zero numerical plots of the normalized absorption coefficient ( /0 ) versus dimensionless detuning parameter ( k ) for different values of lower decay rate ( a = 0.0 , 0.005 and 0.1 ) when the ddi has been taken into account (  =  = 2 )
in conclusion , we have studied the effect the quantum interference and ddi on the absorption of a mpc doped with an ensemble of four - level qds , for both cases where ddi was neglected or accounted for , while the system approached a steady state .
a single driving laser field which induces a dipole moment in each qd was applied to measure the absorption .
the density matrix method and linear - response theory have been used to calculate the absorption .
it is found that when the resonance energy of the lower levels is within the band gap , the system is in an absorbing state . however , when the resonance energy of the lower levels is outside the band gap , the system is in a nonabsorption state .
we anticipate that the results described here will be useful for developing new types of optical switching devices .
this article is distributed under the terms of the creative commons attribution noncommercial license which permits any noncommercial use , distribution , and reproduction in any medium , provided the original author(s ) and source are credited .
 in this work , the absorption coefficient of a metallic photonic crystal doped with nanoparticles has been obtained using numerical simulation techniques . 
 the effects of quantum interference and the concentration of doped particles on the absorption coefficient of the system have been investigated . 
 the nanoparticles have been considered as semiconductor quantum dots which behave as a four - level quantum system and are driven by a single coherent laser field . 
 the results show that changing the position of the photonic band gap about the resonant energy of the two lower levels directly affects the decay rate , and the system can be switched between transparent and opaque states if the probe laser field is tuned to the resonance frequency . 
 these results provide an application for metallic nanostructures in the fabrication of new optical switches and photonic devices . 
Introduction
Metallic Photonic Crystals
Active Medium Embedded in a MPC
System Hamiltonian and Density Matrix Equations
Numerical Simulation
Conclusions
Open Access