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Paper text:
photonic crystal ( phc ) cavities @xcite have high quality factors and small mode volume . with such cavities
, light can be confined in a very small volume with low loss rate .
thus , the interaction between light and matter can be greatly enhanced , which is important for investigations of cavity quantum electrodynamics  @xcite .
phc waveguide can guide light with very low losses @xcite .
the coupled phc cavity - waveguide structures have been used to realize various types of optical devices , such as optical switches @xcite and optical sensors @xcite . to improve the device performance ,
fano resonance with asymmetric and steep spectral line shape has been intensively used , which arises from the interference between discrete resonance states and continuum states  @xcite . by modulating the coupling between phc cavity and waveguide
, fano resonance can be achieved due to the constructive and destructive interference of discrete cavity resonance states with broadband continuum waveguide states @xcite .    in this paper , we design a coupled phc cavity @xcite and waveguide structure using finite - difference time - domain ( fdtd ) simulation @xcite . by placing partially reflecting elements in the phc waveguide , fano type transmission line shape is achieved .
recently , gain materials such as quantum dots  @xcite , nanocrystals @xcite and rare - earth ions @xcite have been used as light - emitting centers to achieve single - photon sources , display devices and all - optical diodes , respectively .
additionally , lasers have been demonstrated with coupling these light sources to microcavities @xcite . here ,
such gain materials are introduced into phc cavities to enhance fano resonance . with introducing the gain materials into cavity ,
the transmission line shape of the fano resonance is greatly steepened and a large contrast is achieved by a small frequency shift . in our structure ,
the transmissivity in the waveguide is increased by a factor of six , which is potentially applicable for optical switches and optical sensors .
we employed a free - software package meep @xcite to design the phc structure .
hexagonal lattice of air holes was patterned on a planar dielectric slab .
the slab thickness is 0.6 @xmath1 , where _ a _ is the lattice constant .
the air hole radius is @xmath2 , and the slab dielectric is 12.96 . with the mit photonic - bands ( mpb ) package @xcite , the phc slab band structure has been calculated , which has te - like photonic bandgap from 0.2455 to 0.3229 ( _ a_/@xmath3 ) , where @xmath3 is wavelength in the vacuum . with one air hole missing in the phc slab , an h1 cavity is formed @xcite .
the radii of the nearest six air holes around the h1 cavity are reduced to 0.24 _ a _ to increase the quality factors of the cavity modes .
three lattice constant _ a _ away from the center of h1 cavity , one array of air holes are removed to form a w1 waveguide .
shows the coupled phc cavity - waveguide structure .    to simulate the spectral response of the coupled cavity - waveguide structure , a gaussian source with frequency center 0.2980 ( _ a_/@xmath4 ) and frequency width 0.02 ( _ a_/@xmath4 )
is located at the left end of waveguide and the energy flux has been detected at the right end . shows the normalized transmission spectrum of the light .
as waveguide mode frequencies are far away from the cavity resonant frequencies , light transmits from one end of the waveguide to another end with almost 100% transmission without any coupling with the cavity . when waveguide modes are resonant or nearly resonant with the cavity modes , light couples into the cavity resulting in dips in the transmission spectrum .
three dips in the transmission spectrum in fig .
1(b ) correspond to three resonant modes of the h1 cavity .
shows the z components of the magnetic fields of the cavity modes corresponding to the three dips in the transmission spectrum in .
the mode of shows a hexapole mode corresponding to the dip denoted by _
while the modes of and ( c ) are nearly degenerate quadrupole modes for _ b _ and _ c _ in .
the cavity mode field distributions indicate that the cavity couples with the waveguide via the evanescent field .
the cavity resonance frequencies of , ( b ) and ( c ) are 0.2961 , 0.2996 and 0.3002 ( _ a_/@xmath4 ) and the corresponding quality factors are 2890 , 5960 and 2770 , respectively .
a cavity mode with low quality factor has a large coupling strength with the waveguide , thus a deeper dip appears in the transmission spectrum with a broad linewidth .      to realize fano resonance , partially reflecting elements are introduced into the coupled phc cavity - waveguide structure .
two air hole barriers have been placed in the center of the waveguide , as shown in .
the amplitude transmissivity @xmath5 is determined by @xcite @xmath6 where @xmath7 is the resonant frequency of the cavity mode .
@xmath8 is the amplitude transmission coefficient of the partially reflecting elements .
@xmath9 is the intrinsic cavity loss rate .
@xmath10 is the cavity decay rate due to coupling into one port of the waveguide . when there is no partially reflecting elements existing ( @xmath11 ) , the transmission line shape is lorentzian symmetric . when @xmath12 , the light from the incoming port of waveguide to the outgoing port has two pathways .
one with light transmitting through the cavity provides discrete modes , and the other with light passing through the partially reflecting elements gives rise to continuum modes .
the constructive and destructive interferences between these modes induce an asymmetric fano type transmission line shape .
the amplitude transmission coefficient @xmath8 determines whether the transmission line shape is symmetric lorentzian type or asymmetric fano type .
the simulated and normalized transmission spectrum is shown in .
two fano type peaks are observed in the spectrum .
when the waveguide modes are non - resonant with the cavity modes , the light can not be coupled into the cavity and is reflected mostly by the two air hole barriers .
the light with the cavity mode frequency can couple into the cavity and transmit to the other port of the waveguide .
therefore , fano type peaks appear in the transmission spectrum .
the simulated cavity mode distributions show that , there are three cavity modes in the frequency range of the transmission spectrum in .
the z components of the magnetic fields of the three modes are shown in , ( b ) and ( c ) .
the cavity resonance frequencies of the three modes are 0.2956 , 0.2998 and 0.2998 ( _ a_/@xmath4 ) and the corresponding quality factors are 1830 , 2980 and 4570 , respectively .
the resonant frequencies of cavity modes in and ( c ) are so close that the cavity modes can not be distinguished in the transmission spectrum in
.      the asymmetric and steep line shape of fano resonance is beneficial for optical switches and optical sensors .
the steep line shape of fano resonance can significantly reduce the frequency shift required for on / off switching .
optical sensors with high sensitivity can be constructed based on the spectral shift associated with asymmetric line shapes .
recently , investigations have been performed to enhance the fano resonance .
@xcite reported that the fano resonance can be modified by adjusting the partially reflecting elements . in this work ,
we introduce gain material into the phc cavity to enhance the fano resonance .    in the two dimensional phc slab structure ,
it is straightforward to couple quantum dots into a cavity @xcite , with which the optical gain can be obtained by pumping the quantum dot either optically or electrically . as the quantum dot transition and the cavity mode
are tuned into resonance , the loss of the cavity mode is compensated by the photons emitted by the quantum dot , resulting in increasing of the quality factor of the cavity mode . to simulate the gain supplied by quantum dot , we consider the material structure in maxwell s equations which is determined by the relative permittivity @xmath13 .
it should be noted that @xmath14 does not depend only on position , but also on frequency ( material dispersion ) , electric field ( nonlinearity ) , and the orientation of the field ( anisotropy ) @xcite . material dispersion is generally associated with absorption loss or gain in the material , which is given by the following expression : @xmath15 \end{split}\ ] ] where @xmath16 is the instantaneous dielectric function , @xmath17 is the electric conductivity , @xmath18 and @xmath19 are user - specified constants , and @xmath20 is a user - specified function of position giving the strength of the _ n_-th resonance .
note that the imaginary part of @xmath14 can be expressed by conductivity @xmath17 as @xmath21 . here , we only consider the absorption loss or gain in a narrow bandwidth , thus the imaginary part of @xmath14 can be set to some known experimental value , and a dispersionless real part of @xmath14 can be set with a value in our bandwidth of interest . in fdtd simulation , the imaginary part im@xmath22 is set by choosing the conductivity @xmath17 at our frequency @xmath23 of interest .
the absorption loss ( gain ) in the material is associated with positive ( negative ) value of the imaginary part of @xmath14 .
a series of negative values of im@xmath14 has been used to simulate the gain , which can be achieved by pumping quantum dot in the cavity .    in our coupled phc cavity - waveguide structure ( shown in ) ,
gain material is located at the center of the h1 cavity with a radius of 0.6 @xmath24 and thickness of 0.6 @xmath24 .
the frequency @xmath23 is set around the resonant frequency of the degenerated quadrupole modes .
the imaginary part of the dielectric function of the gain material introduced into the h1 cavity is set as -0.001 , -0.01 , -0.0135 and -0.01415 at frequency 0.2998 @xmath25 .
the normalized transmission spectra are shown in .
as the gain increases , the normalized transmissivity becomes larger and larger , and the linewidth of the transmission line shape becomes narrower .
light trapped in the cavity is fed by the gain so that the light energy flux is enlarged .
then the enlarged light energy flux is coupled into waveguide , resulting in enlarged transmissivity . without gain material in the cavity ,
the normalized transmissivity corresponding to the degenerated quadrupole modes is 0.246 .
when gain material with im@xmath14=0.01415 is introduced into the cavity , the normalized transmissivity is 1.564 , which is about six times larger than that of the case without gain .
it can be clearly seen that the line shapes become steeper as the gain increases .    with the coupled cavity - waveguide structure , we further analysed the transmission spectra in the cases with and without gain material in h1 cavity . with the same frequency shift ,
the transmission contrast with gain material is much larger than the case without the gain .
the large transmission contrast with a small frequency shift is very promising for realizing optical switches @xcite .
in addition , the resonant mode can be easily perturbed if there are nano - particles adhere to the h1 cavity , which gives a frequency shift in the transmission spectrum .
the narrow and steep fano asymmetric transmission lineshape could make the coupled structure much more attractive as a nano - particle sensor @xcite .
we have theoretically proposed that the fano resonance in a coupled phc cavity - wavegudie structure can be greatly enhanced when gain material is introduced into the cavity .
the fano spectral line shape becomes steeper and steeper as the gain increases .
with im@xmath26 the transmission is enhanced by six times in our case . the optical gain in the cavity supplies photons with frequency of the cavity mode .
the supplied photons compensate the losses of cavity mode . as a result ,
the quality factor of the cavity mode is increased , corresponding to a narrower linewidth . the discrete cavity mode with narrower linewidth interferes with the continuum modes , generating a steeper fano asymmetric linedwidth .
the doubly degenerate modes at 0.2998 @xmath25 are chosen to enhance fano asymmetric lineshape because the field of the degenerate modes distributes in a large volume in the cavity , which can enhance the gain . with the gain enhanced fano resonance , a large contrast can be achieved by only a small frequency shift , which could be useful in ultralow - energy and high - contrast optical switches .
the steepened transmission line shape can be also sensitive to nano - particle adhering to the designed structures , which enables such coupled phc structure to be used as optical sensors .
* acknowledgments * + this work was supported by the national basic research program of china under grant no .
2013cb328706 and 2014cb921003 ; the national natural science foundation of china under grant no .
91436101 , 11174356 and 61275060 ; the strategic priority research program of the chinese academy of sciences under grant no .
xdb07030200 ; and the hundred talents program of the chinese academy of sciences .    * author contributions * + x.x . and k.j .
conceived the project .
y.z . designed the structure and carried out the calculations .
, k.q . , j.t .
, y.s . and x.x .
discussed the results and the experimental implementations .
y.z . and x.x .
co - wrote the paper and all authors reviewed the manuscript .
systems with coupled cavities and waveguides have been demonstrated as optical switches and optical sensors . to optimize the functionalities of these optical devices , fano resonance with asymmetric and steep spectral line shape has been used . 
 we theoretically propose a coupled photonic crystal cavity - waveguide structure to achieve fano resonance by placing partially reflecting elements in waveguide . 
 to enhance fano resonance , optical gain material is introduced into the cavity . as the gain increases , the transmission line shape becomes steepened and the transmissivity can be six times enhanced , giving a large contrast by a small frequency shift . 
 it is prospected that the gain enhanced fano resonance is very useful for optical switches and optical sensors .    
 beijing national laboratory for condensed matter physics , institute of physics , chinese academy of sciences , beijing 100190 , p.  r.  china + @xmath0 to whom correspondence should be addressed . 
 e - mail : xlxu@iphy.ac.cn