## Abstract

We study a photonic crystal (PhC) heterostructure cavity consisting of gain medium in a three-dimensional (3D) PhC sandwiched between two identical passive multilayers. For this structure, based on Korringa-Kohn-Rostoker method, we observe a decrease in the lasing threshold of two orders of magnitude, as compared with a stand-alone 3D PhC. We attribute this remarkable decrease in threshold gain to the overlap of the defect cavity mode with the reduced group velocity region of the PhC’s dispersion, and the associated enhancement in the distributed feedback from the ordered layers of the PhC. The obtained results show the potency for designing PhC-based, compact on-chip lasers with ultra-low thresholds.

© 2014 Optical Society of America

## 1. Introduction

Photonic crystals (PhCs) allow one to control and manipulate the spontaneous emission of light near a band gap, owing to the redistribution of the photonic density of states (DOS) [1, 2] instigated as a result of the periodically varying permittivity. The enhanced light emission from a PhC can be obtained using two methods. The first method relies on localizing the light at a defect, having a small mode volume, and thus creating an ultrasmall high-quality microcavity [3, 4]. This method was used to demonstrate lasing in [5–8]. The second method relies on exploiting the enhanced distributed feedback in the PhC caused by the reduction in the group velocity near the band edges, thus prolonging the interaction of light with the gain medium [9–11]. Several groups have utilized this method to achieve low-threshold lasing [12, 13]. Recently, we theoretically analyzed the band edge lasing in a three-dimensional (3D) PhC and experimentally demonstrated both the lasing and modification in the spontaneous emission from a PhC made of Rhodamine-B-doped polystyrene colloidal microspheres [14–16].

In this paper, we propose and analyze a microcavity composed of a heterostructure, in which both methods mentioned above are combined to give a drastic decrease in the lasing threshold. The design consists of a 3D PhC containing gain medium sandwiched between two identical passive multilayer stacks. The PhC heterostructure cavity suggested in our work has the advantages of low lasing threshold and tunability of the lasing wavelength. The latter is achieved by changing either the number of layers or the periodicity in the sandwiched 3D PhC. The lasing threshold characteristics are calculated using the Korringa-Kohn-Rostoker (KKR) method [17, 18]. We unambiguously demonstrate that the lasing threshold is significantly reduced for the cavity modes near the band edges of the sandwiched 3D PhC, when compared to other cavity modes. The dependence of the lasing threshold and wavelength on the number of layers in the sandwiched 3D PhC and the multilayer stack is also calculated and analyzed.

## 2. Proposed design

The gain distribution has very minimal effects on the lasing behavior. Therefore, without loss of generality, we consider a uniform gain medium in this paper. As the main focus of this work is on the study of the effect of photonic bandgap on the lasing threshold of the PhC, the frequency dependence of the permittivity is not included. Under these assumptions the gain can be modeled using a complex-valued permittivity *ε* = *ε′* + *iε″* with *ε″* < 0 [10].

Figures 1(a) and 1(b) show two common designs of PhCs for lasing applications [7, 15, 19–21]. The former is a 3D PhC with a face-centered cubic structure made up of dielectric spheres that are uniformly doped with the gain medium surrounded by the air voids, whereas the latter is a cavity formed by multilayers with a defect-active layer. In the proposed structure shown in Fig. 1(c), the sandwiched medium is an active 3D PhC. The structures are assumed to be infinite in extent in the *xy* plane and of finite thickness in the z direction. It is well known that the available DOS is higher at the defect mode frequency due to the multilayer cavity, and also near the band edge frequencies of the sandwiched 3D PhC [7,11]. When the cavity defect mode becomes resonant with the band edge region of the 3D PhC, the net availability of DOS for the mode increases as compared to a stand-alone configuration. As a consequence, a drastic decrease in the lasing threshold is expected for these modes. We quantitatively substantiate this prediction in Section 4.

In calculations, we assume that the 3D PhC is made of polystyrene spheres (*ε′* = 2.53) doped with a gain medium. The multilayers are composed of 5 double layers each, with *ε*_{1} = 7 (TiO_{2}), *ε*_{2} = 2.37 (SiO_{2}) in the visible range [22], and thicknesses *t*_{1} = 0.25*a* and *t*_{2} = 0.16*a*, where *a* is the lattice constant. The period and the number of layers of the multilayer structure are chosen in such a way that the stopband is broad enough to cover the stopband of the 3D PhC. The stop band of the PhC is centered at the normalized frequency *ωa*/2*πc* = 0.6.

## 3. Numerical method

The optical characteristics of the proposed design were calculated using the KKR method [17] by exploiting the spherical symmetry of the colloidal spehers constituting building blocks of the PhC. This method enables one to calculate the complex photonic band structure associated with a given crystallographic plane of the PhC, for evaluating the reflection and transmission matrix elements. Using these matrices, the reflectance and transmittance of the finite-thickness PhC can be calculated [17].

Consider a plane wave
${\sum}_{i}{\left[{E}_{in}\right]}_{{g}^{\prime}i}^{+}\text{exp}(i{K}_{{g}^{\prime}}^{+}(r-{A}_{L})){\widehat{u}}_{i}$, incident on the PhC heterostructure from the top as shown in Fig. 1. The three components *i* = *x*, *y*, *z* of the electric field vector are obtained from polarization direction and magnitude of the field associated with a given beam *g′* [17]. The reflected wave is given by
${\sum}_{gi}{\left[{E}_{rf}\right]}_{gi}^{-}\text{exp}\left(i{K}_{g}^{-}(r-{A}_{L})\right){\widehat{u}}_{i}$ and the transmitted wave is given by
${\sum}_{gi}{\left[{E}_{tr}\right]}_{gi}^{+}\text{exp}\left(i{K}_{g}^{+}(r-{A}_{R})\right){\widehat{u}}_{i}$, where

*A*(

_{L}*A*) is the appropriate origin on the top (bottom) side of the structure and the

_{R}*Q*,

^{I}*Q*,

^{II}*Q*,

^{III}*Q*are the matrix elements of the transmission/reflection matrix with a definite sequence in the ordering of indices:

^{IV}*g*

_{1}

*x*,

*g*

_{1}

*y*,

*g*

_{1}

*z*,

*g*

_{2}

*x*,

*g*

_{2}

*y*,

*g*

_{2}

*z*,...... They are obtained using the matrix elements of the single layer via layer doubling method, which is given explicitly in [17].

*û*is the unit vector (

_{i}*i*,

*i′*=

*x*,

*y*,

*z*) and

*g*(

*g′*) is the two-dimensional (2D) reciprocal lattice vector in the

*xy*plane given by where

*m*

_{1},

*m*

_{2}= 0, ±1, ±2, ±3,.... and

*b*,

_{x}*b*are defined by Here

_{y}*a*(

_{j}*j*=

*x*,

*y*) are the primitive vectors of the lattice in

*xy*plane with lattice constant

*a*. The wave vector of the incident plane wave with parallel wave vector component

*q*

_{||}=

*k*

_{||}+

*g*is given by

*μ*, and

*ε*are the complex permeability and complex dielectric constant of the sphere and

*c*is the speed of light in vacuum. The parameter

*ε*with a negative imaginary part gives gain, and the positive imaginary part of

*ε*gives absorbance of the medium.

*k*

_{||}is the reduced wave vector in the surface Brillouin zone and

*g′*is one of the reciprocal lattice vectors. +, − defines the sign of the

*z*component of the wave vector.

The transmittance (*T*) and the reflectance (*R*) of the PhC heterostructure can be obtained using the calculated transmitted and reflected wave using Eqs. (1),(2) for a corresponding incident wave. The ratio between the flux of transmitted (reflected) wave and the flux of incident wave is called as *T*(*R*). It can be obtained by integrating the Poynting vector over the *xy* plane with a time average over the period 2*π/ω* on each side of the slab. The transmittance and reflectance are given by

This method works for structures with non-overlapping spheres. In an ideal crystal without absorption, the *T* and *R* never exceed one. In a crystal with a gain medium, the *T* and *R* may be greater than unity due to stimulated emission, which will be discussed in the next section. In the calculations, we assumed that the PhC is perfectly crystalline and neglected the spontaneous emission in estimating the lasing threshold. The losses due to the domain cracks, as well as the spontaneous emission, may slightly increase the lasing threshold value without significantly affecting the lasing wavelength.

## 4. Results and discussion

Convergent results in the calculation of *T* and *R* were obtained by using 41 reciprocal two-dimensional plane wave vectors, which were expanded in spherical waves with angular momenta *l* = 1, 2,...,7. The multilayer stack gives a broad stopband shown in Fig. 2 (a), which is seen to be broad enough to cover the stopband of the 3D PhC [see pink curve in Fig. 2(b)]. The group velocity (black curve) and the dispersion relation of the propagating eigenmodes of the 3D PhC calculated using the plane wave expansion method [23–25] are presented in Fig. 2(b) for the (111) direction. The group velocity is seen to be close to zero at the edges of the photonic stopband. The structure shown in Fig. 1(b) leads to the defect modes with high transmittance, as shown in Fig. 2(c). The defect layer has a thickness of *t _{u}* = 17.44

*a*(equal to

*t*with 30 layers) and permittivity of 2.53 (which is the same as that of the 3D PhC).

_{PhC}The transmission spectra of the structure in Fig. 1(c) are shown in Fig. 2(d). They are calculated by taking real permittivities (*ε″* = 0) for the sandwiched 3D PhC with 30 layers. One can observe that the allowed mode is absent in the range of frequencies corresponding to the stopband of the sandwiched 3D PhC (region II). Region I in Fig. 2(d) shows the modes with reduced group velocity of the heterostructure cavity.

While an increase in the number of layers of the sandwiched 3D PhC may result in the frequency shift of the cavity modes, one can expect that the modes near the stopband edges would have smaller group velocities in comparison with other modes. The transmission spectra of the heterostructure cavity for the sandwiched 3D PhC with 10, 20, and 30 layers are shown in Fig. 3 by green, black, and red curves, respectively. One can see that there is always a defect mode present in the reduced group velocity (region I), regardless of the number of layers in the PhC. As mentioned earlier, there is no allowed mode in the range of frequencies of region II. Once the gain medium is introduced into the building blocks of the sandwiched 3D PhC in such a way that its emission band overlaps the stopband edges, a large gain enhancement can be observed for these modes as compared to other cavity modes. This is due to the combined effect of the reduced group velocity and the increase in the DOS of the defect mode.

To confirm this enhancement, the transmission spectrum calculated by assuming *ε″* = −0.0005 for the sandwiched 3D PhC is shown in Fig. 4(a). The transmittance of the defect modes near the band edges is much greater than unity due to the emission. For comparison, Fig. 4(b) shows the transmission spectrum of a similar PhC heterostructure cavity by assuming a purely real permittivity (*ε″* = 0). It can be clearly seen from Fig. 4 that the enhancement of gain for the cavity modes [shown by arrows in Figs. 4(a) and 4(b)] near the band edges of the 3D PhC is much larger than that for the other modes. This is an expected result due to the large values of DOS.

Owing to the significant enhancement in the emission from the cavity modes, one can expect a low-threshold lasing due to the increased distributed feedback experienced by the defect modes inside the gain medium. The lasing threshold is calculated by assuming that the population inversion of the uniformly doped gain medium in the sandwiched 3D PhC is achieved by means of optical pumping and that the system is ready to emit [12].

We obtained the lasing threshold numerically as given in [12]. Since the lasing is a process of light emission without input signal similar to the oscillations in electric circuits, the onset of lasing is equivalent to unbounded points of either reflectance, or transmittance, or the sum of reflectance and transmittance of the system. The lasing threshold *ε″ _{th}* calculated this way may serve as a measure of the population inversion [12].

The transmittance for the high-frequency band edge cavity mode of the PhC heterostructure cavity is plotted in logarithmic scale (color coded) as a function of *ε″* and the normalized frequency in Fig. 5(a). It is clearly seen to diverge at *ωa*/2*πc* = 0.6272, with the lasing threshold of *ε″*_{th} = −0.00069.

Figure 5(b) shows the unbounded transmittance/reflectance points (filled circles) for all the PhC heterostructure cavity modes depicted in Fig. 4. As the stopband edge is approached, the magnitude of *ε″*_{th} for the cavity modes decreases drastically due to the reduced group velocity. A decrease of more than two orders of magnitude in *ε″*_{th} is observed for the cavity mode near the low-frequency band edge (at *ωa*/2*πc* = 0.582), as opposed to a cavity mode far from the band edge (at *ωa*/2*πc* = 0.466). Although the cavity confinement effect is present for the mode at the normalized frequency of 0.466, it is far from the stopband and does not have the bandgap effect of the 3D PhC. Thus, the cavity mode at this frequency is equivalent to the cavity mode of the design shown in Fig. 1 (b). A lower value of *ε″*_{th} for the mode at the low-frequency edge (at *ωa*/2*πc* = 0.582), as compared to the mode at the high-frequency edge (at *ωa*/2*πc* = 0.627), is expected [14] and can be seen in Fig. 5(b). This is due to the fact that the mode at the low-frequency edge stores its energy in the high-permittivity medium, which provides gain in the present design. The filled squares in Fig. 5(b) show the numerically evaluated *ε″*_{th} at the two edges of the stopband for the stand-alone 3D PhC.

When we compare the *ε″*_{th} value at the band edges, a significant decrease in the lasing threshold of PhC heterostructure cavity is seen due to the increased feedback provided by the multilayer in comparison to a stand-alone 3D PhC. The *ε″*_{th} in active PhCs can be calculated by [12]

*f*is the filling factor of the dielectric medium containing gain and

*V*is the group velocity of the mode with frequency

_{g}*ω*.

*ε̄′*is the spatial average of the dielectric function

*ε′*(

*r*), which is the real part of the permittivity of the medium containing gain. It is given by Here

*V*is the volume of unit cell. The values of

_{o}*V*of the stand-alone 3D PhC, shown earlier in Fig. 2(b) are plotted as open squares in Fig. 5(b) for the band edge frequencies. Using these

_{g}*V*values,

_{g}*ε″*can be estimated using Eq. (8) for the 3D PhC. These are found to be in good agreement with the values obtained numerically.

_{th}Direct calculation of the *V _{g}* for the cavity modes of the PhC heterostructure cavity shown in Fig. 1(c) is difficult due to the composite nature of the structure. Hence, using the numerically obtained

*ε″*of this structure, the

_{th}*V*is estimated using Eq. (8), and shown as open circles in Fig. 5(b). One can note that the group velocity of the cavity modes near the stopband of the 3D PhC is decreased by more than an order of magnitude as compared to the

_{g}*V*in stand-alone 3D PhC. Thus the drastic reduction in the lasing threshold in PhC heterostructure cavity is supported by the lowered group velocity of its modes.

_{g}The imaginary part of the complex permittivity allows one to estimate the gain cofficient (*γ*) from the relation
$\left|{\epsilon}^{\u2033}\right|{k}_{0}=\gamma \sqrt{{\epsilon}^{\prime}}$ [26], where *k*_{0} is the free-space propagation constant. The threshold gain cofficient (*γ*_{th}) obtained by varying the number of periodic multilayers on either side of the 3D PhC is shown by the black curve in Fig. 6(a). The number of periodic layers in the PhC is chosen to be 25 and *a* = 367 nm. The threshold gain can be reduced by more than an order of magnitude via increasing the periodic bilayers from two to five in the multilayer stack. With a further increase in the number of bilayers, only a small change in *γ*_{th} can be observed. It is interesting to note that five bi-layers periodically stacked on each side of the 3D PhC are sufficient to obtain a significant reduction in *γ*_{th}. The reflection calculated for the same mode from a stand-alone multilayer stack with an equivalent number of periodic layers, shown by the blue curve, confirms this conclusion.

We further study the dependence of *γ*_{th} on the number N of ordered layers of the sandwiched active 3D PhC, which determines the cavity length, with a five-bilayer periodic stack on either side. The results are shown by the filled circles in Fig. 6(b). The open circles show *γ*_{th} for the stand-alone 3D PhC. One can observe a reduction of two orders of magnitude in the gain coefficient of the heterostructure PhC cavity. The change in lasing wavelength, which occurs due to the variation in the number of sandwiched 3D PhC layers, is marked by stars in Fig. 6(b). It implies that one can attain the lasing with reduced threshold values regardless of the number of layers in the sandwiched 3D PhC. Moreover, one can select the lasing mode with a lower threshold from the different cavity modes available, by overlapping that particular mode with the band edge region via a change in the periodicity of the sandwiched 3D PhC.

The structure proposed in this work can be fabricated by using well-developed methods such as self-assembly for the 3D PhCs [27] and deposition techniques for multilayers [7]. Multilayers can be fabricated on substrates such as glass and an active 3D PhC can be grown using the self-assembly method on the multilayer structure [5]. The second multilayer can then be deposited over it. Even though we chose colloidal 3D PhC containing gain medium, the mechanism of lowered threshold will be applicable even in structures with woodpile arrangement.

## 5. Conclusion

We have proposed and analyzed a PhC heterostructure cavity consisting of a gain-medium doped 3D PhC sandwiched between passive multilayers. A decrease of two orders of magnitude in the threshold gain as compared to a stand-alone 3D PhC was achieved. We explained this drastic decrease in the threshold gain by the overlapping of the defect cavity mode with the reduced group velocity region of the PhC, which enhances the distributed feedback from the ordered layers of the PhC. We also studied the effect of the number of layers on the threshold gain. The proposed cavity design holds an immense potential for realizing miniaturized PhC based compact chip lasers with an ultra-low threshold.

## Acknowledgments

M. S. Reddy gratefully acknowledges the mentorship of the Prof. S. Dhar, Department of Physics, IIT Bombay. The work of R. Vijaya was supported by the Instrument Research and Development Establishment, Dehradun, India under the DRDO Nanophotonics program (ST-12/IRD-124). R. Vijaya acknowledges the Director of IRDE for granting the permission to publish this work. The work of I. D. Rukhlenko and M. Premaratne is supported by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055 and Discovery Grant DP110100713, respectively.

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