Low-threshold lasing in photonic-crystal heterostructures M. Srinivas Reddy,1 Ramarao Vijaya,2,∗ Ivan D. Rukhlenko,3,4 and Malin Premaratne3 1 IITB-Monash
Research Academy, CSE Building 2 nd Floor, IIT Bombay, Powai, Mumbai 400076, India 2 Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India 3 Advanced Computing and Simulation Laboratory (A χ L), Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3800, Australia 4 Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 Saint Petersburg, Russia ∗ [email protected]
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 OCIS codes: (230.5298) Photonic crystals; (140.3490) Lasers, distributed-feedback; (000.4430) Numerical approximation and analysis.
References and links 1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). 2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486– 2489 (1987). 3. R. Herrmann, T. Snner, T. Hein, A. Lffler, M. Kamp, and A. Forchel, “Ultrahigh-quality photonic crystal cavity in GaAs,” Opt. Lett. 31, 1229–1231 (2006). 4. H. Y. Ryu, and M. Notomi, “Enhancement of spontaneous emission from the resonant modes of a photonic crystal slab single-defect cavity,” Opt. Lett. 28, 2390–2392 (2003). 5. L.-T. Shi, F. Jin, M.-L. Zheng, X.-Z. Dong, W.-Q. Chen, Z.-S. Zhao, and X.-M. Duan, “Threshold optimization of polymeric opal photonic crystal cavity as organic solid-state dye-doped laser,” Appl. Phys. Lett. 98, 093304 (2011). 6. F. Jin, Y. Song, X.-Z. Dong, W.-Q. Chen, and X.-M. Duan, “Amplified spontaneous emission from dye-doped polymer film sandwiched by two opal photonic crystals,” Appl. Phys. Lett. 91, 031109 (2007). 7. J. Yoon, W. Lee, J. M. Caruge, M. Bawendi, E. L. Thomas, S. Kooi, and P. N. Prasad, “Defect-mode mirrorless lasing in dye-doped organic/inorganic hybrid onedimensional photonic crystal,” Appl. Phys. Lett. 88, 091102 (2006). 8. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. OBrien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). 9. N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. 89, 815–823 (2001).
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6229
10. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999). 11. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994). 12. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999). 13. P.-H. Weng, T.-T. Wu, T.-C. Lu, and S.-C. Wang, “Threshold gain analysis in GaN-based photonic crystal surface emitting lasers,” Opt. Lett. 36, 1908–1910 (2011). 14. M. S. Reddy, R. Vijaya, I. D. Rukhlenko, and M. Premaratne, “Low-threshold lasing in active opal photonic crystals,” Opt. Lett. 38, 1046–1048 (2013). 15. M. S. Reddy, S. Kedia, R. Vijaya, A. K. Ray, S. Sinha, I. D. Rukhlenko, and M. Premaratne, “Analysis of lasing in dye-doped photonic crystals,” IEEE Photonics J. 5, 4700409 (2013). 16. M. S. Reddy, R. Vijaya, I. D. Rukhlenko, and M. Premaratne, “Spatial and spectral distributions of emission from dye-doped photonic crystals in reflection and transmission geometries,” J. Nanophotonics 6, 063526 (2012). 17. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). 18. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). 19. M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mat. 12, 2126, (2002). 20. S. Furumi, “Recent advances in polymer colloidal crystal lasers,” Nanosale 4, 5564-5571 (2012). 21. F. Yu. Sychev, I. E. Razdolski, T. V. Murzina, O. A. Aktsipetrov, T. Trifonov, and S. Cheylan, “Vertical hybrid microcavity based on a polymer layer sandwiched between porous silicon photonic crystals,” Appl. Phys. Lett. 95, 163301 (2009). 22. http://refractiveindex.info 23. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173-190 (2001). 24. S. Satpathy, Ze Zhang and M.R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structures,” Phys. Rev. Lett. 64, 1239-1242 ( 1990). 25. K. Sakoda, Optical properties of Photonic crystals (Springer-Verlag, Berlin, 2001) 26. D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain-assisted waveguiding in metaldielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010). 27. Q. Yan, Z. Zhou, and X. S. Zhao, “Inward-growing self-assembly of colloidal crystal films on horizontal substrates,” Langmuir 21, 3158–3164 (2005).
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)
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6230
Fig. 1. Schematic of different designs studied: (a) stand-alone 3D PhC made of colloids, (b) multilayer stack with a defect layer, and (c) heterostructure with a 3D PhC sandwiched between multilayers. Brown represents the active medium. In (a) and (c), the colloids are surrounded by air voids. Green arrow shows the direction of incidence of the pumping beam.
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 ε1 = 7 (TiO2 ), ε2 = 2.37 (SiO2 ) in the visible range [22], and thicknesses t1 = 0.25a and t2 = 0.16a, 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 ω a/2π c = 0.6.
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6231
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 ∑i [Ein ]+ g i exp(iKg (r − AL ))uˆi , 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 ∑gi [Er f ]− gi exp(iKg (r − AL ))uˆi and the transmitted wave + + is given by ∑gi [Etr ]gi exp(iKg (r − AR ))uˆi , where I + [Etr ]+ gi = ∑ Qgi;g i [Ein ]g i ,
(1)
III + [Er f ]− gi = ∑ Qgi;g i [Ein ]g i ,
(2)
i
i
AL (AR ) is the appropriate origin on the top (bottom) side of the structure and the QI , QII , QIII , QIV are the matrix elements of the transmission/reflection matrix with a definite sequence in the ordering of indices: g1 x, g1 y, g1 z, g2 x, g2 y, g2 z, ...... They are obtained using the matrix elements of the single layer via layer doubling method, which is given explicitly in [17]. uˆi is the unit vector (i, i = x, y, z) and g (g ) is the two-dimensional (2D) reciprocal lattice vector in the xy plane given by g = m1 bx + m2 by
(3)
where m1 , m2 = 0, ±1, ±2, ±3, .... and bx , by are defined by bi · a j = 2πδi j .
(4)
Here 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 √
Kg± = (k + g, ±[q2 − (k + g)2 ]1/2 ),
(5)
where q = μεω /c. Here 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 2π /ω on each side of the slab. The transmittance and reflectance are given by T=
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+ ∗ + ∑g,i [Etr ]+ gi ([Etr ]gi ) Kgz + ∗ + , ∑i [Ein ]+ g i ([Ein ]g i ) Kg z
(6)
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6232
I
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Reflectance, R
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Transmittance, T
Transmittance, T
L
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(a)
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(d)
0.8 0.6 0.4 0.2 0.50
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Normalized frequency (a/2c)
Fig. 2. (a) Reflection spectrum of the stand-alone multilayer stack, (b) group velocity (black curves) and dispersion relation (pink curves) in the [111] direction of the stand-alone 3D PhC, (c) transmission spectrum of cavity formed by sandwiched homogeneous medium (ε = 2.53) with thickness (tu ) equal to the thickness (tPhC ) of 3D PhC, and (d) transmission spectrum of proposed cavity with 30 layers of sandwiched 3D PhC. Five bilayers on either side of the homogeneous medium and 3D PhC are used in calculating (c) and (d). Region I shows the range of frequencies with reduced group velocity whereas region II represents the stopband of the 3D PhC without an allowed mode in the new cavity spectrum.
R=
+ ∗ + ∑g,i [Er f ]+ gi ([Er f ]gi ) Kgz + ∗ + . ∑i [Ein ]+ g i ([Ein ]g i ) Kg z
(7)
Here ∗ denotes the complex conjugation. 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 twodimensional 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)]. #204848 - $15.00 USD (C) 2014 OSA
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6233
Transmittance, T
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 tu = 17.44a (equal to tPhC with 30 layers) and permittivity of 2.53 (which is the same as that of the 3D 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
I
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II
I
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Normalized frequency (a/2c)
Fig. 3. Transmission spectrum of the heterostructure PhC cavity with 10 (green curve), 20 (black curve), and 30 (red curve) layers of sandwiched 3D PhC and five bilayers on either side of the 3D PhC. Region I shows the range of frequencies with reduced group velocity and region II represents the stopband of the 3D PhC, as mentioned in Fig. 2.
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6234
Transmittance, T
30
(a)
20 10
Transmittance, T
0 0.8
(b)
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Fig. 4. (a) Transmission as a function of normalized frequency for heterostructure PhC cavity, calculated by assuming the complex-valued permittivity (ε = −0.0005) for the dielectric spheres composing the sandwiched 3D PhC. (b) Transmission spectrum of the heterostructure cavity with passive sandwiched 3D PhC (ε = 0). We used 30 layers in the sandwiched 3D PhC with five bilayers on either side in calculations. One can see from (a) that the emission of cavity modes near the band edges of the 3D PhC (marked with arrows) is enhanced as compared to the other cavity modes, due to the increased lightmatter interaction.
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 calculated this way may reflectance and transmittance of the system. The lasing threshold εth 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 ω a/2π c = 0.6272, with the lasing threshold of εth = −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 εth for the cavity modes decreases drastically due to the reduced group velocity. A decrease of more than two orders of magnitude in εth is observed for the cavity mode near the low-frequency band edge (at ω a/2π c = 0.582), as opposed to a cavity mode far from the band edge (at ω a/2π c = 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 εth for the mode at the low-frequency edge (at ω a/2π c = 0.582), as compared to the mode at the high-frequency edge (at ω a/2π c = 0.627), is expected [14] and can be seen in Fig. 5(b). This is due to the fact that the mode at the lowfrequency 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 εth at the two edges of the stopband for the stand-alone 3D PhC. When we compare the εth 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 εth in active PhCs can be calculated by [12]
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6235
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0.50 0.55 0.60 0.65 Normalized frequency (a/2c)
(a)
(b)
Fig. 5. (a) Transmittance (in logarithmic scale) for cavity mode near the high-frequency band edge [right arrow in Fig. 4(a)] of the heterostructure cavity and (b) the lasing threshold for cavity modes (filled circles) of the heterostructure PhC cavity and the frequencies near the band edges for the stand-alone 3D PhC (filled squares). The lasing threshold decreases by two orders of magnitude for the cavity mode in the vicinity of the stopband edges. The calculated group velocity of the cavity modes of the heterostructure PhC and in the standalone 3D PhC are marked as open circles and open squares, respectively.
εth =−
8ε¯ vg 1 +Vg /c ), log( f ω tPhC 1 −Vg /c
(8)
where f is the filling factor of the dielectric medium containing gain and Vg is the group velocity of the mode with frequency ω . ε¯ 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
ε¯ =
1 V0
V0
drε (r).
(9)
Here Vo is the volume of unit cell. The values of Vg 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 can be estimated using Eq. (8) for the 3D PhC. These are found to be in good Vg values, εth agreement with the values obtained numerically. Direct calculation of the Vg 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 of this structure, the V is estimated using Eq. (8), and shown as open circles in obtained εth g 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 Vg 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. The imaginary part of the complex permittivity allows one to estimate the gain cofficient √ (γ ) from the relation |ε |k0 = γ ε [26], where k0 is the free-space propagation constant. The threshold gain cofficient (γth ) 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 γth 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 γth . The reflection calculated for the same mode from a stand-alone multilayer stack with an equivalent number of periodic layers, shown by the
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6236
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1.0 0.9 800 0.8 400 (a) 0.7 0 0.6 2 3 4 5 6 Number of periodic bi-layers on each side of 3D-PhC
1200
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Fig. 6. (a) Variation of threshold gain of the heterostructure cavity (black curve) and reflection of the bare multilayer stack (blue curve) as function of number of periodic layers in the multilayer stack. (b) Threshold gain of 3D PhC (open circles) and heterostructure PhC cavity (filled circles) as functions of number N of layers in 3D PhC. The lasing wavelengths are shown by star symbols. The number of periodic multilayers on each side of the PhC equals 5 and a = 367 nm.
blue curve, confirms this conclusion. We further study the dependence of γth 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 γth 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 (ST12/IRD-124). R. Vijaya acknowledges the Director of IRDE for granting the permission to
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6237
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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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6238
Research Academy, CSE Building 2 nd Floor, IIT Bombay, Powai, Mumbai 400076, India 2 Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India 3 Advanced Computing and Simulation Laboratory (A χ L), Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Victoria 3800, Australia 4 Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 Saint Petersburg, Russia ∗ [email protected]
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 OCIS codes: (230.5298) Photonic crystals; (140.3490) Lasers, distributed-feedback; (000.4430) Numerical approximation and analysis.
References and links 1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). 2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486– 2489 (1987). 3. R. Herrmann, T. Snner, T. Hein, A. Lffler, M. Kamp, and A. Forchel, “Ultrahigh-quality photonic crystal cavity in GaAs,” Opt. Lett. 31, 1229–1231 (2006). 4. H. Y. Ryu, and M. Notomi, “Enhancement of spontaneous emission from the resonant modes of a photonic crystal slab single-defect cavity,” Opt. Lett. 28, 2390–2392 (2003). 5. L.-T. Shi, F. Jin, M.-L. Zheng, X.-Z. Dong, W.-Q. Chen, Z.-S. Zhao, and X.-M. Duan, “Threshold optimization of polymeric opal photonic crystal cavity as organic solid-state dye-doped laser,” Appl. Phys. Lett. 98, 093304 (2011). 6. F. Jin, Y. Song, X.-Z. Dong, W.-Q. Chen, and X.-M. Duan, “Amplified spontaneous emission from dye-doped polymer film sandwiched by two opal photonic crystals,” Appl. Phys. Lett. 91, 031109 (2007). 7. J. Yoon, W. Lee, J. M. Caruge, M. Bawendi, E. L. Thomas, S. Kooi, and P. N. Prasad, “Defect-mode mirrorless lasing in dye-doped organic/inorganic hybrid onedimensional photonic crystal,” Appl. Phys. Lett. 88, 091102 (2006). 8. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. OBrien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). 9. N. Susa, “Threshold gain and gain-enhancement due to distributed-feedback in two-dimensional photonic-crystal lasers,” J. Appl. Phys. 89, 815–823 (2001).
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10. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999). 11. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994). 12. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999). 13. P.-H. Weng, T.-T. Wu, T.-C. Lu, and S.-C. Wang, “Threshold gain analysis in GaN-based photonic crystal surface emitting lasers,” Opt. Lett. 36, 1908–1910 (2011). 14. M. S. Reddy, R. Vijaya, I. D. Rukhlenko, and M. Premaratne, “Low-threshold lasing in active opal photonic crystals,” Opt. Lett. 38, 1046–1048 (2013). 15. M. S. Reddy, S. Kedia, R. Vijaya, A. K. Ray, S. Sinha, I. D. Rukhlenko, and M. Premaratne, “Analysis of lasing in dye-doped photonic crystals,” IEEE Photonics J. 5, 4700409 (2013). 16. M. S. Reddy, R. Vijaya, I. D. Rukhlenko, and M. Premaratne, “Spatial and spectral distributions of emission from dye-doped photonic crystals in reflection and transmission geometries,” J. Nanophotonics 6, 063526 (2012). 17. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: Frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998). 18. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000). 19. M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mat. 12, 2126, (2002). 20. S. Furumi, “Recent advances in polymer colloidal crystal lasers,” Nanosale 4, 5564-5571 (2012). 21. F. Yu. Sychev, I. E. Razdolski, T. V. Murzina, O. A. Aktsipetrov, T. Trifonov, and S. Cheylan, “Vertical hybrid microcavity based on a polymer layer sandwiched between porous silicon photonic crystals,” Appl. Phys. Lett. 95, 163301 (2009). 22. http://refractiveindex.info 23. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173-190 (2001). 24. S. Satpathy, Ze Zhang and M.R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structures,” Phys. Rev. Lett. 64, 1239-1242 ( 1990). 25. K. Sakoda, Optical properties of Photonic crystals (Springer-Verlag, Berlin, 2001) 26. D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain-assisted waveguiding in metaldielectric nanowires,” Opt. Lett. 35, 4190–4192 (2010). 27. Q. Yan, Z. Zhou, and X. S. Zhao, “Inward-growing self-assembly of colloidal crystal films on horizontal substrates,” Langmuir 21, 3158–3164 (2005).
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)
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6230
Fig. 1. Schematic of different designs studied: (a) stand-alone 3D PhC made of colloids, (b) multilayer stack with a defect layer, and (c) heterostructure with a 3D PhC sandwiched between multilayers. Brown represents the active medium. In (a) and (c), the colloids are surrounded by air voids. Green arrow shows the direction of incidence of the pumping beam.
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 ε1 = 7 (TiO2 ), ε2 = 2.37 (SiO2 ) in the visible range [22], and thicknesses t1 = 0.25a and t2 = 0.16a, 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 ω a/2π c = 0.6.
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6231
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 ∑i [Ein ]+ g i exp(iKg (r − AL ))uˆi , 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 ∑gi [Er f ]− gi exp(iKg (r − AL ))uˆi and the transmitted wave + + is given by ∑gi [Etr ]gi exp(iKg (r − AR ))uˆi , where I + [Etr ]+ gi = ∑ Qgi;g i [Ein ]g i ,
(1)
III + [Er f ]− gi = ∑ Qgi;g i [Ein ]g i ,
(2)
i
i
AL (AR ) is the appropriate origin on the top (bottom) side of the structure and the QI , QII , QIII , QIV are the matrix elements of the transmission/reflection matrix with a definite sequence in the ordering of indices: g1 x, g1 y, g1 z, g2 x, g2 y, g2 z, ...... They are obtained using the matrix elements of the single layer via layer doubling method, which is given explicitly in [17]. uˆi is the unit vector (i, i = x, y, z) and g (g ) is the two-dimensional (2D) reciprocal lattice vector in the xy plane given by g = m1 bx + m2 by
(3)
where m1 , m2 = 0, ±1, ±2, ±3, .... and bx , by are defined by bi · a j = 2πδi j .
(4)
Here 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 √
Kg± = (k + g, ±[q2 − (k + g)2 ]1/2 ),
(5)
where q = μεω /c. Here 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 2π /ω on each side of the slab. The transmittance and reflectance are given by T=
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+ ∗ + ∑g,i [Etr ]+ gi ([Etr ]gi ) Kgz + ∗ + , ∑i [Ein ]+ g i ([Ein ]g i ) Kg z
(6)
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6232
I
0.6
II
I
0.4
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(b)
Wave vector
Reflectance, R
0.4 0.2
Transmittance, T
Transmittance, T
L
Group velocity, Vg /c
(a)
0.8
(c)
0.8 0.6 0.4 0.2
(d)
0.8 0.6 0.4 0.2 0.50
0.55
0.60
0.65
0.70
0.75
Normalized frequency (a/2c)
Fig. 2. (a) Reflection spectrum of the stand-alone multilayer stack, (b) group velocity (black curves) and dispersion relation (pink curves) in the [111] direction of the stand-alone 3D PhC, (c) transmission spectrum of cavity formed by sandwiched homogeneous medium (ε = 2.53) with thickness (tu ) equal to the thickness (tPhC ) of 3D PhC, and (d) transmission spectrum of proposed cavity with 30 layers of sandwiched 3D PhC. Five bilayers on either side of the homogeneous medium and 3D PhC are used in calculating (c) and (d). Region I shows the range of frequencies with reduced group velocity whereas region II represents the stopband of the 3D PhC without an allowed mode in the new cavity spectrum.
R=
+ ∗ + ∑g,i [Er f ]+ gi ([Er f ]gi ) Kgz + ∗ + . ∑i [Ein ]+ g i ([Ein ]g i ) Kg z
(7)
Here ∗ denotes the complex conjugation. 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 twodimensional 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)]. #204848 - $15.00 USD (C) 2014 OSA
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6233
Transmittance, T
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 tu = 17.44a (equal to tPhC with 30 layers) and permittivity of 2.53 (which is the same as that of the 3D 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
I
0.8
II
I
0.6 0.4 0.2 0.50
0.55
0.60
0.65
0.70
0.75
Normalized frequency (a/2c)
Fig. 3. Transmission spectrum of the heterostructure PhC cavity with 10 (green curve), 20 (black curve), and 30 (red curve) layers of sandwiched 3D PhC and five bilayers on either side of the 3D PhC. Region I shows the range of frequencies with reduced group velocity and region II represents the stopband of the 3D PhC, as mentioned in Fig. 2.
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6234
Transmittance, T
30
(a)
20 10
Transmittance, T
0 0.8
(b)
0.6 0.4 0.2 0.45
0.50
0.55
0.60
0.65
Normalized frequency (a/2c)
Fig. 4. (a) Transmission as a function of normalized frequency for heterostructure PhC cavity, calculated by assuming the complex-valued permittivity (ε = −0.0005) for the dielectric spheres composing the sandwiched 3D PhC. (b) Transmission spectrum of the heterostructure cavity with passive sandwiched 3D PhC (ε = 0). We used 30 layers in the sandwiched 3D PhC with five bilayers on either side in calculations. One can see from (a) that the emission of cavity modes near the band edges of the 3D PhC (marked with arrows) is enhanced as compared to the other cavity modes, due to the increased lightmatter interaction.
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 calculated this way may reflectance and transmittance of the system. The lasing threshold εth 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 ω a/2π c = 0.6272, with the lasing threshold of εth = −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 εth for the cavity modes decreases drastically due to the reduced group velocity. A decrease of more than two orders of magnitude in εth is observed for the cavity mode near the low-frequency band edge (at ω a/2π c = 0.582), as opposed to a cavity mode far from the band edge (at ω a/2π c = 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 εth for the mode at the low-frequency edge (at ω a/2π c = 0.582), as compared to the mode at the high-frequency edge (at ω a/2π c = 0.627), is expected [14] and can be seen in Fig. 5(b). This is due to the fact that the mode at the lowfrequency 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 εth at the two edges of the stopband for the stand-alone 3D PhC. When we compare the εth 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 εth in active PhCs can be calculated by [12]
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6235
3.6
-1
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75
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-1
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65
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Stop band
2.2
"th
"
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0.62720 0.62724 Normalized frequency (a/2c)
0.62728
-2
10
-3
10
0.45
Vg /c
80x10-5 80
0.50 0.55 0.60 0.65 Normalized frequency (a/2c)
(a)
(b)
Fig. 5. (a) Transmittance (in logarithmic scale) for cavity mode near the high-frequency band edge [right arrow in Fig. 4(a)] of the heterostructure cavity and (b) the lasing threshold for cavity modes (filled circles) of the heterostructure PhC cavity and the frequencies near the band edges for the stand-alone 3D PhC (filled squares). The lasing threshold decreases by two orders of magnitude for the cavity mode in the vicinity of the stopband edges. The calculated group velocity of the cavity modes of the heterostructure PhC and in the standalone 3D PhC are marked as open circles and open squares, respectively.
εth =−
8ε¯ vg 1 +Vg /c ), log( f ω tPhC 1 −Vg /c
(8)
where f is the filling factor of the dielectric medium containing gain and Vg is the group velocity of the mode with frequency ω . ε¯ 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
ε¯ =
1 V0
V0
drε (r).
(9)
Here Vo is the volume of unit cell. The values of Vg 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 can be estimated using Eq. (8) for the 3D PhC. These are found to be in good Vg values, εth agreement with the values obtained numerically. Direct calculation of the Vg 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 of this structure, the V is estimated using Eq. (8), and shown as open circles in obtained εth g 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 Vg 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. The imaginary part of the complex permittivity allows one to estimate the gain cofficient √ (γ ) from the relation |ε |k0 = γ ε [26], where k0 is the free-space propagation constant. The threshold gain cofficient (γth ) 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 γth 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 γth . The reflection calculated for the same mode from a stand-alone multilayer stack with an equivalent number of periodic layers, shown by the
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Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6236
4
586
10
584
3
10
2
10
(b) 582 580 10 15 20 25 30
Wavelength (nm)
Threshold gain (cm-1)
Reflectance, R
Threshold gain (cm-1)
1.0 0.9 800 0.8 400 (a) 0.7 0 0.6 2 3 4 5 6 Number of periodic bi-layers on each side of 3D-PhC
1200
Number of layers, N
Fig. 6. (a) Variation of threshold gain of the heterostructure cavity (black curve) and reflection of the bare multilayer stack (blue curve) as function of number of periodic layers in the multilayer stack. (b) Threshold gain of 3D PhC (open circles) and heterostructure PhC cavity (filled circles) as functions of number N of layers in 3D PhC. The lasing wavelengths are shown by star symbols. The number of periodic multilayers on each side of the PhC equals 5 and a = 367 nm.
blue curve, confirms this conclusion. We further study the dependence of γth 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 γth 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 (ST12/IRD-124). R. Vijaya acknowledges the Director of IRDE for granting the permission to
#204848 - $15.00 USD (C) 2014 OSA
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6237
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.
#204848 - $15.00 USD (C) 2014 OSA
Received 17 Jan 2014; revised 20 Feb 2014; accepted 21 Feb 2014; published 10 Mar 2014 24 March 2014 | Vol. 22, No. 6 | DOI:10.1364/OE.22.006229 | OPTICS EXPRESS 6238
