- Email: [email protected]
Enhancement of spontaneous emission induced by all-dielectric hyperbolic metamaterial at quasi-Dirac points
Optics Communications 393 (2017) 113–117
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Enhancement of spontaneous emission induced by all-dielectric hyperbolic metamaterial at quasi-Dirac points
MARK
⁎
Subir Majumdera, , Shyamal Kumar Bhadraa,b a
Fiber Optics and Photonic Division, CSIR-Central Glass and Ceramics Research Institute, 196, Raja S. C. Mullick Road, Kolkata 700032, India Raman Centre for Atomic, Molecular and Optical Sciences, Indian Association for the Cultivation of Science, 2A & 2B, Raja S. C. Mullick Road, Kolkata 700032, India b
A R T I C L E I N F O
A BS T RAC T
PACS: 42.70.Qs 78.67.Pt
We report an all dielectric hyperbolic metamaterial which is sufficiently capable of enhancing spontaneous emission (SE) of a quantum emitter. This hyperbolic metamaterial consists of specially designed dielectric rod based photonic crystal exhibiting Dirac-like and a semi-Dirac cone in its band structure. Rigorous numerical simulation has been done to verify this significant enhancement of SE.
Keywords: Semi Dirac cone Hyperbolic metamaterial Spontaneous emission Photonic crystal
1. Introduction Like the electromagnetic counterpart, engineered photonic density of states (PDOS) opens up new avenue to manipulate optical properties and thus it leads to development of new photonic niche structures. Modification of environment of an emitter leads to change in PDOS and in the Wigner-Weisskopf approximation; its SE is directly proportional to the local density of states (LDOS) [1–3] of the system. Change in such SE of an atom (or quantum dots, dipole source [3] etc.) in a cavity (or a nano-antenna [4,5]) due to modification of its environment is known as Purcell effect and it was first described by E. M. Purcell in 1946 [6] while studying magnetic nuclear resonance absorption. An increased LDOS can help to enhance SE in various optical devices [7,8] and this process emerges as a promising tool to achieve single photon sources [9–11]. Enhancement of SE is estimated by a factor known as Purcell factor [2,3,5],
F =
3 3 ⎛λ⎞ ⎜ ⎟ Q / Vmode 2⎝ ⎠ 4π n 2
with the mode volume, Vmode =
∫ ε E dr max(ε E 2 )
(1)
where λ is resonant wavelength and n is the mode index. For a large quality factor Q, other contributions from other associated modes to the LDOS are negligibly small at resonance frequency; hence the ratio Q / Vmode is widely approximated as Purcell factor. It is to be noted here ⁎
that the expression of Vmode is valid when ( ∫ ε E 2 dr ) is not small. A microcavity with high quality factor Q and an ultra-low mode volume (Vmode ), thus achieving the highest value of the ratio Q / Vmode is the prime focus to make use of such an effect [3,12]. Purcell effect is applicable in the case of weak coupling between an emitter (viz. atom, dipole source etc.) and a resonating object (viz. optical cavity or an antenna). It is important to note here that modification of the environment in this weak coupling regime doesn’t modify the resonance frequency of the cavity. It is only the decay rate which changes due to light-matter interaction [4]. Implementation of such an enhancement has been greatly made by the use of hyperbolic metamaterials (HMM) [13]. An HMM shows a hyperbolic dispersion where the iso-frequency surface for a transverse-magnetic (TM) wave in such a medium follows the relation
k y2 kx 2 ω2 + = 2 εy εx c
(2)
where εx . εy <1, i.e. dielectric constants in two orthogonal directions that possess opposite signs. Such a metamaterial is known as indefinite media since it supports propagating waves with unbounded wave vectors (k) due to its hyperbolic geometry. Thus, in principle, it can accommodate indefinite number electromagnetic states and results in a divergence in density of states [14] and contributes to the enhancement of SE. Most of the HMMs are reported to be a periodic combination of metal and dielectrics either in the form of a stratified structures or
Corresponding author. E-mail addresses: [email protected] (S. Majumder), [email protected] (S.K. Bhadra).
http://dx.doi.org/10.1016/j.optcom.2017.02.022 Received 26 December 2016; Received in revised form 3 February 2017; Accepted 8 February 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 393 (2017) 113–117
S. Majumder, S.K. Bhadra
Dirac-like structures can significantly enhances SE. In the present work we have taken a metallic square box with small slit [3]. The inner wall is highly polished so that it becomes a perfect reflector and works as an ideal cavity. The small slit at one sidewall allows modes to escape out in the air to have the system a finite Q, decaying asymptotically with time. By placing an emitter inside the box turned cavity, we have compared its SE enhancement with that of the same cavity which consists of all-dielectric embedded HMM. Introduction of this HMM changes the nature of the cavity as well as the environment of the emitter and the effect of this change has been rigorously studied. 2. Results and discussions Initially we took a (9Λ×9Λ) inner volume metallic box with wall thickness 0.5Λ. It has a slit in one sidewall of width 0.1Λ which appears as a channel waveguide for the outgoing radiating modes from the emitter placed at the centre of the box. Here the parameter Λ is in length scale, explained later. The inner wall is highly polished to make it perfectly reflecting. We have calculated the Purcell factor (Q / Vmode ) by Finite Difference Time Domain (FDTD) method for a range of frequencies radiated by the source and results are shown in Fig. 1. Decent enhancement of SE has been observed which is of the order of ~105. With an attempt to increase this enhancement we tried to modify the surrounding environment of the emitter inside the cavity. We looked for a suitable PC based hyperbolic metamaterial which doesn’t
Fig. 1. : Purcell factor for empty metal box. Shape of the box is shown in the inset. Slit width has been kept at 0.1Λ.
nanowires [15,16]. Recently, Ying Wu [17] reported all-dielectric photonic crystal (PC) with hyperbolic topology where dispersion relation possesses a semi-Dirac cone. This semi-Dirac cone is obtained by a symmetry reduction (from C4 to C2) from the structure used in Dirac-like-cone [18,19]. A structure with hyperbolic geometry should enhance the SE if it is placed in the neighbourhood of a quantum emitter. The structure with Dirac-like cone actually shows a rectangular hyperbola but we will try to show later that both the Semi-Dirac and
Fig. 2. Photonic band structure and density of states of (a) circular, (b) elliptical, (c) square, (d) rectangular rods (ε=8.8) photonic crystal with Brillouin zone in the inset. Points marked as “A” and “C” are Dirac-like where the points “B” and “D” are the Semi-Dirac points; iso-frequency plots around these respective points are in the lower right insets of each case.
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along with photonic density of states (DOS) in the right of each panel. Respective first Brillouin zones are shown in the left lower insets. Points marked as “A” and “C” in Fig. 2 are triply degenerate Dirac-like cone with two cone-like and other one is a flat band. Here the DOS is divergent. Iso-frequency plots of the flat band around these points exhibits a rectangular hyperbolic topology. Such a structure offers zero-index [18] at this frequency. On the other hand points “B” and “D” in Fig. 2 are doubly degenerate semi-Dirac cone with one conelike band and the other one is flat band. It also has a divergent DOS at this point. Choice of such a PC (Fig. 2(b) or (d)) has been made from the prescription of Wu [17] by simple symmetry reduction (from C4 to C2) from the structures in Figs. 2(a) and 2(c) respectively. Such a structure has dielectric constant with opposite sign in orthogonal directions, it exhibits zero-index along Γ-X1 direction (Fig. 2b) whereas a band gap along Γ-X direction at semi-Dirac frequency, which are reflected in the transmission characteristics as reported in [17]. Iso-frequency plots around the point “B” and “D” show a hyperbolic topology which is in lower inset of Figs. 2(b) and 2(d) respectively. Sudden divergence in the DOS at and around these frequencies suggests that there should be an enhancement in number of electromagnetic states, which are well predicted for the hyperbolic geometry due to its unbounded locus and made it our choice for studying Purcell enhancement. We have inserted these PC into the proposed cavity and allowed the emitter to emit (Fig. 3). An HMM in the environment of the emitter is expected to enhance its SE. We have used FDTD method to calculate Purcell factor which is shown in Fig. 4. The emitter is placed at the centre of the cavity (right insets of both the panel in Fig. 4) and we have performed FDTD simulations over a long period of time. It shows a sufficient enhancement of SE near the Dirac-like and semi-Dirac point
Fig. 3. Structure to study Purcell enhancement. Dielectric rods are in brown where circular, elliptical, square and rectangular rods have been placed to study. The emitter is located at the center marked in red.
have any metallic component. We have adopted a PC of a 6×6 array of alumina (ε=8.8) rods (circular, elliptical, square, rectangular) placed inside the cavity. For the circular rods, diameter is 0.4411Λ, where for the elliptical rods; minor axis is 0.4175Λ with an eccentricity of 1.3, Λ being the lattice parameter of the PC. Square rods have side length 0.393Λ where the rectangular rods have side lengths 0.38Λ & 0.45Λ respectively. Fig. 2 shows the photonic band structure (calculated by plane wave expansion (PWE) method) for the first Brillouin zone of (a) circular, (b) elliptic, (c) square and (d) rectangular rods PC
Fig. 4. Purcell factor of the system i.e. the cavity with (a) circular, (b) elliptical, (c) square, (d) rectangular rods PC. Magnified Purcell curves are in the left inset and the position of the dipole source w.r.t. the PC in the right inset in each case.
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Optics Communications 393 (2017) 113–117
S. Majumder, S.K. Bhadra
Fig. 5. Dependence of resonance Purcell factor with slit width variation of Metallic box as shown in Fig. 3 with PC of (a) circular, (b) elliptical, (c) square, (d) rectangular rods in air.
3. Conclusion
due to the modification of its environment (left inset of Fig. 4) and at a frequency near these points, the system exhibits a resonance cavity mode which is evident from the much enhanced Purcell factor. In all these cases, slit width has been kept fixed at 0.1Λ. Q value of an empty metal box (Fig. 1) has more or less similar order of magnitude to that of the cavity with HMM PC, but modification of environment of the emitter by the introduction of the HMM PC reduces the mode volume (Vmode ) to some extent which leads to the enhancement of Purcell factor. This is in agreement with the fact that hyperbolic geometry of the PC due to its unbounded locus can accommodate infinite number of modes, hence results in exhibiting enhanced SE. At the resonance condition, cavity itself shows huge mode confinement, hence Purcell factor shows large enhancement. It is obvious that the existence of the slit in the sidewall perturbs the solution of a closed cavity and makes the cavity to have finite Q. Hence role of the slit width is important here in order to optimize the output of the composite cavity. With the variation of the slit width, FDTD study showed decreasing Purcell factor with increasing slit width, provided the location of the emitter is at the centre (i.e. at X or A as in Fig. 4) of the cavity (Fig. 5) which is much anticipated. There are some cases as observed while considering the width variation close to the 0.1Λ that Purcell factor is increased with increasing width at the close vicinity of 0.1Λ, but that is possibly because of local numerical fluctuations, intensive study to these cases may invoke further course of study. Realistic values for such a device may be for Λ=18.75 mm, for which the device will have output at X-bands with possible application in radar systems where it will act as a suitable patch antenna [20] of smaller size. Possible source (emitter) of such a system may be a 6H-SiC system with Si vacancy defect reported recently [21]. Fabrication of PC of similar dimension (Λ=17 mm) has been reported earlier in ref. 18.
In conclusion, we find that an all-dielectric hyperbolic metamaterial (HMM) is capable of enhancing spontaneous emission (SE) of a quantum emitter as predicted long ago by Purcell for atomic system. This kind of HMM in PC possessing Dirac-like or semi-Dirac cone exhibits hyperbolic dispersion around these special degeneracy points. Hyperbolic topology of these structures can accommodate large number of modes due to its unbounded locus compared to the closed one (e.g. elliptic topology), and thus they are capable of exhibiting enhanced Purcell factor. Our work showed that spontaneous emission enhancement is possible even in the absence of metallic component in HMM. Owing to simple configuration based on photonic crystal structure; one can fabricate such structure in a range of frequencies even in the near-IR [2] or visible range as well for future emerging applications. 4. Methods We have used MPB [22] software for all PWE calculations and MEEP [23] for FDTD calculations. PWE calculations are made within the first Brillouin Zone only. All MEEP calculations are made for 2D PC in the resolution of 100. Acknowledgement The authors would like to thank Dr. K. Muraleedharan, Director, CSIR-CGCRI for support and encouragement to carry out the work and DST, Govt. of India for financial support. Part of the work of S. K. Bhadra is supported by CSIR-ES Scheme-21 (1017)/15/EMR-II and SKB is thankful to Prof. S. Bhattacharya, Director, IACS for unstinted support. 116
Optics Communications 393 (2017) 113–117
S. Majumder, S.K. Bhadra
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.optcom.2017.02.022.
[11] E. Esteban, T.V. Teperik, J.J. Greffet, Phys. Rev. Lett. 104 (2) (2010) 26802. [12] A.F. Koenderink, Opt. Lett. 35 (24) (2010) 4208. [13] Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E. Narimanov, V.M. Shalaev, Appl. Phys. B 100 (2010) 215–218. [14] Z. Jacob, I.I. Smolyaninov, E. Narimanov, Appl. Phys. Lett. 100 (2012) 181105. [15] K.V. Sreekanth, A. De Luca, G. Strangi, Anisotropic Nanomaterials, NanoScience and Technology (Chapter 12), Springer International Publishing Switzerland, 2015, pp. 447–467. [16] P. Shekhar, J. Atkinson, Z. Jacob, Nano Converg. 1 (2014) 14. [17] Y. Wu, Opt. Exp. 22 (1) (2014) 1906–1917. [18] X.Q. Huang, Y. Lai, Z.H. Hang, H.H. Zheng, C.T. Chan, Nat. Mater. 10 (2011) 582. [19] Y. Fu, L. Xu, Z.H. Hang, H. Chen, Appl. Phys. Lett. 104 (2014) 193509. [20] J.Q. Howell, Microstrip Antennas, IEEE International Symposium on Antennas and Propagation, Williamsburg, Virginia, 1972, pp. 177–180. [21] H. Kraus, V.A. Soltamov, D. Riedel, S. Väth, F. Fuchs, A. Sperlich, P.G. Baranov, V. Dyakonov, G.V. Astakhov, Nat. Phys. 10 (2014) 157. [22] S.G. Johnson, J.D. Joannopoulos, Opt. Exp. 8 (3) (2001) 173–190. [23] A. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J.D. Joannopoulos, S.G. Johnson, Comp. Phys. Commun. 181 (2010) 687–702.
References [1] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997. [2] D. Englund, D. Fattal, W. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, J. Vuckovic, Phys. Rev. Lett. 95 (2005) 013904. [3] A. Taflov, A. Oskooi, S.G. Johnson, (Chapter 4) Advances in FDTD Computational Electrodynamics: photonics and Nanotechnology, Artech House Publishers, Boston, 2013. [4] A. Krasnok, A. Slobozhanyuk, C. Simovski, S. Tretyakov, A. Poddubny, A. Mirohnichenko, Y. Kivishar, P. Belov, Sci. Rep. 5 (2015) 12956. [5] N. Kumar, Spontaneous Emission Rate Enhancement Using Optical Antennas (Ph.D. thesis), University of California at Berkeley, 2013. [6] E.M. Purcell, Phys. Rev. 69 (1946) 681. [7] C. Santori, D. Fattal, J. Vuckovic, G.S. Solomon, Y. Yamamoto, Nature 419 (2002) 594. [8] J. McKeever, A. Boca, A.D. Boozer, J.R. Buck, H.J. Kimble, Nature 425 (2003) 268. [9] A. Kiraz, M. Atature, I. Imamoglu, Phys. Rev. A 69 (2004) 032305. [10] M. Patterson, S. Hughes, D. Dalacu, R.L. Williams, Phys. Rev. B. 80 (12) (2009) 125307.
117
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Enhancement of spontaneous emission induced by all-dielectric hyperbolic metamaterial at quasi-Dirac points
MARK
⁎
Subir Majumdera, , Shyamal Kumar Bhadraa,b a
Fiber Optics and Photonic Division, CSIR-Central Glass and Ceramics Research Institute, 196, Raja S. C. Mullick Road, Kolkata 700032, India Raman Centre for Atomic, Molecular and Optical Sciences, Indian Association for the Cultivation of Science, 2A & 2B, Raja S. C. Mullick Road, Kolkata 700032, India b
A R T I C L E I N F O
A BS T RAC T
PACS: 42.70.Qs 78.67.Pt
We report an all dielectric hyperbolic metamaterial which is sufficiently capable of enhancing spontaneous emission (SE) of a quantum emitter. This hyperbolic metamaterial consists of specially designed dielectric rod based photonic crystal exhibiting Dirac-like and a semi-Dirac cone in its band structure. Rigorous numerical simulation has been done to verify this significant enhancement of SE.
Keywords: Semi Dirac cone Hyperbolic metamaterial Spontaneous emission Photonic crystal
1. Introduction Like the electromagnetic counterpart, engineered photonic density of states (PDOS) opens up new avenue to manipulate optical properties and thus it leads to development of new photonic niche structures. Modification of environment of an emitter leads to change in PDOS and in the Wigner-Weisskopf approximation; its SE is directly proportional to the local density of states (LDOS) [1–3] of the system. Change in such SE of an atom (or quantum dots, dipole source [3] etc.) in a cavity (or a nano-antenna [4,5]) due to modification of its environment is known as Purcell effect and it was first described by E. M. Purcell in 1946 [6] while studying magnetic nuclear resonance absorption. An increased LDOS can help to enhance SE in various optical devices [7,8] and this process emerges as a promising tool to achieve single photon sources [9–11]. Enhancement of SE is estimated by a factor known as Purcell factor [2,3,5],
F =
3 3 ⎛λ⎞ ⎜ ⎟ Q / Vmode 2⎝ ⎠ 4π n 2
with the mode volume, Vmode =
∫ ε E dr max(ε E 2 )
(1)
where λ is resonant wavelength and n is the mode index. For a large quality factor Q, other contributions from other associated modes to the LDOS are negligibly small at resonance frequency; hence the ratio Q / Vmode is widely approximated as Purcell factor. It is to be noted here ⁎
that the expression of Vmode is valid when ( ∫ ε E 2 dr ) is not small. A microcavity with high quality factor Q and an ultra-low mode volume (Vmode ), thus achieving the highest value of the ratio Q / Vmode is the prime focus to make use of such an effect [3,12]. Purcell effect is applicable in the case of weak coupling between an emitter (viz. atom, dipole source etc.) and a resonating object (viz. optical cavity or an antenna). It is important to note here that modification of the environment in this weak coupling regime doesn’t modify the resonance frequency of the cavity. It is only the decay rate which changes due to light-matter interaction [4]. Implementation of such an enhancement has been greatly made by the use of hyperbolic metamaterials (HMM) [13]. An HMM shows a hyperbolic dispersion where the iso-frequency surface for a transverse-magnetic (TM) wave in such a medium follows the relation
k y2 kx 2 ω2 + = 2 εy εx c
(2)
where εx . εy <1, i.e. dielectric constants in two orthogonal directions that possess opposite signs. Such a metamaterial is known as indefinite media since it supports propagating waves with unbounded wave vectors (k) due to its hyperbolic geometry. Thus, in principle, it can accommodate indefinite number electromagnetic states and results in a divergence in density of states [14] and contributes to the enhancement of SE. Most of the HMMs are reported to be a periodic combination of metal and dielectrics either in the form of a stratified structures or
Corresponding author. E-mail addresses: [email protected] (S. Majumder), [email protected] (S.K. Bhadra).
http://dx.doi.org/10.1016/j.optcom.2017.02.022 Received 26 December 2016; Received in revised form 3 February 2017; Accepted 8 February 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
Optics Communications 393 (2017) 113–117
S. Majumder, S.K. Bhadra
Dirac-like structures can significantly enhances SE. In the present work we have taken a metallic square box with small slit [3]. The inner wall is highly polished so that it becomes a perfect reflector and works as an ideal cavity. The small slit at one sidewall allows modes to escape out in the air to have the system a finite Q, decaying asymptotically with time. By placing an emitter inside the box turned cavity, we have compared its SE enhancement with that of the same cavity which consists of all-dielectric embedded HMM. Introduction of this HMM changes the nature of the cavity as well as the environment of the emitter and the effect of this change has been rigorously studied. 2. Results and discussions Initially we took a (9Λ×9Λ) inner volume metallic box with wall thickness 0.5Λ. It has a slit in one sidewall of width 0.1Λ which appears as a channel waveguide for the outgoing radiating modes from the emitter placed at the centre of the box. Here the parameter Λ is in length scale, explained later. The inner wall is highly polished to make it perfectly reflecting. We have calculated the Purcell factor (Q / Vmode ) by Finite Difference Time Domain (FDTD) method for a range of frequencies radiated by the source and results are shown in Fig. 1. Decent enhancement of SE has been observed which is of the order of ~105. With an attempt to increase this enhancement we tried to modify the surrounding environment of the emitter inside the cavity. We looked for a suitable PC based hyperbolic metamaterial which doesn’t
Fig. 1. : Purcell factor for empty metal box. Shape of the box is shown in the inset. Slit width has been kept at 0.1Λ.
nanowires [15,16]. Recently, Ying Wu [17] reported all-dielectric photonic crystal (PC) with hyperbolic topology where dispersion relation possesses a semi-Dirac cone. This semi-Dirac cone is obtained by a symmetry reduction (from C4 to C2) from the structure used in Dirac-like-cone [18,19]. A structure with hyperbolic geometry should enhance the SE if it is placed in the neighbourhood of a quantum emitter. The structure with Dirac-like cone actually shows a rectangular hyperbola but we will try to show later that both the Semi-Dirac and
Fig. 2. Photonic band structure and density of states of (a) circular, (b) elliptical, (c) square, (d) rectangular rods (ε=8.8) photonic crystal with Brillouin zone in the inset. Points marked as “A” and “C” are Dirac-like where the points “B” and “D” are the Semi-Dirac points; iso-frequency plots around these respective points are in the lower right insets of each case.
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S. Majumder, S.K. Bhadra
along with photonic density of states (DOS) in the right of each panel. Respective first Brillouin zones are shown in the left lower insets. Points marked as “A” and “C” in Fig. 2 are triply degenerate Dirac-like cone with two cone-like and other one is a flat band. Here the DOS is divergent. Iso-frequency plots of the flat band around these points exhibits a rectangular hyperbolic topology. Such a structure offers zero-index [18] at this frequency. On the other hand points “B” and “D” in Fig. 2 are doubly degenerate semi-Dirac cone with one conelike band and the other one is flat band. It also has a divergent DOS at this point. Choice of such a PC (Fig. 2(b) or (d)) has been made from the prescription of Wu [17] by simple symmetry reduction (from C4 to C2) from the structures in Figs. 2(a) and 2(c) respectively. Such a structure has dielectric constant with opposite sign in orthogonal directions, it exhibits zero-index along Γ-X1 direction (Fig. 2b) whereas a band gap along Γ-X direction at semi-Dirac frequency, which are reflected in the transmission characteristics as reported in [17]. Iso-frequency plots around the point “B” and “D” show a hyperbolic topology which is in lower inset of Figs. 2(b) and 2(d) respectively. Sudden divergence in the DOS at and around these frequencies suggests that there should be an enhancement in number of electromagnetic states, which are well predicted for the hyperbolic geometry due to its unbounded locus and made it our choice for studying Purcell enhancement. We have inserted these PC into the proposed cavity and allowed the emitter to emit (Fig. 3). An HMM in the environment of the emitter is expected to enhance its SE. We have used FDTD method to calculate Purcell factor which is shown in Fig. 4. The emitter is placed at the centre of the cavity (right insets of both the panel in Fig. 4) and we have performed FDTD simulations over a long period of time. It shows a sufficient enhancement of SE near the Dirac-like and semi-Dirac point
Fig. 3. Structure to study Purcell enhancement. Dielectric rods are in brown where circular, elliptical, square and rectangular rods have been placed to study. The emitter is located at the center marked in red.
have any metallic component. We have adopted a PC of a 6×6 array of alumina (ε=8.8) rods (circular, elliptical, square, rectangular) placed inside the cavity. For the circular rods, diameter is 0.4411Λ, where for the elliptical rods; minor axis is 0.4175Λ with an eccentricity of 1.3, Λ being the lattice parameter of the PC. Square rods have side length 0.393Λ where the rectangular rods have side lengths 0.38Λ & 0.45Λ respectively. Fig. 2 shows the photonic band structure (calculated by plane wave expansion (PWE) method) for the first Brillouin zone of (a) circular, (b) elliptic, (c) square and (d) rectangular rods PC
Fig. 4. Purcell factor of the system i.e. the cavity with (a) circular, (b) elliptical, (c) square, (d) rectangular rods PC. Magnified Purcell curves are in the left inset and the position of the dipole source w.r.t. the PC in the right inset in each case.
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S. Majumder, S.K. Bhadra
Fig. 5. Dependence of resonance Purcell factor with slit width variation of Metallic box as shown in Fig. 3 with PC of (a) circular, (b) elliptical, (c) square, (d) rectangular rods in air.
3. Conclusion
due to the modification of its environment (left inset of Fig. 4) and at a frequency near these points, the system exhibits a resonance cavity mode which is evident from the much enhanced Purcell factor. In all these cases, slit width has been kept fixed at 0.1Λ. Q value of an empty metal box (Fig. 1) has more or less similar order of magnitude to that of the cavity with HMM PC, but modification of environment of the emitter by the introduction of the HMM PC reduces the mode volume (Vmode ) to some extent which leads to the enhancement of Purcell factor. This is in agreement with the fact that hyperbolic geometry of the PC due to its unbounded locus can accommodate infinite number of modes, hence results in exhibiting enhanced SE. At the resonance condition, cavity itself shows huge mode confinement, hence Purcell factor shows large enhancement. It is obvious that the existence of the slit in the sidewall perturbs the solution of a closed cavity and makes the cavity to have finite Q. Hence role of the slit width is important here in order to optimize the output of the composite cavity. With the variation of the slit width, FDTD study showed decreasing Purcell factor with increasing slit width, provided the location of the emitter is at the centre (i.e. at X or A as in Fig. 4) of the cavity (Fig. 5) which is much anticipated. There are some cases as observed while considering the width variation close to the 0.1Λ that Purcell factor is increased with increasing width at the close vicinity of 0.1Λ, but that is possibly because of local numerical fluctuations, intensive study to these cases may invoke further course of study. Realistic values for such a device may be for Λ=18.75 mm, for which the device will have output at X-bands with possible application in radar systems where it will act as a suitable patch antenna [20] of smaller size. Possible source (emitter) of such a system may be a 6H-SiC system with Si vacancy defect reported recently [21]. Fabrication of PC of similar dimension (Λ=17 mm) has been reported earlier in ref. 18.
In conclusion, we find that an all-dielectric hyperbolic metamaterial (HMM) is capable of enhancing spontaneous emission (SE) of a quantum emitter as predicted long ago by Purcell for atomic system. This kind of HMM in PC possessing Dirac-like or semi-Dirac cone exhibits hyperbolic dispersion around these special degeneracy points. Hyperbolic topology of these structures can accommodate large number of modes due to its unbounded locus compared to the closed one (e.g. elliptic topology), and thus they are capable of exhibiting enhanced Purcell factor. Our work showed that spontaneous emission enhancement is possible even in the absence of metallic component in HMM. Owing to simple configuration based on photonic crystal structure; one can fabricate such structure in a range of frequencies even in the near-IR [2] or visible range as well for future emerging applications. 4. Methods We have used MPB [22] software for all PWE calculations and MEEP [23] for FDTD calculations. PWE calculations are made within the first Brillouin Zone only. All MEEP calculations are made for 2D PC in the resolution of 100. Acknowledgement The authors would like to thank Dr. K. Muraleedharan, Director, CSIR-CGCRI for support and encouragement to carry out the work and DST, Govt. of India for financial support. Part of the work of S. K. Bhadra is supported by CSIR-ES Scheme-21 (1017)/15/EMR-II and SKB is thankful to Prof. S. Bhattacharya, Director, IACS for unstinted support. 116
Optics Communications 393 (2017) 113–117
S. Majumder, S.K. Bhadra
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.optcom.2017.02.022.
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