Relative permittivity dependence of photonic band gaps for unit cells of the basic structural unit of two-dimensional decagonal photonic quasicrystals Jianjun Liu a,b,n , Zhigang Fan a , Min Kuang c , Guiming He a , Chunying Guan d , Libo Yuan d a School of Astronautics, Harbin Institute of Technology, Harbin 150001, China b School of Electronic Science, Northeast Petroleum University, Daqing 163318, China c School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China d College of Science, Harbin Engineering University, Harbin 150001, China article info Article history: Received 13 January 2011 Received in revised form 15 September 2012 Accepted 17 September 2012 Available online 11 October 2012 Keywords: Photonic quasicrystals Photonic band gap Basic structural unit Unit cell abstract In the basic structural unit of two-dimensional (2D) decagonal photonic quasicrystals (PQCs), the photonic band gaps (PBGs) of the four square unit cells in two constructions have been calculated as functions of relative permittivity. The different unit cells possess different PBG structures and different relative permittivity threshold values. For a given unit cell, the type of PBG and the variation in the center frequency of the PBG depend on the construction and the effective relative permittivity, respectively. We propose a qualitative relationship, F er ¼ F er q, e 1=2 reff , for the PBG impact factors, where the construction-dependent factor q determines the PBG type. Depending on the relative values of the relative permittivity of the scatterers (e ra ) and the background medium (e rb ), either the TM band gap dominates (e ra 4e rb ) or the TE band gap and complete band gap dominate (e ra oe rb ). The variation in the center frequency of the PBG depends on the inverse square root of the effective relative permittivity e reff , i.e., e 1=2 reff : Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. 1. Introduction Photonic crystals (PCs), namely, materials possessing a photo- nic band gap (PBG) and capable of photon localization, have a wide range of potential applications [1–4], because of their unique ability to control the flow of photons and guide the propagation of light. Applications include omnidirectional reflec- tors [5], PC fibers and devices [6,7], nanocavity and laser [8,9], large angle bends and low loss waveguides [10], and correspond- ing coupled devices [11]. The generated type, width, and corre- sponding frequency of the PBGs are closely related to the PC configuration and the composition of the substrate materials as well as size, shape, and arrangement of scatterers. For periodic photonic crystals (PPCs), substrate materials with high relative permittivities are required to generate a complete PBG. In con- trast, quasi-periodic photonic crystals (QPCs; also known as photonic quasicrystals (PQCs)) do not require such high relative permittivities. A two-dimensional (2D) dodecagonal PQC made of an array of, for example, air holes in either silicon nitride (e r ¼ 4.08, n ¼ 2.02) or glass (e r ¼ 2.10, n ¼ 1.45), generates a com- plete band gap [12]. Also, a 2D octagonal PQC of an array of dielectric rods (e r ¼ 2.4, n ¼ 1.55) surrounded by air generates a complete band gap [13]. Indeed, a complete PBG can be obtained for PQCs with relative permittivity threshold values as small as e r ¼ 1.6 (n ¼ 1.26) [14]. However, PBGs of a 2D PQC with a threshold value relative permittivity were reported mostly for a single unit cell scope [12,14]. These results were specialized with no generality. In this paper, four square unit cells forming the basic structural unit of the 2D decagonal PQC structure were chosen. Considering two constructions (dielectric-cylinders-in-air and air-cylinders- in-dielectric) and values for the relative permittivity of the dielectric in the range of 1–30, band structures for the decagonal PQCs were calculated using the plane wave expansion (PWE) method. The results from this calculation will be more general. 2. Algorithm and model The density of states [15], transmission spectra [16,17,12] and band structures [18,19] are general means to characterize PBGs of 2D PQCs. Because the density of states and the spectral lines depend continuously on frequency (or wavelength), it is difficult to determine the boundary between the band-gap and the non- band-gap regions. Therefore, values for the band gap width (or boundary) and the corresponding frequency range are hard to determine. These values are more easily obtained from the band Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/optcom Optics Communications 0030-4018/$ - see front matter Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.09.055 n Corresponding author at: School of Astronautics, Harbin Institute of Technology, Harbin 150001, China. Tel./fax: þ86 451 86402784. E-mail address: [email protected] (J. Liu). Optics Communications 288 (2013) 52–55
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Optics Communications 288 (2013) 52–55
Contents lists available at SciVerse ScienceDirect
Optics Communications
0030-40
http://d
n Corr
Harbin
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journal homepage: www.elsevier.com/locate/optcom
Relative permittivity dependence of photonic band gaps for unit cells of thebasic structural unit of two-dimensional decagonal photonic quasicrystals
Jianjun Liu a,b,n, Zhigang Fan a, Min Kuang c, Guiming He a, Chunying Guan d, Libo Yuan d
a School of Astronautics, Harbin Institute of Technology, Harbin 150001, Chinab School of Electronic Science, Northeast Petroleum University, Daqing 163318, Chinac School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, Chinad College of Science, Harbin Engineering University, Harbin 150001, China
a r t i c l e i n f o
Article history:
Received 13 January 2011
Received in revised form
15 September 2012
Accepted 17 September 2012Available online 11 October 2012
In the basic structural unit of two-dimensional (2D) decagonal photonic quasicrystals (PQCs), the
photonic band gaps (PBGs) of the four square unit cells in two constructions have been calculated as
functions of relative permittivity. The different unit cells possess different PBG structures and different
relative permittivity threshold values. For a given unit cell, the type of PBG and the variation in the
center frequency of the PBG depend on the construction and the effective relative permittivity,
respectively. We propose a qualitative relationship, Fer ¼ Fer q,e�1=2reff
� �, for the PBG impact factors, where
the construction-dependent factor q determines the PBG type. Depending on the relative values of the
relative permittivity of the scatterers (era) and the background medium (erb), either the TM band gap
dominates (era4erb) or the TE band gap and complete band gap dominate (eraoerb). The variation in
the center frequency of the PBG depends on the inverse square root of the effective relative permittivity
ereff, i.e., e�1=2reff :
Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.
1. Introduction
Photonic crystals (PCs), namely, materials possessing a photo-nic band gap (PBG) and capable of photon localization, have awide range of potential applications [1–4], because of theirunique ability to control the flow of photons and guide thepropagation of light. Applications include omnidirectional reflec-tors [5], PC fibers and devices [6,7], nanocavity and laser [8,9],large angle bends and low loss waveguides [10], and correspond-ing coupled devices [11]. The generated type, width, and corre-sponding frequency of the PBGs are closely related to the PCconfiguration and the composition of the substrate materials aswell as size, shape, and arrangement of scatterers. For periodicphotonic crystals (PPCs), substrate materials with high relativepermittivities are required to generate a complete PBG. In con-trast, quasi-periodic photonic crystals (QPCs; also known asphotonic quasicrystals (PQCs)) do not require such high relativepermittivities. A two-dimensional (2D) dodecagonal PQC made ofan array of, for example, air holes in either silicon nitride(er¼4.08, n¼2.02) or glass (er¼2.10, n¼1.45), generates a com-plete band gap [12]. Also, a 2D octagonal PQC of an array of
012 Published by Elsevier B.V. All
arbin Institute of Technology,
dielectric rods (er¼2.4, n¼1.55) surrounded by air generates acomplete band gap [13]. Indeed, a complete PBG can be obtainedfor PQCs with relative permittivity threshold values as small aser¼1.6 (n¼1.26) [14].
However, PBGs of a 2D PQC with a threshold value relativepermittivity were reported mostly for a single unit cell scope[12,14]. These results were specialized with no generality. In thispaper, four square unit cells forming the basic structural unit ofthe 2D decagonal PQC structure were chosen. Consideringtwo constructions (dielectric-cylinders-in-air and air-cylinders-in-dielectric) and values for the relative permittivity of thedielectric in the range of 1–30, band structures for the decagonalPQCs were calculated using the plane wave expansion (PWE)method. The results from this calculation will be more general.
2. Algorithm and model
The density of states [15], transmission spectra [16,17,12] andband structures [18,19] are general means to characterize PBGs of2D PQCs. Because the density of states and the spectral linesdepend continuously on frequency (or wavelength), it is difficultto determine the boundary between the band-gap and the non-band-gap regions. Therefore, values for the band gap width (orboundary) and the corresponding frequency range are hard todetermine. These values are more easily obtained from the band
J. Liu et al. / Optics Communications 288 (2013) 52–55 53
structure, which is usually calculated using the PWE method[18,19]. This method has many advantages, including a simpletheoretical analysis, a clear physical conceptual basis, and aresearch tool in new photonic crystals. For this reason, the PWEmethod is adopted for our PBG calculations.
Exploiting central symmetry of the 2D PQC and locating allscattering centers along the top and bottom edges of the squareunit cell, the four unit cell structures are obtained (see Fig. 1). Weeasily establish L1¼1.6180a, L2¼3.6180a, L3¼4.2361a, andL4¼5.2361a. The unit cell areas are S1¼2.6179a2, S2¼13.0899a2,S3¼17.9445a2, and S4¼27.4167a2.
Fig. 1. Square unit cells in the basic structural unit of a 2D decagonal PQC.
Fig. 2. (Color online.) Dielectric permittivity dependence of PBGs for the different sq
3. Results and discussion
Without loss of generality, two constructions considered are(1) dielectric cylinders (er) arranged in air (eair¼1) and (2) aircylinders arranged in dielectric. According to the unit cell areas, weset the number of bands as 3, 14, 18, and 28 in the calculation. Therelative permittivity er of the dielectric material is allowed to vary inthe range 1–30 and the radius of scatterer is set at r¼0.3a for allunit cells.
3.1. PBGs with dielectric cylinders arranged in air
PBGs for the different unit cells with dielectric cylinders are shownin Fig. 2, from which the following observations can be made:(1) Different unit cells generate different PBG structures with theTM band gap dominating; (2) To generate a TM band gap, the relativepermittivity of the dielectric cylinder must be above threshold valuesof e1min ¼ 5:34 (the center frequency is o01¼0.1891(2pc/a)), e2min ¼
2:88 (o02¼0.4184(2pc/a)), e3min ¼ 7:20 (o03¼0.2786(2pc/a)), ande4min ¼ 4:80 (o04¼0.3392(2pc/a)); (3) As the relative permittivity ofthe scatterer increases, the widths of the TM band gaps change onlyslightly, whereas the center frequencies of the band gaps decrease assmooth hyperbola-like curves; and (4) The TM band gap curves fromall four unit cells are parallel to each other.
3.2. PBGs with air cylinders arranged in dielectric
PBGs for the different unit cells with air cylinders (r¼0.3a)arranged in a dielectric medium (er in the range of 1–30 areshown in Fig. 3. The results produce the following observations:(1) Different unit cells produce different PBG structures althoughin all the TE band gap and complete band gap dominate; (2) To
uare unit cells of a decagonal PQC for a dielectric-cylinders-in-air construction.
Fig. 3. (Color online.) Dielectric permittivity dependence of PBGs for the different square unit cells of a decagonal PQC for an air-cylinders-in-dielectric construction.
J. Liu et al. / Optics Communications 288 (2013) 52–5554
generate the TE band gaps and complete band gaps, the respectivethreshold values for the relative permittivity (correspondingcenter frequencies follow in parentheses) of the backgrounddielectric material are e1min ¼ 5:70 and 7.08 (o01¼0.2976(2pc/a) and 0.2802(2pc/a)), e2min ¼ 4:38 and 10.74 (o02¼0.2602(2pc/a) and 0.2426(2pc/a)), e3min ¼ 7:14 and 12.12 (o03¼0.2118(2pc/a) and 0.1922(2pc/a)), and e4min ¼ 5:76 and 6.06 (o04¼0.2720(2pc/a) and 0.2747(2pc/a)); (3) Compared with the TE band gaps,the complete band gaps appear in the high frequency region; (4)As the relative permittivity of the background dielectric increases,the band gap widths vary little whereas the correspondingcenter frequencies decrease in smooth ‘‘hyperbola-like’’ curves;and (5) For all unit cells, the band gap curves are parallel toeach other.
3.3. Qualitative relationship of PBG impact factors with changing the
relative permittivity of the dielectric
From electromagnetic theory, k¼o ðemÞ1=2¼o e0erm0mr
� �1=2�
o e1=2r =c, and o¼ kc e�1=2
r , for a given unit cell. The change in thecenter frequency of the band gaps for the two constructions can bequalitatively obtained. On increasing the value of the relativepermittivity of the dielectric material (cylinders for Fig. 2 andbackground for Fig. 3), the effective relative permittivity of the unitcell increases whereas o decreases and the corresponding centerfrequency decreases. The effective relative permittivity can becalculated using the Maxwell–Garnett formula [20]
ereff ¼ erbþ3f c erbera�erb
eraþ2erb�f c era�erbð Þ, ð1Þ
where era is the relative permittivity of the scatterer, erb the relativepermittivity of the background medium, and fc is the fill factor ofscatterers in the unit cell.
If the radius of the scatterer is r¼0.3a, the values of the four unitcells fc is 0.4249, 0.3796, 0.3597, and 0.3517; fill factor decreases asthe unit cell scope increases. For both the dielectric-cylinders-in-airconstruction, era¼er and erb¼1 and the air-cylinders-in-dielectricconstruction, era¼1 and erb¼er, the relationship between e�1=2
reff andthe relative permittivity of the dielectric given in formula (1) isplotted in Fig. 4 for each unit cell. Clearly, increasing the relativepermittivity of the dielectric decreases monotonically e�1=2
reff for eachunit cell in smooth ‘‘hyperbola-like’’ curves.
If changes in PBG features are characterized by e-1=2reff , which
from our previous work [21] depends on ‘‘effective massm¼q1epr2’’ and ‘‘environmental factor q2’’, the qualitative rela-tionship of PBG impact factors can be expressed as
Fer ¼ Fer q,e�1=2reff
� �ð2Þ
where q is a construction-dependent factor for the unit cell andereff is the effective relative permittivity of the unit cell. Fromformula (2), the relationship between the PBG and the relativepermittivity of the dielectric can be transformed into a relation-ship between the PBG and the effective relative permittivity of theunit cell. Also by formula (2), the construction-dependent factor q
determines the type of PBG. If era4erb, the unit cell readilygenerates a TM band gap, whereas if eraoerb, the unit cellgenerates a TE band gap and a complete band gap. The effectiverelative permittivity ereff determines the variation in the centerfrequency of the PBG, i.e., the center frequency changes withincreasing ereff or decreasing e�1=2
reff in a smooth ‘‘hyperbola-like’’ curve.
Fig. 4. Relationship between e�1=2reff and relative permittivity er of the dielectric for each of the unit cells in the two PQC constructions.
J. Liu et al. / Optics Communications 288 (2013) 52–55 55
It should be emphasized that the relationship between thesePBG impact factors for these 2D PQCs is solely qualitative (formula(2)). It summarizes only the main factors that affect the PBGcharacteristics, and does not fully characterize the specific infor-mation of the PBG. The qualitative relationship is also effectivewhen interpreting the impact factor of the PBG characteristics of2D PPC [1].
4. Conclusion
In the basic structural unit of a 2D decagonal PQC, PBGs of thefour square unit cells used in two constructions with variablerelative permittivity have been calculated. Different unit cellspossess different PBG structures and have different relativepermittivity threshold values. For a given unit cell, the type ofthe PBG generated and the variation in its center frequency aredependent on the construction and the effective relative permit-tivity, respectively. With era4erb, the TM band gap dominated; incontrast, with eraoerb the TE band gap and complete band gapdominate. In both cases, the center frequency variation changeswith increasing ereff or decreasing e-1=2
reff as smooth ‘‘hyperbola-like’’curves.
References
[1] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade., Photonic Crystals:Molding the Flow of Light, Second Edition, Princeton University Press,3Market Place, Woodstock, Oxfordshire OX20 1SY, 2008.
