近二十年來出現一個稱為光子晶體的名詞，光子晶體的本質為一週期結構，通常由兩種或兩種以上的介電材料所組成，透過適當的設計，光子晶體可以擁有隔絕電磁波傳遞的能力，因與電子能階中存在的電子能隙現象相似，人們稱之為光子能隙。若將缺陷引入光子晶體當中，即移除一個或多個填充物，會造成光子晶體的週期性被破壞，進而產生缺陷模態，若為點缺陷，則可形成共振腔，具有侷限能量的能力，若為線缺陷，則可形成光波導，使得原本不能通過結構的頻率得以通過。在選定材料、填充方式與幾何尺寸之後，光子晶體的色散特性即為固定，若欲調整其光學性質，則須重新製作整個結構，在應用上相當不便，因此有學者提出利用具有可調性的材料組成光子晶體，將可以避免掉這個問題，磁光材料即屬於可調性材料的一環，透過外加磁場的方式，可以改變磁光材料的材料參數，進而影響光子晶體的光學性質。本文將磁光材料與阿基米德晶格相結合，分析光子晶體在不同尺寸與填充方式下的色散特性，藉以求得最佳的設計方式，而外加磁場對於色散性質的改變亦屬於我們討論的範疇。我們緊接著利用阿基米德晶格本身的對稱性，設計一環形共振腔，透過平面波展開法與有限元素軟體分析其共振頻率與電磁場分佈，在引入磁場這個變因之後，探討其造成的模態分裂現象，此外，我們改變共振腔內部的結構藉以優化共振腔的效能。文末，我們與一般常見介電材料所組成的光子晶體相互比較，羅列其優缺。

SUMMARY

We numerically analyze photonic crystals with Archimedean tilings arrangement, which were consisted of dielectric materials and magneto-optical (MO) materials. In the beginning, the dispersion curves of three different type of photonic crystals were calculated by plane wave expansion method. Among them, the one which is composed of air holes in the MO background is suitable for applications. We futher remove six air holes of the photonic crystal to form a ring cavity, and the whispering-gallery-mode is observed in it. When the magnetic field is applied along the out-of-plane direction, the permittivity of MO materials will be changed and the resonant frequency of whispering-gallery-mode will split. Besides, we introduce different size of circular hole and ring-shape hole into the center of the ring cavity to optimize the efficacy. Finally, a common photonic crystal which is consisted of air holes in the silicon background at the same filling ratio is used to compare the performance.

Key words: Magneto-optical materials, whispering-gallery-mode, Archimedean tilings

INTRODUCTION

Over past decades, people pay a lot of attention to a kind of periodic dielectric structure named photonic crystal. In certain regions of the frequency, the optical wave can not pass the photonic crystals, and these regions are called photonic band gaps. When point defects or line defects are introduced into the perfect photonic crystals, it will cause highly localized resonance and guide modes relatively at the frequency within the band gap, and thus many novel applications such as sensors, filters and coupler are realized.
Archimedean tilings are constituted by regular convex polygons which are not identical necessarily. There are not any intervals or overlaps between these regular convex polygons, therefore, the vertices are all identical. Although there exists many kinds of Archimedean tilings, once the parameters of photonic crystal are determined, the wave propagation behaviors are fixed. Thus, how to tune the properties of photonic crystals becomes an important issue. Magneto-optical materials are one kind of tunable materials. When the external magnetic field is employed, the permittivity or the permeability will be changed. Based on this phenomenon, tunable photonic crystal devices can be designed. In this work, we combine photonic crystals with Archimedean tilings, and the wave propagation properties, i.e. width of band gaps and resonant frequencies of photonic crystal which is composed of magneto-optical materials with Archimedean tilings arrangement are studied. The influence of external magnetic field is analyzed, too. Computations are performed by using plane wave expansion and finite element software.

MATERIALS AND METHODS

In this section, we will illustrate the plane wave expansion method which is used to calculate the dispersion curves of photonic crystals.
The Maxwell equations are shown below:
(1)
(2)
Where "E" ⃑ is the intensity of electric field, "H" ⃑ is the intensity of magnetic field, "ε" _"r" is the relative permittivity, "ε" _"0" is permittivity in vacuum, and "μ" _"0" is permeability in vacuum.
Assume that the electric field and the magnetic field are both time harmonic functions and can be defined as:
(3)
(4)
When we introduce equations (3) and (4) into equation (1) and (2), the Helmholtz’s equation can be derived, as shown below:
(5)
According to Bloch’s theorem, the magnetic field in the periodic structure can be expressed as:
(6)
Where "G" ⃑ is the reciprocal lattice vector, "k" ⃑ is Bloch’s wavevector, and ("e" _"λ" ) ̂ are two unit vectors which are perpendicular to ("k" ⃑+"G" ⃑).
Since "1" /("ε" _"r" "(" "r" ⃑")" ) is a periodic function, it can be expanded into Fourier series, as shown below:
(7)
(8)
Where Ω represents the unit cell, and V is the volume of the unit cell.
By substituting equation (7) and (8) into equation (5), we can obtain an eigenfunction defined as:
(9)
Equation (9) can be separated into two different eigenfunctions as shown as equation (10) and (11) relatively according to E-polarization(TE) and H-polarization(TM).
(10)
(11)
If the external magnetic field is applied along z direction, the permittivity of the MO materials can be defined as[72]:
"ε" ̂"=" [■("ε" _"xx" &"ε" _"xy" &"0" @"ε" _"yx" &"ε" _"yy" &"0" @"0" &"0" &"ε" _"zz" )] (12)
And the equation (9) will be modified as shown below:
∑_("G" ^"'" )▒|"k" ⃑"+" "G" ⃑ ||"k" ⃑"+" "G" ⃑^"'" |[■("e" ̂_"2" "∙κ(" "G" ⃑-"G" ⃑^"'" ")" "e" ̂_"2" ^"'" &-"e" ̂_"2" "∙κ(" "G" ⃑-"G" ⃑^"'" ")" "e" ̂_"1" ^"'" @-"e" ̂_"1" "∙κ(" "G" ⃑-"G" ⃑^"'" ")" "e" ̂_"2" ^"'" &"e" ̂_"1" "∙κ(" "G" ⃑-"G" ⃑^"'" ")" "e" ̂_"1" ^"'" )]{■("h" _("1," "G" ^"'" )@"h" _("2," "G" ^"'" ) )} "=" "ω" ^"2" /"c" ^"2" {■("h" _("1," "G" ^"'" )@"h" _("2," "G" ^"'" ) )} (13)
Equation (13) can be separated into two different eigenfunctions as shown as equation (14) and (15) relatively according to E-polarization(TE) and H-polarization(TM).
∑_("G" ^"'" )▒〖"H" ⃑_"k" ^"TM" ("G" ⃑"," "G" ⃑^"'" ) "h" _("1," "G" ⃑^"'" ) "=" "ω" ^"2" /"c" ^"2" "h" _("1," "G" ⃑ ) 〗 (14)
"H" ⃑_"k" ^"TM" ("G" ⃑"," "G" ⃑^"'" )="ε" _"xx" ^"-1" ("G" ⃑-"G" ⃑^"'" )("k" _"y" "+" "G" _"y" )("k" _"y" "+" "G" _"y" ^"'" )
" +" "ε" _"yy" ^"-1" ("G" ⃑-"G" ⃑^"'" )("k" _"x" "+" "G" _"x" )("k" _"x" "+" "G" _"x" ^"'" )
-"ε" _"xy" ^"-1" ("G" ⃑-"G" ⃑^"'" )("k" _"y" "+" "G" _"y" )("k" _"x" "+" "G" _"x" ^"'" )
-"ε" _"yx" ^"-1" ("G" ⃑-"G" ⃑^"'" )("k" _"x" "+" "G" _"x" )("k" _"y" "+" "G" _"y" ^"'" )
∑_("G" ^"'" )▒〖"H" ⃑_"k" ^"TE" ("G" ⃑"," "G" ⃑^"'" ) "h" _("2," "G" ⃑^"'" ) "=" "ω" ^"2" /"c" ^"2" "h" _("2," "G" ⃑ ) 〗 (15)
"H" ⃑_"k" ^"TE" ("G" ⃑"," "G" ⃑^"'" )="ε" _"zz" ^"-1" "(" "G" ⃑-"G" ⃑^"'" ")" |"k" ⃑"+" "G" ⃑ ||"k" ⃑"+" "G" ⃑^"'" |
The two equations above indicate that the off-diagonal elements of the dielectric tensor will affect the dispersion relationship of photonic crystals only when the optical wave is TM polarized. Thus, we just discuss the behavior of TM polarized waves.

RESULTS AND DISCUSSION

The unit cell of the photonic crystal with Archimedean tilings arrangement is shown in figure 1. It is consisted of air holes arranged in MO background whose permittivity is 4.75. The largest width of band gap whose value is 0.025(a/λ) appears when the radius of air holes are 0.38a,as shown in figure 2, where a is the side of the regular polygon. We futher remove six air holes of the photonic crystal to creat a ring cavity, as shown in figure 3. The plane wave is incident from left, and there exists five eigenmode in the range of band gap. Moreover, the whispering-gallery-mode occurs when the normalized frequency is 0.351673(a/λ), and its quality factor is 3885. If the wavelength of this frequency is designed to locate at 1550 nm, then the value of side of polygon is 545.093 nm. When applying an external magnetic field to the photonic crystal, the resonant frequency of the whispering-gallery-mode will separate. Figure 4 shows the wavelength spectrum when g is 0.4. As long as the magnetic field is large enough, the two resonant peak can be identified, and the quality factor can maintain at a high level. By appropriately introducing circular hole or ring-shape hole into the center of the ring cavity, the performances of the cavity will be better. Finally, a common photonic crystal which is consisted of air holes arranged in silicon background is analyzed. Because the contrast of impedance between silicon and air is larger than that between MO materials and air. Thus, the value of quality factor and separation of the latter are more higher than the former.

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