The optical transmission characteristics in metallic photonic crystals

Materials Chemistry and Physics 124 (2010) 856–860

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The optical transmission characteristics in metallic photonic crystals Arafa H. Aly ∗ , Hussein A. Elsayed, Hany S. Hamdy Physics department, Faculty of Sciences, Beni-Suef University, Egypt

a r t i c l e

i n f o

Article history: Received 16 February 2010 Received in revised form 11 July 2010 Accepted 1 August 2010 Keywords: Transmittance, Photonic crystal, Metallic

a b s t r a c t We theoretically studied electromagnetic wave propagation in a one-dimensional metal/dielectric photonic crystal (1D MDPC) consisting of alternating metallic and dielectric materials by using the transfer matrix method in visible and infrared regions. We have investigated the photonic band gap by using four kinds of metals: silver, lithium, gold and copper. We discuss the details of the calculated results in terms of the thickness of the metallic layer and different kinds of metals, and the plasma frequency. Our results have a potential for applications in optical devices because it is easy and cheap to manufacture. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Photonic crystals (PCs) have periodic structures exhibiting the appearance of forbidden frequency regions—the so-called photonic band gaps (PBGs). Since the first pioneering work [1] in this field, many new interesting ideas have been developed dealing with onedimensional (1D), two-dimensional (2D), and three-dimensional (3D) PCs. Researchers have proposed many new and unique applications of photonic devices which may revolutionize the field of photonics in much the same way as semiconductors revolutionized electronics. The existence of photonic band gaps (PBGs) in PCs, owing to multiple Bragg scatterings, provides a guide to numerous interesting phenomena: for example, the suppression of spontaneous emission [2] and photon localization [3,4]. Many physical features can be modified with the presence of PBGs. Researchers suggest, theoretically, that Planck blackbody radiation can be suppressed in the PBG regions of 1D PCs [5,6]. Reinforcement and suppression of thermal emission and absorption were reported experimentally in Si-based 3D PCs [7] and theoretically in 1D dielectric PCs [8], respectively. There have been many reports on metallic PCs, most concentrated on microwave, millimeter wave, and far-infrared frequency [9–15]. At these frequencies, metals act like nearly perfect reflectors with no crucial absorption problems. Studies of metallic PCs in an optical frequency mainly focus on the reflectance and absorption phenomena, as metals are highly reflective, as well as absorptive in this region [16–18]. Metals are more reflective than semiconductors over a wide range of wavelengths, so they tend to exhibit

∗ Corresponding author. E-mail address: [email protected] (A.H. Aly). 0254-0584/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2010.08.006

pronounced PBGs with fewer layers than semiconductor-based PCs. Metallic photonic crystals can possess more extensive PBGs than their semiconductor counterparts. Recently, researchers have, based on theoretical and experimental work reported that it is possible to make a 1D metal/dielectric photonic crystal (MDPC) that contains hundreds of nanometers thick metal that is transparent to visible wavelength [19–22]. It is due to the transmission resonance phenomena as in the Fabry–Perot cavity. This MDPC is conductive and transparent in the visible range, blocking ultraviolet and infrared wavelength. Thus, it could be used for sensor protection, UV-protective films, and transparent conductive produce panels [23]. The reinforcement of optical transmission in this structure is very important in improving the performance of the above-mentioned devices. In this paper, we discuss our study of the optical properties of the 1D MDPC: transmittance independent on the kind of metal employed in the structure. Furthermore, in our study, we showed the dependence of the transmittance of a metal/dielectric multilayer on the thickness of the metallic layer. The proposed structures can be potentially useful for applications because MDPCs have many advantages over dielectric-based photonic crystals; metal components can be used for wiring of future optoelectronic merger circuits. The PBGs of MPCs can be more robust against structural disorders.

2. Basic equations In order to investigate the dispersion relation and the transmittance characteristics of the incident electromagnetic waves, the Maxwell’s equation is solved numerically by the transfer matrix method [24]. The structure used is shown in Fig. 1, composed of metal and dielectric layers that are repeated for (N) period being situated

A.H. Aly et al. / Materials Chemistry and Physics 124 (2010) 856–860

857

Fig. 1. A metallic-dielectric structure; the thicknesses of metal and dielectric are denoted by d1 and d2 , respectively, and the corresponding refractive indices are separately indicated by no , n1 , n2 ,and n , where no = 1 is taken and n is the index of substrate layer.

between two media; the substrate and the air. The refractive index of metallic layer can be calculated using the Drude model i.e.,



n21

=1−



ωp2

(1)

ω(ω − i)

where ω is the frequency of the incidence radiation, ωp and ␥ are frequency-independent parameters and the thickness of metallic layer is d1 . The plasmon frequency is defined by: ωp2 =

ne2 ε0 meff

(2)

n2 n(x) = .. ⎪ ⎪ ⎪ ⎪. ⎩ n

x ≺ x0 x0 ≺ x ≺ x1 x1 ≺ x ≺ x2 .. . x2N ≺ x

with with .. . with

x1 = x0 + d1 x2 = x1 + d2 .. . x2N = x2N−1 + d2

(3)

⎧ −i k (x−x ) i k0x (x−x0 ) 0x 0 ⎪ ⎨ A0 e−i k1x (x−x1 ) + B0 ei k1x (x−x1 ) x ≺ x0 A1 e

+ B1 e

x0 ≺ x ≺ x1

A2 e

+ B2 e

x  x2

−i k2x (x−x2 ) i k2x (x−x2 ) ⎪ ⎩ A2 e−i k2x (x−x2 ) + B2 ei k2x (x−x2 ) x1 ≺ x ≺ x2

Am−1 Bm−1





−1 = Dm−1 Dm

Am  Bm





−1 = Dm−1 Dm Pm

Am Bm





Dm =

 Dm

1 nm cos m

cos m nm

m =

1 −nm cos m

cos m −nm



for TE waves

(6-a)

(6-b)

0 cos m − i sin m



(7)

(m = 1, 2, 3, . . .)

The relation between the amplitudes of the plane wave can be written as A0 B0





= M(a)

A2 B2



= D0−1 D1 P1 D1−1 D2 P2 D2−2 D3

(8)

where M(a) is the matrix for one period, since a = (d1 + d2 ) and a is the lattice constant. This matrix can take the following form



M(a) =

m11 m21

m12 m22



(9)

The elements m11 , m12 , m21 and m22 were computed in the case of TE waves which are given as 1 2



1 + 2 m12 =

1 2

1 + 2 m21 =

1 2 −

m22 =

1 2 −

1+



3

0

1−



cos 2 +

3

0



1−



1

3

0

1

2

0



1+



1

−

3

0

1

0





1

1

2

0

1

0

−



1

2

0

−



1



2

0

+



2

1

1 3

0 2



 (i sin 2 ) (i sin 1 )



 (i sin 2 ) cos 1

1 3

0 2



 (i sin 2 ) (i sin 1 )



 (i sin 2 ) cos 1

−

3

2



(i sin 2 ) cos 

−

3

2

2



+

3

2

2



cos 2 −

3

2

2

2

cos 2 −

+





cos 2 +

cos 2 −



3



cos 2 +

cos 2 −



3



+



cos 2 +

3 1 −

0

1



2



3 1 +

0

1



1 3

0 2





+

 (i sin 2 ) (i sin 1 )

 (i sin 2 ) cos 1

1 3

0 2



 (i sin 2 ) (i sin 1 )

where

i = ni cos i ; where i = 0, 1, 2, 3

(10)

For a system of N periods we can modify the previous expression as [12]





for TM waves

2dm nm cos m

(5)

with m = 1, 2, . . . 2 N + 1.where matrices D (dynamical matrix) and P (propagation matrix) can be written as

cos m + i sin m 0

where

(4)

where km x is the x-component of the wave vector, km x = nm ωc cos m (m = 1, 2, 3 . . .). Am and Bm represent the amplitude of the plane wave at each interface x = xm . It is well known that the distribution of the electric field was taken for N = 1, and we will obtain the general case for N period. The amplitudes of the plane waves at different layers can be related by



Pm =

m11 =

The electric field of a general plane-wave solution can be written as E = E(x)ei(ω t−ˇ z) , where ˇ is the z-component of the wave vector; ˇ = nm ωc sin(m ) (m = 1, 2, 3, . . . ),  m is the angle of the incident ray in each layer and c is the speed of light and the electric field distribution E(x) can be written as:

E(x) =





where n is the electron density, meff is the effective mass of the electron, e is the electronic charge and the permittivity of the vacuum is ε0 . For the dielectric layer the refractive index is n2 and the layer thickness is d2 . The electromagnetic waves are propagating through the multilayer structure along x-direction (Fig. 1), and the structure can be defined along this direction as

⎧ n0 ⎪ ⎪ ⎪ ⎪ ⎨ n1

Since the propagation matrix can be written in another form for simplifying

M(Na) =

M11 M21

M12 M22



(11)

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A.H. Aly et al. / Materials Chemistry and Physics 124 (2010) 856–860

Fig. 2. Transmittance spectra dependence on the visible frequencies, d1 = 5, d2 = 196 nm and N = 6; (a) Ag–Al2 O3 , (b) Li–Al2 O3 , (c) Au–Al2 O3 and (d) Cu–Al2 O3 .

whose elements M11 , M12 , M21 and M22 can take the form M11 = m11 UN−1 ( ) − UN−2 ( ), M12 = m12 UN−1 ( ) M21 = m21 UN−1 ( ) and M22 = m22 UN−1 ( ) − UN−2 ( )

(12)

with  = 0.5(m11 + m22 ) and UN ( ) =

sin((N + 1) cos−1  )



(13)

1 − 2

where UN ( ) is Chebyshev polynomials of the second kind,  is the Bloch phase for a single period [25]. We can obtain the reflection and the transmission coefficients r=

(M11 + M12 f ) f0 − (M21 + M22 f ) (M11 + M12 f ) f0 + (M21 + M22 f )

t=

2f0 (M11 + M12 f ) f0 + (M21 + M22 f )

where

f0 =

ε0 n0 cos 0 and 0

f =

and

ε0 n cos  0 

(14)

(15)

Finally we can calculate the transmittance and the reflectance using the following expressions

 

R = r 2 

and, T =

f1 2 |t | f0

(16)

3. Results and discussions In our paper we theoretically studied electromagnetic wave propagation in a one-dimensional metal/dielectric photonic crystal (1D MDPC) consisting of alternating metallic and dielectric materials by using the transfer matrix method in visible and infrared regions. The magnitude of the refractive index for the dielectric medium was obtained from references [26]. Also the value of thickness for the dielectric layer was kept equal to 196 nm for all our results

and the refractive indices of metals were given as a function of the Plasmon frequency that had been calculated by using Drude model. The magnitude of the skin depth can be described as a function of the wavelength and the extinction coefficient of the metal i.e., ı = /4 k. [27]. The thickness of metallic layer is the significant parameter which had been chosen according to the value of the skin depth of each metal (i.e. d2 ≺ ı for all metals).In Fig. 2 we studied four different types of photonic crystals composed of layers of the dielectric material (Al2 O3 ) and the metals silver (Ag), lithium (Li), gold (Au) and copper (Cu). We have calculated our results (Fig. 2) in visible wavelength and we have obtained a complete PBG between 400 and 750 nm, which covers almost all of the visible wavelength range; this indicates that such PCs are good reflectors. In the case of Ag (Fig. 2a) we can see the magnitude of transmittance (T) near the unity at < 400 nm and in the case of Li (Fig. 2b), T is near 0.4 but in the case of Au (Fig. 2c), T is reduced to 0.1 with the same PBG range 400–750 nm. For the case of Cu (Fig. 2d) the transmittance is reduced below 0.03 at < 400 nm and we have obtained the same PBG between 400 and 750 nm. Fig. 3 shows the photonic band gap in visible wavelength 400–750 nm corresponding to the Ag–Al2 O3 structure for the cases of metal layer thickness of 3, 5, and 7 nm, respectively. In Fig. 3 we have shown the clearest and completed PBG in the visible wavelength and also the effect of the metal thickness on the transmittance values as well as Fig. 3 shows the transmittance depends on the thickness within the range 300–400 nm.Under the same conditions as in the previous results we have studied the four types of PCs in the infrared (IR) wavelengths (Fig. 4); Ag, Li, Au and Cu which are composed of the dielectric material (Al2 O3 ) to form the new type of photonic crystals. We performed our calculation at IR wavelengths (Fig. 4) to obtain a complete PBG between 1700 and 3000 nm. In the case of Ag (Fig. 4a) we can see the magnitude of transmittance near the unity at < 1700 nm and T in the case of Li (Fig. 4b) is near 0.8 but in the case of Au (Fig. 4c) the magnitude of transmittance is reduced to 0.5 with the same PBG range 1700–3000 nm. In Fig. 4(d) the magnitude of transmittance in the case of Cu is reduced below

A.H. Aly et al. / Materials Chemistry and Physics 124 (2010) 856–860

Fig. 3. Transmittance spectra dependence on the visible frequencies at different thicknesses, with d2 = 196 nm and the structure is Ag–Al2 O3 .

0.3 at < 1700 nm and we obtained the same PBG between 1700 and 3000 nm. Fig. 5 shows the photonic band gap in IR wavelength (1700–3000 nm) corresponding to the Ag–Al2 O3 structure for the cases of metal layer thickness of 3, 5, and 7 nm, respectively. Also Fig. 5 shows an oscillation transmittance curve in the 1D MDPC within (800–1700 nm). This oscillatory behavior of the transmittance spectrum, resulting from the periodic structure of metals and dielectrics, can be reduced slightly by varying the thickness of metals and/or dielectric layers. We have shown that the magnitude of transmittance increases when the thickness of metallic layer decreases (Figs. 3 and 5), thus we found the high value of transmittance at d1 = 3 nm, is near unity (red line) while the magnitude of transmittance at d1 = 7 nm (blue line) is 0.7 in Fig. 3. This is why the thickness of the layer is one of important parameters and should be highly concerned in designing photonic or any optical devices depend on photonic as well. Moreover, the thickness of each layer plays a significant role in this case and more attention

859

Fig. 5. Transmittance spectra dependence on the IR frequencies at different thicknesses, with d2 = 196 nm and the structure is Ag–Al2 O3 .

should be paid to when we design photonic devices or any optical devices. The magnitude of transmittance along the visible wavelength and IR was found different from metal to another, which depends on the surface characteristics for each metal. Our results indicating to Copper is lossy and is not recommended for optical photonic crystals. Silver, Lithium and Gold are acceptable although slightly lower in performance. 4. Conclusion Using a well-known transfer matrix method, we calculated the transmission spectra of the 1D metal-dielectric periodic structure for different metals (Ag, Au, Li and Cu) in both the visible region (Figs. 2 and 3) and the near infrared region (Figs. 4 and 5). We can conclude that the magnitude of the transmittance for the different structures was found to be near zero within visible wavelengths,

Fig. 4. Transmittance spectra dependence on the IR frequencies, d1 = 5, d2 = 196 nm and N = 6; (a) Ag–Al2 O3 , (b) Li–Al2 O3 , (c) Au–Al2 O3 and (d) Cu–Al2 O3 .

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A.H. Aly et al. / Materials Chemistry and Physics 124 (2010) 856–860

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