- Email: [email protected]
Research in light transmission characteristics of 1-dimensional photonic crystal
Optik 123 (2012) 314–318
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Research in light transmission characteristics of 1-dimensional photonic crystal Yi Lin ∗ , Huan Xu 1 School of Physical Electronics, University of Electronic Science & Technology of China, Chengdu 610054, China
a r t i c l e
i n f o
Article history: Received 14 October 2010 Accepted 23 March 2011
Keywords: Photonic crystals Photonic bandgap Transfer matrix Light transmission characteristics
a b s t r a c t In this paper, the principal and characteristics of one-dimensional photonic crystal are introduced, its transmission matrix is derived; and its light transmission characteristics are also analyzed through numerical calculation by means of transfer matrix method, including impacts of dielectric layer thickness, permittivity, and dielectric periodicity, etc. on light transmission characteristics of 1-D photonic crystals. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals are composed of periodic dielectric layers or metallo-dielectric nanostructures that affect the propagation of electromagnetic waves (EM) in the same way as the periodic potential in a semiconductor crystal affects the electron motion by defining allowed and forbidden electronic energy bands. Essentially, photonic crystals contain regularly repeating internal regions of high and low dielectric constant. Photons (behaving as waves) propagate through this structure – or not – depending on their wavelengths. Wavelengths of light that are allowed to travel are known as modes, and groups of allowed modes form bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena such as inhibition of spontaneous emission, high-reflecting omni-directional mirrors and low-loss-wave guiding, amongst others. Some familiar research methods include transfer matrix method, plane-wave expansion method and finite-difference timedomain (FDTD) method, etc. While it may not be easy for the results to be correctly or accurately calculated using the plane-wave expansion method when many plane waves are needed for baroque phonic crystals; it may cause dispersion to the wave mode being simulated during numerical calculation of the Maxwell equations using FDTD method. In contrast to the former two methods, transfer matrix method is much more effective for metallo-dielectric
∗ Corresponding author. Tel.: +86 13666208357. E-mail addresses: [email protected] (Y. Lin), [email protected] (H. Xu). 1 Tel.: +86 15882401343. 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.03.027
nanostructures whose permittivity varies with frequency as the amount of calculation is greatly decreased while good accuracy is also achieved because the transfer matrix is very small, and also it is convenient to calculate the reflection and transmission coefficients [1,5]. So we adopt the transfer matrix method to discuss our topic.
2. Theoretical derivation of the transfer matrix of 1-D crystals Transfer matrix method is based on the Maxwell equations, whose physical interpretation is to transfer the tangential components of the electric and magnetic field intensity of the light from one side of the dielectric layer to the other side. Consider a periodic dielectric structure composed of two alternate pile-up dielectric layers whose permittivity is εa , εb respectively. As shown in Fig. 1, the thickness of the two dielectric layers is a, b respectively, thus the period is d = a + b. A beam of light with angular frequency ω and incident angle beams into the periodic dielectric structure. The interaction between the dielectric layers is completely determined by the transfer matrix. Take a single layer as an example, for TE mode, there is no free charge or conduction current in the dielectric, from the tangent component continuity of the electromagnetic field on any interface in the media, we can get the transfer matrix of the single layer:
M(Z) =
cos ıN −iN sin ıN
−i/N sin ıN cos ıN
(1)
Y. Lin, H. Xu / Optik 123 (2012) 314–318
a
where
d=a+b
b
ıN =
ω θ
N =
Z Fig. 1. 1-D periodic dielectric structure.
ω √ z ε cos c
ε/ cos (TE wave) /ε cos
(2)
(TM wave)
c is the velocity of light, z is the thickness of a dielectric layer, ε, is the permittivity and magnetic permeability of the Nth dielectric
Dispersion
3.5
315
T-F Relation
1
π
0.9
3
0.8
Transmissivity(T)
Band Gap(kd)
2.5 2 1.5 1
0.7 0.6 0.5 0.4 0.3 0.2
0.5 0
0.1 0
1
2
3
6
7
8
9
0
10
2
3
0.9 0.8
0.7
0.7
0.6 0.5 0.4 0.3
5
6
7
8
9
0
10
0.7
0.7
Transmissivity(T)
0.8
0.5 0.4 0.3
4
5
6
Frequency(ω/ω0)
(c)
n=5
7
8
9
10
10
7
8
9
10
T-F Relation
0.3
0.1
6
9
0.4
0.2
5
8
0.5
0.1
4
7
0.6
0.2
3
3
1 0.9
2
2
(e) εa / εb = 10
0.8
1
1
(b) n=10
0.9
0
0
Frequency(ω/ω0)
0.6
10
T-F Relation
Frequency(ω/ω0)
T-F Relation
1
9
0.3
0.1 4
8
0.4
0.2
3
7
0.5
0.1 2
6
0.6
0.2
1
5
(d) εa / εb = 5 1
0
4
(a) n=10
0.8
0
1
Frequency(ω/ω0)
0.9
0
0
Frequency(ω/ω0)
Transmissivity(T)
Transmissivity(T)
5
T-F Relation
1
Transmissivity(T)
4
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
(f) εa / εb = 20
Fig. 2. Photonic band gap of the 1-D photonic crystal when na a = nb b.
316
Y. Lin, H. Xu / Optik 123 (2012) 314–318
layer respectively. For the 1-D periodic structure, by successively using (1), we can get the transfer matrix of the structure: A B C D
0.9
(3)
3. Light transmission characteristics of 1-D photonic crystals After we get the transfer matrix of 1-D photonic crystal with N layers, the field intensity of the structure can be presented [2,3]:
E0 H0
= M(Z)
EN+1 HN+1
0.8
Transmissivity(T)
M(Z) = M1 (Z1 )M2 (Z2 − Z1 )· · ·MN (ZN − ZN−1 ) =
(4)
0.6 0.5 0.4 0.3
0.1 0
0
1
2
3
(a)
5
6
7
8
9
10
7
8
9
10
7
8
9
10
θ=π/6
T-F Relation
1
(5)
(6)
Reflectivity: R = |r|2
(7)
0.8
Transmissivity(T)
20 0 (A + BN+1 ) + (C + DN+1 )
0.7 0.6 0.5 0.4 0.3 0.2
Transmissivity:
0.1
T = |t|2
(8)
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
We can easily get: R+T =1
(b)
(9)
For normal incidence ( = 0), we can get the dispersion relation of the 1-D periodic structure:
+ b b a a
0.9
sin ıa sin ıb
0.8
(10)
where k is the wave number,
⎧ ε0 εa ⎪ ⎪ ⎨ a = c√ 0 , ε ωb b ε0 εb ⎪ ⎩ ıb = ⎪ ⎩ b = c 0 ⎧ √ ⎨ ıa = ωa εa
θ=π/4
T-F Relation
1
Transmissivity(T)
1 cos kd = cos ıa cos ıb − 2
4
Frequency(ω/ω0)
0.9
0 (A + BN+1 ) − (C + DN+1 ) 0 (A + BN+1 ) + (C + DN+1 )
Transmission coefficient: t=
0.7
0.2
where E0 , H0 and EN+1 , HN+1 is the electric and magnetic field intensity on the left side of the 1st interface and the right side of the Nth interface of the 1-D periodic structure respectively. Further, we can get the following parameters of the whole structure for TE mode. Reflection coefficient: r=
T-F Relation
1
0.7 0.6 0.5 0.4 0.3 0.2 0.1
4. Numerical calculation and analysis Based on the previous discussion and analysis, we can find the photonic band gap of the 1-D photonic crystal by calculating (3), (7), (8) and (10). For normal incidence ( = 0), the characteristics for TE mode should be the same as that for TM mode. The fundamental frequency (the central frequency of the first forbidden gap) of the photonic crystal is: c ck0 = (11) n na a + nb b √ √ where na = εa a , nb = εb b is the refractive index of dielectric a, b respectively. In order to simplify calculation, unless specified, we assume that all the dielectrics are non-magnetic, and that the photonic crystal is in vacuo, i.e. = 0 , = 0 ; suppose the relative dielectric constant of dielectric a, b is εa = 5, εb = 1 (though some papers evaluate permeability as 0 = 1 and dielectric constant as ε0 = 1, the penman ω0 =
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
(c)
θ=π/3
Fig. 3. Impact of the incident angle on the band gap.
find that it is actually different from that when 0 = 4 × 10−7 H/m, ε0 = 8.85 × 10−12 F/m, so we take the later); number of periods n = 10, i.e. 20 dielectric layers in total. We’ll discuss the characteristics both when na a = nb b and na a = / nb b. 4.1. Discussion when na a = nb b Let the dielectric layer thickness a = 100 nm, then b = (na /nb )a = 223.6 nm, we get the photonic band gap of the
Y. Lin, H. Xu / Optik 123 (2012) 314–318
T-F Relation
0.9
0.9
0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
5
6
T-F Relation
1
Transmissivity(T)
Transmissivity(T)
1
317
7
8
9
0
10
0
1
2
3
Frequency(ω/ω0)
4
(a) g=1.5
0.9 0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
3
4
5
6
7
8
9
0
10
0
1
2
3
Frequency(ω/ω0)
4
T-F Relation
0.8
0.7
0.7
Transmissivity(T)
Transmissivity(T)
0.9
0.6 0.5 0.4 0.3
5
6
7
8
9
10
7
8
9
10
g=3
0.3
0.1 4
10
0.4
0.1 3
9
0.5
0.2
2
8
0.6
0.2
1
6
T-F Relation
1
0.8
0
5
(e)
0.9
0
7
g=2
Frequency(ω/ω0)
(b) g=2.5 1
10
0.3
0.1
2
9
0.4
0.2
1
8
0.5
0.1 0
7
0.6
0.2
0
6
T-F Relation
1 0.9
Transmissivity(T)
Transmissivity(T)
(d)
T-F Relation
1
5
Frequency(ω/ω0)
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
Frequency(ω/ω0)
(c) g=3.5
(f) g=4
Fig. 4. Photonic band gap of the 1-D photonic crystal when na a = / nb b.
1-D photonic crystal as shown in Fig. 2 (only the variables are shown in the figure, similarly hereinafter). Fig. 2(a) and (b) show the photonic band gap from different angles. Fig. 2(a) shows the relation between kd and frequency, the dashed line represents , obviously 0 ≤ kd ≤ , so the area above and below 0 in the Y-direction is forbidden, which indicates the forbidden gap, where kd is an imaginary number. Fig. 2(b) shows the relation between transmissivity T and frequency, we can see that T = 0 at and near the odd times of the fundamental frequency (forbidden band), which means light cannot go through the photonic crystal; whereas T is relatively high at other frequencies (conduc-
tion band) with local recurrent fluctuations. In general, T reaches the peak at the even times of the fundamental frequency; this is in line with the results in [4], which proves the correctness of transfer matrix method. As R + T = 1, the relation between R and frequency is complimentary with Fig. 2(b). So we will use only the T–R relation to analyze the characteristics in the following discussion. Fig. 2(c) shows the band gap when n = 5 while other parameters remain the same. In contrast to Fig. 2(b), we see that the less the number of periods, the less local recurrent fluctuation, and the sparser the image, but the amplitude of the fluctuation remains the same and the band gap is similar.Fig. 2(d), (e) and (g) show the
318
Y. Lin, H. Xu / Optik 123 (2012) 314–318
band gaps when εa /εb = 5, 10, 20 respectively while other parameters remain the same. We can know that the larger ratio εa /εb , the larger forbidden band. Therefore, in order to achieve lager forbidden band, we can consider choosing dielectrics that have larger dielectric constant ratio εa /εb , vice versa. To consider the incident angle ’s impact on the photonic band gap, let = /6, /4, /3, we get the corresponding band gaps as shown in Fig. 3, the gap “gets larger” as the incident angle increases. The so-called “gets larger” is actually extension in the X-direction. The width ratio of the conduction band to that of the forbidden band is the same. The influence that the incident angle has on the photonic crystal is to enlarge its fundamental frequency ω0 . 4.2. Discussion when na a = / nb b Suppose the thickness of dielectric layer a = 100 nm, b = g · (na /nb )a; when the coefficient g = 1.5, 2, 2.5, 3, 3.5, 4, correspondingly b = 335.4, 447.2, 559.0, 670.8, 782.6, 894.4 nm, the corresponding photonic band gaps are shown in Fig. 4. We can see that some band gaps of the photonic crystal break up, and the period of the band gap depends on g: when g is a half-integer, say g = (2m + 1)/2, the period of the photonic band gap T = 2(g + 1)ω0 = (2m + 3)ω0 ; when g is an integer, say g = m, the period of the photonic band gap T = (g + 1)ω0 = (m + 1)ω0 , (m = 0, 1, 2, . . .). The band gap breaks at other frequencies that do not satisfy either of the conditions.
5. Conclusion In this paper, the principal and main application of 1-D photonic crystal are introduced, its characteristic matrix is derived and its light transmission characteristics are also analyzed through numerical calculation by means of transfer matrix method, including impacts of dielectric layers’ thickness, permittivity, and dielectric periodicity, etc. on light transmission characteristics of 1-D photonic crystals. We draw the conclusion that the number of periods of the photonic crystal does not affect the photonic band gap, but the less the number of periods, the less recurrent fluctuation of the transmissivity; the larger difference of the dielectric constant of the dielectric layers, the larger width of forbidden band; the fundamental frequency of the photonic crystal increases with the incident angle; the photonic band gap remains the same at some frequencies and breaks up at the other when na a = / nb b. References [1] B. Liu, Research in characteristics of 1-D photonic crystals that can be used in optical communication, Master Thesis, University of Electronic Science and Technology of Xi’an, 2007, 5–7. [2] H. Wang, et al., An eigen matrix method for obtaining the band structure of photonic crystals, Acta Phys. Sin. 50 (11) (2001). [3] C. Kuang, et al., Transfer matrix for analyzing properties of light propagation in 1-dimension photonic crystals, Laser J. 24 (4) (2003). [4] G. Gu, et al., Properties of light propagation in 1-D periodic dielectric structure, Acta Opt. Sin. 20 (6) (2006). [5] Y. Gao, Research in propagation characteristics of 1-D photonic crystals, Master Thesis, Qufu Normal University, 2007, 9–11.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Research in light transmission characteristics of 1-dimensional photonic crystal Yi Lin ∗ , Huan Xu 1 School of Physical Electronics, University of Electronic Science & Technology of China, Chengdu 610054, China
a r t i c l e
i n f o
Article history: Received 14 October 2010 Accepted 23 March 2011
Keywords: Photonic crystals Photonic bandgap Transfer matrix Light transmission characteristics
a b s t r a c t In this paper, the principal and characteristics of one-dimensional photonic crystal are introduced, its transmission matrix is derived; and its light transmission characteristics are also analyzed through numerical calculation by means of transfer matrix method, including impacts of dielectric layer thickness, permittivity, and dielectric periodicity, etc. on light transmission characteristics of 1-D photonic crystals. © 2011 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals are composed of periodic dielectric layers or metallo-dielectric nanostructures that affect the propagation of electromagnetic waves (EM) in the same way as the periodic potential in a semiconductor crystal affects the electron motion by defining allowed and forbidden electronic energy bands. Essentially, photonic crystals contain regularly repeating internal regions of high and low dielectric constant. Photons (behaving as waves) propagate through this structure – or not – depending on their wavelengths. Wavelengths of light that are allowed to travel are known as modes, and groups of allowed modes form bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena such as inhibition of spontaneous emission, high-reflecting omni-directional mirrors and low-loss-wave guiding, amongst others. Some familiar research methods include transfer matrix method, plane-wave expansion method and finite-difference timedomain (FDTD) method, etc. While it may not be easy for the results to be correctly or accurately calculated using the plane-wave expansion method when many plane waves are needed for baroque phonic crystals; it may cause dispersion to the wave mode being simulated during numerical calculation of the Maxwell equations using FDTD method. In contrast to the former two methods, transfer matrix method is much more effective for metallo-dielectric
∗ Corresponding author. Tel.: +86 13666208357. E-mail addresses: [email protected] (Y. Lin), [email protected] (H. Xu). 1 Tel.: +86 15882401343. 0030-4026/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2011.03.027
nanostructures whose permittivity varies with frequency as the amount of calculation is greatly decreased while good accuracy is also achieved because the transfer matrix is very small, and also it is convenient to calculate the reflection and transmission coefficients [1,5]. So we adopt the transfer matrix method to discuss our topic.
2. Theoretical derivation of the transfer matrix of 1-D crystals Transfer matrix method is based on the Maxwell equations, whose physical interpretation is to transfer the tangential components of the electric and magnetic field intensity of the light from one side of the dielectric layer to the other side. Consider a periodic dielectric structure composed of two alternate pile-up dielectric layers whose permittivity is εa , εb respectively. As shown in Fig. 1, the thickness of the two dielectric layers is a, b respectively, thus the period is d = a + b. A beam of light with angular frequency ω and incident angle beams into the periodic dielectric structure. The interaction between the dielectric layers is completely determined by the transfer matrix. Take a single layer as an example, for TE mode, there is no free charge or conduction current in the dielectric, from the tangent component continuity of the electromagnetic field on any interface in the media, we can get the transfer matrix of the single layer:
M(Z) =
cos ıN −iN sin ıN
−i/N sin ıN cos ıN
(1)
Y. Lin, H. Xu / Optik 123 (2012) 314–318
a
where
d=a+b
b
ıN =
ω θ
N =
Z Fig. 1. 1-D periodic dielectric structure.
ω √ z ε cos c
ε/ cos (TE wave) /ε cos
(2)
(TM wave)
c is the velocity of light, z is the thickness of a dielectric layer, ε, is the permittivity and magnetic permeability of the Nth dielectric
Dispersion
3.5
315
T-F Relation
1
π
0.9
3
0.8
Transmissivity(T)
Band Gap(kd)
2.5 2 1.5 1
0.7 0.6 0.5 0.4 0.3 0.2
0.5 0
0.1 0
1
2
3
6
7
8
9
0
10
2
3
0.9 0.8
0.7
0.7
0.6 0.5 0.4 0.3
5
6
7
8
9
0
10
0.7
0.7
Transmissivity(T)
0.8
0.5 0.4 0.3
4
5
6
Frequency(ω/ω0)
(c)
n=5
7
8
9
10
10
7
8
9
10
T-F Relation
0.3
0.1
6
9
0.4
0.2
5
8
0.5
0.1
4
7
0.6
0.2
3
3
1 0.9
2
2
(e) εa / εb = 10
0.8
1
1
(b) n=10
0.9
0
0
Frequency(ω/ω0)
0.6
10
T-F Relation
Frequency(ω/ω0)
T-F Relation
1
9
0.3
0.1 4
8
0.4
0.2
3
7
0.5
0.1 2
6
0.6
0.2
1
5
(d) εa / εb = 5 1
0
4
(a) n=10
0.8
0
1
Frequency(ω/ω0)
0.9
0
0
Frequency(ω/ω0)
Transmissivity(T)
Transmissivity(T)
5
T-F Relation
1
Transmissivity(T)
4
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
(f) εa / εb = 20
Fig. 2. Photonic band gap of the 1-D photonic crystal when na a = nb b.
316
Y. Lin, H. Xu / Optik 123 (2012) 314–318
layer respectively. For the 1-D periodic structure, by successively using (1), we can get the transfer matrix of the structure: A B C D
0.9
(3)
3. Light transmission characteristics of 1-D photonic crystals After we get the transfer matrix of 1-D photonic crystal with N layers, the field intensity of the structure can be presented [2,3]:
E0 H0
= M(Z)
EN+1 HN+1
0.8
Transmissivity(T)
M(Z) = M1 (Z1 )M2 (Z2 − Z1 )· · ·MN (ZN − ZN−1 ) =
(4)
0.6 0.5 0.4 0.3
0.1 0
0
1
2
3
(a)
5
6
7
8
9
10
7
8
9
10
7
8
9
10
θ=π/6
T-F Relation
1
(5)
(6)
Reflectivity: R = |r|2
(7)
0.8
Transmissivity(T)
20 0 (A + BN+1 ) + (C + DN+1 )
0.7 0.6 0.5 0.4 0.3 0.2
Transmissivity:
0.1
T = |t|2
(8)
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
We can easily get: R+T =1
(b)
(9)
For normal incidence ( = 0), we can get the dispersion relation of the 1-D periodic structure:
+ b b a a
0.9
sin ıa sin ıb
0.8
(10)
where k is the wave number,
⎧ ε0 εa ⎪ ⎪ ⎨ a = c√ 0 , ε ωb b ε0 εb ⎪ ⎩ ıb = ⎪ ⎩ b = c 0 ⎧ √ ⎨ ıa = ωa εa
θ=π/4
T-F Relation
1
Transmissivity(T)
1 cos kd = cos ıa cos ıb − 2
4
Frequency(ω/ω0)
0.9
0 (A + BN+1 ) − (C + DN+1 ) 0 (A + BN+1 ) + (C + DN+1 )
Transmission coefficient: t=
0.7
0.2
where E0 , H0 and EN+1 , HN+1 is the electric and magnetic field intensity on the left side of the 1st interface and the right side of the Nth interface of the 1-D periodic structure respectively. Further, we can get the following parameters of the whole structure for TE mode. Reflection coefficient: r=
T-F Relation
1
0.7 0.6 0.5 0.4 0.3 0.2 0.1
4. Numerical calculation and analysis Based on the previous discussion and analysis, we can find the photonic band gap of the 1-D photonic crystal by calculating (3), (7), (8) and (10). For normal incidence ( = 0), the characteristics for TE mode should be the same as that for TM mode. The fundamental frequency (the central frequency of the first forbidden gap) of the photonic crystal is: c ck0 = (11) n na a + nb b √ √ where na = εa a , nb = εb b is the refractive index of dielectric a, b respectively. In order to simplify calculation, unless specified, we assume that all the dielectrics are non-magnetic, and that the photonic crystal is in vacuo, i.e. = 0 , = 0 ; suppose the relative dielectric constant of dielectric a, b is εa = 5, εb = 1 (though some papers evaluate permeability as 0 = 1 and dielectric constant as ε0 = 1, the penman ω0 =
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
(c)
θ=π/3
Fig. 3. Impact of the incident angle on the band gap.
find that it is actually different from that when 0 = 4 × 10−7 H/m, ε0 = 8.85 × 10−12 F/m, so we take the later); number of periods n = 10, i.e. 20 dielectric layers in total. We’ll discuss the characteristics both when na a = nb b and na a = / nb b. 4.1. Discussion when na a = nb b Let the dielectric layer thickness a = 100 nm, then b = (na /nb )a = 223.6 nm, we get the photonic band gap of the
Y. Lin, H. Xu / Optik 123 (2012) 314–318
T-F Relation
0.9
0.9
0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
0.6 0.5 0.4 0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
4
5
6
T-F Relation
1
Transmissivity(T)
Transmissivity(T)
1
317
7
8
9
0
10
0
1
2
3
Frequency(ω/ω0)
4
(a) g=1.5
0.9 0.8
0.8
0.7
0.7
0.6 0.5 0.4 0.3
3
4
5
6
7
8
9
0
10
0
1
2
3
Frequency(ω/ω0)
4
T-F Relation
0.8
0.7
0.7
Transmissivity(T)
Transmissivity(T)
0.9
0.6 0.5 0.4 0.3
5
6
7
8
9
10
7
8
9
10
g=3
0.3
0.1 4
10
0.4
0.1 3
9
0.5
0.2
2
8
0.6
0.2
1
6
T-F Relation
1
0.8
0
5
(e)
0.9
0
7
g=2
Frequency(ω/ω0)
(b) g=2.5 1
10
0.3
0.1
2
9
0.4
0.2
1
8
0.5
0.1 0
7
0.6
0.2
0
6
T-F Relation
1 0.9
Transmissivity(T)
Transmissivity(T)
(d)
T-F Relation
1
5
Frequency(ω/ω0)
0
0
1
2
3
4
5
6
Frequency(ω/ω0)
Frequency(ω/ω0)
(c) g=3.5
(f) g=4
Fig. 4. Photonic band gap of the 1-D photonic crystal when na a = / nb b.
1-D photonic crystal as shown in Fig. 2 (only the variables are shown in the figure, similarly hereinafter). Fig. 2(a) and (b) show the photonic band gap from different angles. Fig. 2(a) shows the relation between kd and frequency, the dashed line represents , obviously 0 ≤ kd ≤ , so the area above and below 0 in the Y-direction is forbidden, which indicates the forbidden gap, where kd is an imaginary number. Fig. 2(b) shows the relation between transmissivity T and frequency, we can see that T = 0 at and near the odd times of the fundamental frequency (forbidden band), which means light cannot go through the photonic crystal; whereas T is relatively high at other frequencies (conduc-
tion band) with local recurrent fluctuations. In general, T reaches the peak at the even times of the fundamental frequency; this is in line with the results in [4], which proves the correctness of transfer matrix method. As R + T = 1, the relation between R and frequency is complimentary with Fig. 2(b). So we will use only the T–R relation to analyze the characteristics in the following discussion. Fig. 2(c) shows the band gap when n = 5 while other parameters remain the same. In contrast to Fig. 2(b), we see that the less the number of periods, the less local recurrent fluctuation, and the sparser the image, but the amplitude of the fluctuation remains the same and the band gap is similar.Fig. 2(d), (e) and (g) show the
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band gaps when εa /εb = 5, 10, 20 respectively while other parameters remain the same. We can know that the larger ratio εa /εb , the larger forbidden band. Therefore, in order to achieve lager forbidden band, we can consider choosing dielectrics that have larger dielectric constant ratio εa /εb , vice versa. To consider the incident angle ’s impact on the photonic band gap, let = /6, /4, /3, we get the corresponding band gaps as shown in Fig. 3, the gap “gets larger” as the incident angle increases. The so-called “gets larger” is actually extension in the X-direction. The width ratio of the conduction band to that of the forbidden band is the same. The influence that the incident angle has on the photonic crystal is to enlarge its fundamental frequency ω0 . 4.2. Discussion when na a = / nb b Suppose the thickness of dielectric layer a = 100 nm, b = g · (na /nb )a; when the coefficient g = 1.5, 2, 2.5, 3, 3.5, 4, correspondingly b = 335.4, 447.2, 559.0, 670.8, 782.6, 894.4 nm, the corresponding photonic band gaps are shown in Fig. 4. We can see that some band gaps of the photonic crystal break up, and the period of the band gap depends on g: when g is a half-integer, say g = (2m + 1)/2, the period of the photonic band gap T = 2(g + 1)ω0 = (2m + 3)ω0 ; when g is an integer, say g = m, the period of the photonic band gap T = (g + 1)ω0 = (m + 1)ω0 , (m = 0, 1, 2, . . .). The band gap breaks at other frequencies that do not satisfy either of the conditions.
5. Conclusion In this paper, the principal and main application of 1-D photonic crystal are introduced, its characteristic matrix is derived and its light transmission characteristics are also analyzed through numerical calculation by means of transfer matrix method, including impacts of dielectric layers’ thickness, permittivity, and dielectric periodicity, etc. on light transmission characteristics of 1-D photonic crystals. We draw the conclusion that the number of periods of the photonic crystal does not affect the photonic band gap, but the less the number of periods, the less recurrent fluctuation of the transmissivity; the larger difference of the dielectric constant of the dielectric layers, the larger width of forbidden band; the fundamental frequency of the photonic crystal increases with the incident angle; the photonic band gap remains the same at some frequencies and breaks up at the other when na a = / nb b. References [1] B. Liu, Research in characteristics of 1-D photonic crystals that can be used in optical communication, Master Thesis, University of Electronic Science and Technology of Xi’an, 2007, 5–7. [2] H. Wang, et al., An eigen matrix method for obtaining the band structure of photonic crystals, Acta Phys. Sin. 50 (11) (2001). [3] C. Kuang, et al., Transfer matrix for analyzing properties of light propagation in 1-dimension photonic crystals, Laser J. 24 (4) (2003). [4] G. Gu, et al., Properties of light propagation in 1-D periodic dielectric structure, Acta Opt. Sin. 20 (6) (2006). [5] Y. Gao, Research in propagation characteristics of 1-D photonic crystals, Master Thesis, Qufu Normal University, 2007, 9–11.