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Photonic crystal as a refractometric sensor operated in reflection mode
Accepted Manuscript Photonic crystal as a refractometric sensor operated in reflection mode
Sofyan A. Taya, Somaia A. Shaheen, Anas A. Alkanoo PII:
S0749-6036(16)31046-1
DOI:
10.1016/j.spmi.2016.11.057
Reference:
YSPMI 4692
To appear in:
Superlattices and Microstructures
Received Date:
21 September 2016
Revised Date:
26 November 2016
Accepted Date:
27 November 2016
Please cite this article as: Sofyan A. Taya, Somaia A. Shaheen, Anas A. Alkanoo, Photonic crystal as a refractometric sensor operated in reflection mode, Superlattices and Microstructures (2016), doi: 10.1016/j.spmi.2016.11.057
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Research Highlights > One-dimensional ternary photonic crystal is investigated as refractometric sensor. > The transmission of an incident wave from a ternary photonic crystal is derived using Chebyshev polynomials. > The transmissivity is investigated with the angle of incidence, wavelength and number of periods. > The peak angular shift can reach up to Δθ = 7.05o for specific values of the layer thicknesses.
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Photonic crystal as a refractometric sensor operated in reflection mode Sofyan A. Taya*, Somaia A. Shaheen and Anas A. Alkanoo Physics Department, Islamic University of Gaza, P.O.Box 108, Gaza, Palestine. *Corresponding author Email: [email protected] Tel. 00970 8 2644400 (ext 1201), Fax: 00970 8 2644800 Abstract In this work, one dimensional ternary photonic crystal is investigated as refractometric sensor. Using Chebyshev polynomials of the second kind, the transmission of an incident wave from a ternary photonic crystal is studied in details. The variation of the transmissivity with the angle of incidence and wavelength of incident light for different values of number of periods is investigated. Water and air are assumed to be analyte layers. It is found that for water as an analyte, the peak angular shift is Δθ = 1.6o and the peak wavelength shift is Δλ = 2.6 nm for a change in the index of refraction Δn = 0.02. Moreover, the peak angular shift can reach up to Δθ = 7.05o for specific values of the layer thicknesses. Keywords: photonic crystal, transmission, refractometric sensor, sensitivity. 1. Introduction Recently, photonic band gap structures have attracted an increasing interest [1-4] due to their potential applications in optoelectronics and communications. Moreover, they can find enormous applications in optical instrumentation. Photonic band gap structures are multilayer structures formed by two or more materials. It was found that these structures have photonic band gaps or stop bands. The propagation of electromagnetic waves of wavelengths lying in the stop band is prohibited. These bands are dependent upon a number of parameters such as the incidence angle of electromagnetic waves, the index of refraction of materials and filling fraction. If all other parameters are kept unchanged, then any change in the index of refraction of a structural medium will change the stop band and ranges of transmission. By monitoring the transmission profile or the stop band shift, a slight change in the index of refraction of a structural medium can be detected. Ternary photonic bang gap crystals attracted interests of physicist and engineers due to their superior performance over binary photonic band gap crystals in many applications. Among these applications, omni-directional reflection and tunable optical filtering were significant. The ternary photonic crystals can be produced by periodic repetition of three different media instead of two different media as in case of traditional binary photonic crystals. In fact, photonic crystals can be divided into many types according to their construction like binary, ternary, quaternary and so on. Binary photonic crystal has two layers in a periodic structure, ternary photonic crystal has three layers in a periodic structure, quaternary photonic crystal has four layers in a periodic structure and so on. These periods (two, three or four media) are repeated several times to construct photonic band gap crystals. The ternary photonic band gap crystals can be used in many applications such as omni-directional reflectors [5], tunable optical filters [6] and optical sensors [7]. 1
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Highly sensitive optical elements for detection of biological objects are of central significance not only for biological research but also for pharmaceutical manufacturing, public health and environmental observing [8]. Diverse sensing schemes based on mechanical [9], electrochemical [10] and optical [11-20] interactions have been proposed for a wide variety of applications. One of the most significant techniques is based on measuring the change in the index of refraction that analyte molecules experience upon binding. To attain this, physicists have used resonance shifts in surface plasmons [21-27], microresonators [28] or nanoparticle plasmons [29,30]. Each technique depends on the field confinement in a waveguide configuration that acts as a large optical label with characteristic resonance. Moreover, new artificial materials with simultaneously negative electric permittivity and magnetic permeability have been proposed for different applications [31-40]. These materials were referred to as left-handed materials due to the left-handed behavior they exhibit. Among these applications of left-handed materials is slab waveguide sensing [41-47] in which minute changes in the index of refraction of an aqueous cladding can be detected with high sensitivity due to the enhancement of evanescent field attained by left-handed materials. In this paper, one dimensional ternary photonic crystal is studied as an optical sensor for detection small changes in the refractive index of an analyte. The transmission of an incident wave from a ternary photonic crystal is studied with the angle of incidence and wavelength of incident wave. The effects of number of layers and thickness of the layers on the sensor performance are also investigated. 2. Reflection and transmission from ternary photonic crystal Figure 1 shows a schematic diagram of 1-D ternary photonic crystal having three layers of refractive indices 𝑛1, 𝑛2 and 𝑛3 with thicknesses 𝑑1, 𝑑2 and 𝑑3, respectively. The period of the lattice is given by d=d1+d2+d3. x air
n1
n2
n3
....
Өo d1
d2
d1
d2
d3
d3
z
Fig. 1. Structure of one- dimensional ternary photonic crystal sensing element. We consider TE mode in which the characteristic matrix of one period is given by [7,48,49]
M[d] =
∏𝑙 𝑖=1
[
cos 𝛽𝑖 ‒ 𝑖𝑝𝑖sin 𝛽𝑖
2
‒ 𝑖sin 𝛽𝑖
][
𝑀11 = 𝑀 21 cos 𝛽𝑖 𝑝𝑖
𝑀12 𝑀22 ,
]
(1)
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where l=3, β1=
2𝜋𝑛1𝑑1cos 𝜃1 𝜆0 𝜀3
𝑐𝑜𝑠𝜃2 and 𝑝3 =
, β2 =
µ3𝑐𝑜𝑠𝜃3.
2𝜋𝑛2𝑑2cos 𝜃2 𝜆0
, β3 =
2𝜋(𝑛3)𝑑3cos 𝜃3 𝜆0
, 𝑝1 =
𝜀1
𝜀2
1
µ2
µ 𝑐𝑜𝑠𝜃1, 𝑝2 =
and 𝜆𝑜 is the free space wavelength.
Here θ1, θ2 and θ3 are the ray angles inside the layers 1, 2 and 3, respectively which related to the angle of incidence θ0 by
[
cos 𝜃1 = 1 ‒
𝑛20𝑠𝑖𝑛2𝜃0 𝑛21
1 2
] ,cos 𝜃 = [1 ‒
𝑛20𝑠𝑖𝑛2𝜃0
2
1 2
] and cos 𝜃 = [1 ‒ 3
𝑛22
𝑛20𝑠𝑖𝑛2𝜃0 (𝑛3)2
]
1 2
The elements of the matrix M[d] are given by
(
𝑀11 = cos 𝛽1cos 𝛽2cos 𝛽3 ‒
𝑝2sin 𝛽1sin 𝛽2cos 𝛽3 𝑝1
‒
𝑝3cos 𝛽1sin 𝛽2sin 𝛽3 𝑝2
‒
𝑝3sin 𝛽1cos 𝛽2sin 𝛽3 𝑝1
) (2)
𝑀12 = ‒𝑖
(
sin 𝛽1cos 𝛽2cos 𝛽3 𝑝1
+
cos 𝛽1sin 𝛽2cos 𝛽3 𝑝2
+
cos 𝛽1cos 𝛽2sin 𝛽3 𝑝3
‒
𝑝2sin 𝛽1sin 𝛽2sin 𝛽3 𝑝1𝑝2
)
(3)
𝑀21 =‒ 𝑖
(𝑝 sin 𝛽 cos 𝛽 cos 𝛽 1
1
2
𝑝1𝑝3sin 𝛽1sin 𝛽2sin 𝛽3
3 + 𝑝2cos 𝛽1sin 𝛽2cos 𝛽3 + 𝑝3cos 𝛽1cos 𝛽2sin 𝛽3 ‒
𝑝2
(4)
,
(
𝑀22 = cos 𝛽1cos 𝛽2cos 𝛽3 ‒
𝑝1sin 𝛽1sin 𝛽2cos 𝛽3 𝑝2
‒
𝑝2cos 𝛽1sin 𝛽2sin 𝛽3 𝑝3
‒
𝑝1sin 𝛽1cos 𝛽2sin 𝛽3 𝑝3
). (5)
The matrix M[d] in Eq. (1) is unimodular as |𝑀[𝑑]| = 1. For an N period structure, the characteristic matrix of the matrix is given by [7] [𝑀(𝑑)]𝑁 =
[𝑀
11𝑇𝑁 ‒ 1(𝑎) ‒ 𝑇𝑁 ‒ 2(𝑎)
𝑀21𝑇𝑁 ‒ 1(𝑎)
𝑀12𝑇𝑁 ‒ 1(𝑎) 𝑠11 𝑀22𝑇𝑁 ‒ 1(𝑎) ‒ 𝑇𝑁 ‒ 2(𝑎) = 𝑠21
] [
𝑠12 𝑠22
]
(6)
where TN are the Chebyshev polynomials of the second kind 𝑇𝑁(𝑎) =
sin [(𝑁 + 1)𝑐𝑜𝑠 ‒ 1𝑎] 1
[1 ‒ 𝑎 2 ]2
,
(7)
where 1
𝑎 = 2[𝑀11 + 𝑀22]
(8)
3
)
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The transmission coefficient of the multilayer is given by 𝑡 = (𝑆
2𝑝0
11 + 𝑆12𝑝0
)𝑝0 + (𝑆21 + 𝑆22𝑝0).
(9)
where 𝑝𝑜 = 𝑛0cos 𝜃0 = cos 𝜃0.
(10)
with 𝑛0 = 1. The transmissivity for this structure can be written in terms of the transmission coefficients as 𝑇 = |𝑡|2 ,
(11)
3. Numerical Results To study the transmission profile with the angle of incidence, we consider onedimensional ternary photonic crystal with the parameters n1=1.00, n2 =4.00, n3=1.57, d1=150 nm, d2= 160 nm, d3=100 nm and λ=632.8 nm. The layer of index of refraction 𝑛1 is considered the analyte material. The transmission profile is explored in Fig. 2 in which the transmission spectrum for different values of the repetitive periodic layers at normal incidence of light is shown. As can be seen, there exists a photonic band gap. The most remarkable feature of this figure is the appearance of precursors of the band structure characteristic of a multilayer configuration and it arises even for a number of repetitive periodic layers of 4. In each panel of the figure there exist photonic band gaps in which the transmission approaches zero and allowed energy bands in which the transmission is close to unity. Each allowed energy band contains a number of ripples which totally dependent on the number of repetitive period layers. As can be seen from the figure, there is N-1 ripples for N periods. Considering the second valley in which T approaches zero, it gets flatter and deeper as N increases. Considering regions of T=1 (allowed energy bands), the width of the band increases which increasing the wavelength. Taking the forth panel of N = 10, the first allowed band has a width of 43 nm and the last allowed band has a width of 160 nm. The variation of the transmissivity with the angle of incidence for different values of periods is shown in Fig. 3. The angle of incidence was varied from 0o to 90o in a step of 0.01o. The calculations were conducted for periods of N = 4, 6, 8 and 10. The transmission is zero until an angle of incidence of 40o then it increases toward unity. When it reaches unity it shows a number of ripples which depend on the number of periods. When the number of repetitive period layers is 4, the number of ripples is 1. When the number of periods increases to 6, the number of ripples is 2. When the number of repetitive periodic layers becomes 8 and 10, the number of ripples becomes 3 and 4, 4
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respectively. We conclude when the number of periods is N, the number of ripples is ‒ 1.
𝑁 2
We now turn our attention to the use of the proposed structure as a refractometric sensor. We assume two analytes: air (n1=1) and water (n1=1.33). The transmission versus angle of incidence for the proposed one-dimensional ternary photonic crystal is illustrated in Fig. 4 for Δn = 0.02 for both water and air analytes. It is known that the one-dimensional ternary photonic crystal exhibits transmission peaks in the transmission-angle of incidence curve which shift for any variation in the analyte index. To show this transmission peak shift, we plot the transmission curve for pure analyte (solid lines) of indices n1=1.33 (water) and n1=1.00 (air). We also show the transmission peaks at refractive index difference of 0.02 (dotted lines). There is a distinct shift in the transmission peaks due to the analyte index variation. For water as an analyte this shift is Δθ = 1.6o and for air it is Δθ = 1o. Assuming a measurable angle resolution of 0.01o, the resolution of the proposed one-dimensional ternary photonic crystal is ultra-high. The transmission profile is explored further in Fig. 5 which shows the transmission spectrum for different values of n1 with water and air as analyte layers. As is clearly seen there is a peak shift towards larger wavelength as the refractive index of the analyte layer changes. For water analyte (upper panel) the wavelength shift is Δλ = 2.6 nm for Δn = 0.02 and for air analyte this shift is Δλ = 2.2 nm for Δn = 0.02 where these transmission peak shifts are very easily detectable with existing optoelectronic devices. For refractometric applications, as the peak angular shift (Δθ) due to a given refractive index change of the analyte increases, the sensitivity of the sensor to detect the changes in the index of refraction increases. We now optimize d1, d2, and d3 that correspond to the highest Δθ for Δn1 = 0.02 for air analyte. Figure 6 shows the variation of the peak angular shift (Δθ) with d1 (upper left panel), d2 (upper right panel) and d3 (lower panel) for Δn1 = 0.02. As observed from the figure, the angular shift increases monotonically with d1 until it reaches 7.05o for d1 = 500 nm whereas it varies with d2 to reach 2o at d2=145nm. The lower panel shows that Δθ can reach 7.05o for d3=100 nm. Finally, we optimize n2 and n3 that correspond to the highest Δθ for Δn1 = 0.02 for air analyte. Figure 7 shows the variation of the peak angular shift (Δθ) with n2 (left panel) and n3 (right panel) for Δn1 = 0.02. As clearly seen from the figure, the angular shift reaches 1.85o for n2 = 3.55 whereas it reaches 1.00o at n3 = 1.20. It is worth mentioning that the peak disappears for n2 < 3.55 and n2 > 4.10. It also disappears when n3 < 1.20 and n2 > 1.57.
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N=4
Transmission
1.0 0.5
0.5
0.0
0.0
400
600
Transmission
800
N=8
1.0
400
0.5
0.0
0.0 600
600
800
800
N=10
1.0
0.5
400
N=6
1.0
400
600
(nm)
800
(nm)
Fig. 2. Transmission from one dimensional ternary photonic crystal with wavelength. Parameters are the same as in Fig. 2.
N=4
Transmission
1.0 0.5
0.5
0.0
0.0 0
20
40
80
N=8
1.0
Transmission
60
0
0.5
0.0
0.0 20
40
60
20
40
80
0
(deg.)
60
80
N=10
1.0
0.5
0
N=6
1.0
20
40
60
80
(deg.)
Fig. 3. Transmission from one dimensional ternary photonic crystal with the angle of incidence with N = 4, 6, 8, and 10 for d1=150 nm, d2 = 160 nm, d3 = 100 nm, λ = 632.8 nm, n1 = 1.00, n2 = 4.00, and n3 = 1.57.
6
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Transmission
1.0
n1 = 1.33
0.8
N=6
n1 = 1.35
0.6 0.4 40
45
50
55
60
65
70
75
80
Transmission
1.0
N=6
0.8 0.6 0.4
n1 = 1.00
0.2 0.0
n1 = 1.02 40
45
50
55
60
65
70
75
80
(deg.)
Transmission
Fig. 4. Transmission from one dimensional ternary photonic crystal with the angle of incidence for analyte of water of n1 = 1.33 (upper panel) and air of n1 = 1.00 (lower panel).
1.0
n1 = 1.33
0.8
n1 = 1.35
0.6
600 Transmission
N=4
0.4
0.9 0.6
650
700
800
n1 = 1.00 n1 = 1.02
0.3 0.0 600
750
N=4
650
700
750
800
(nm)
Fig. 5. Transmission from one dimensional ternary photonic crystal with wavelength for analyte of water of n1 = 1.33 (upper panel) and air of n1 = 1.00 (lower panel).
7
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7
1.75
5
6
2.00
d2=145 nm d3=100 nm
1.50
4 3
1.25
2
d1=500 nm d3=100 nm
200
300
400
d1(nm) 7
500
120
d1=500 nm d2=145 nm
6
1.00 100
140
160
d2(nm)
5 4 3 0
20
40
60
d3(nm)
80
100
Fig. 6. The angular shift for one dimensional ternary photonic crystal versus d1, d2, and d3 respectively, with N = 4 for λ = 632.8 nm, n1 = 1.00, n2 = 4.00, and n3 = 1.57.
2.0 1.0
1.5
0.9
1.0
0.8
3.6
3.8
4.0
1.2
n2
1.3
1.4
1.5
n3
Fig. 7. The angular shift for one dimensional ternary photonic crystal versus n2 and n3 for N = 4 and λ = 632.8 nm. Conclusion In this work, one dimensional ternary photonic crystal was investigated as a refractometric sensor to detect very small changes in the refractive index of an analyte 8
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material homogeneously distributed in one of the photonic crystal layers. The principle of operation of photonic crystal as a refractometric sensor in reflection mode was explained. The transmission of an incident electromagnetic wave was studied in details using Chebyshev polynomials of the second kind. The variation of the transmissivity with the incidence angle and wavelength of the incident electromagnetic wave was studied for different values of number of periods. The transmissivity shows a number of ripples 𝑁 when plotted with the angle of incidence. The number of ripples is 2 ‒ 1 where N is the number of periods. When the transmissivity is plotted with the light wavelength, the signs of the band structure characteristic of a multilayer configuration appear. For any number of periods, allowed energy bands and stop bands in which the transmission is zero appear. In each allowed energy band, there is a number of ripples (N-1) for N periods. The valley gets flatter and deeper as N increases. When used as an optical sensor for refractometric purposes, the proposed ternary photonic crystal shows an ultra-high sensitivity. For water as an analyte, the peak angular shift is Δθ = 1.6o and for air it is Δθ = 1o for Δn = 0.02. For the same change in the index of refraction of an analyte, the transmission spectrum shows a peak shift towards larger wavelength of Δλ = 2.6 nm for water and Δλ = 2.2 nm for air. Finally, we tried to optimize the peak angular shift (Δθ) to obtain the highest sensitivity. The shift can be reach up to 7.05o for d1 = 500 nm, d2=145nm and d3=100 nm. References [1] S. Sandhu, S. Fan, M. Yanik, M. Povinelli, Advances in theory of photonic crystal, Journal of light wave technology. 24 (2006) 4493-4501. [2] I. Kriegel, F. Scotognella, Disordered one--‐dimensional photonic structures composed by more than two materials with the same optical thickness, Optics Communications. 338 (2015) 523-527. [3] Z. Zare , A. Gharaati, Investigation of band gap width in ternary 1D photonic Crystal with Left-Handed Layer, Acta Physica Polonica A. 125 (2014) 36-38. [4] C. J. Wu, Y. H. Chung, T.J. Yang, B. J. Syu, Band Gap Extension in A OneDimensional Ternary Metal-Dielectric Photonic Crystal, Progress In Electromagnetics Researc. 102 (201081-93). [5] S. K Awasthi, U. Malaviya, S. P. Ojha, Enhancement of omnidirectional totalreflection wavelength range by using one dimensional ternary photonic band gap material, J. Opt. Soc. Am. B. 23 (2006) 2566–2571. [6] S. K Awasthi, S. P. Ojha, Design of a tunable optical filter by using one-dimensional ternary photonic band gap material, Progress In Electromagnetic Research M. 4 (2008) 117–132. [7] A. Banerjee, Enhanced refractometric optical sensing by using one-dimensional ternary photonic crystals, Progress In Electromagnetic Research, PIER. 89 (2009) 11–22. [8] A.P.F. Turner, Biosensors: sense and sensibility, Chem. Soc. Rev. 42 (2013) 3184. [9] J.L. Arlett, E.B. Myers, M.L. Roukes, Comparative advantages of mechanical biosensors, Nature Nanotechnology. 6 (2011) 203–215. [10] N.J. Ronkainen, H.B. Halsall, W.R. Heineman, Electrochemical biosensors, Chem. Soc. Rev. 39 (2010) 1747- 1763. [11] S. A. Taya, T. M. El-Agez, Comparing optical sensing using slab waveguides and total internal reflection ellipsometry, Turk. J. Phys. 35 (2011) 31-36. 9
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12
Sofyan A. Taya, Somaia A. Shaheen, Anas A. Alkanoo PII:
S0749-6036(16)31046-1
DOI:
10.1016/j.spmi.2016.11.057
Reference:
YSPMI 4692
To appear in:
Superlattices and Microstructures
Received Date:
21 September 2016
Revised Date:
26 November 2016
Accepted Date:
27 November 2016
Please cite this article as: Sofyan A. Taya, Somaia A. Shaheen, Anas A. Alkanoo, Photonic crystal as a refractometric sensor operated in reflection mode, Superlattices and Microstructures (2016), doi: 10.1016/j.spmi.2016.11.057
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Research Highlights > One-dimensional ternary photonic crystal is investigated as refractometric sensor. > The transmission of an incident wave from a ternary photonic crystal is derived using Chebyshev polynomials. > The transmissivity is investigated with the angle of incidence, wavelength and number of periods. > The peak angular shift can reach up to Δθ = 7.05o for specific values of the layer thicknesses.
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Photonic crystal as a refractometric sensor operated in reflection mode Sofyan A. Taya*, Somaia A. Shaheen and Anas A. Alkanoo Physics Department, Islamic University of Gaza, P.O.Box 108, Gaza, Palestine. *Corresponding author Email: [email protected] Tel. 00970 8 2644400 (ext 1201), Fax: 00970 8 2644800 Abstract In this work, one dimensional ternary photonic crystal is investigated as refractometric sensor. Using Chebyshev polynomials of the second kind, the transmission of an incident wave from a ternary photonic crystal is studied in details. The variation of the transmissivity with the angle of incidence and wavelength of incident light for different values of number of periods is investigated. Water and air are assumed to be analyte layers. It is found that for water as an analyte, the peak angular shift is Δθ = 1.6o and the peak wavelength shift is Δλ = 2.6 nm for a change in the index of refraction Δn = 0.02. Moreover, the peak angular shift can reach up to Δθ = 7.05o for specific values of the layer thicknesses. Keywords: photonic crystal, transmission, refractometric sensor, sensitivity. 1. Introduction Recently, photonic band gap structures have attracted an increasing interest [1-4] due to their potential applications in optoelectronics and communications. Moreover, they can find enormous applications in optical instrumentation. Photonic band gap structures are multilayer structures formed by two or more materials. It was found that these structures have photonic band gaps or stop bands. The propagation of electromagnetic waves of wavelengths lying in the stop band is prohibited. These bands are dependent upon a number of parameters such as the incidence angle of electromagnetic waves, the index of refraction of materials and filling fraction. If all other parameters are kept unchanged, then any change in the index of refraction of a structural medium will change the stop band and ranges of transmission. By monitoring the transmission profile or the stop band shift, a slight change in the index of refraction of a structural medium can be detected. Ternary photonic bang gap crystals attracted interests of physicist and engineers due to their superior performance over binary photonic band gap crystals in many applications. Among these applications, omni-directional reflection and tunable optical filtering were significant. The ternary photonic crystals can be produced by periodic repetition of three different media instead of two different media as in case of traditional binary photonic crystals. In fact, photonic crystals can be divided into many types according to their construction like binary, ternary, quaternary and so on. Binary photonic crystal has two layers in a periodic structure, ternary photonic crystal has three layers in a periodic structure, quaternary photonic crystal has four layers in a periodic structure and so on. These periods (two, three or four media) are repeated several times to construct photonic band gap crystals. The ternary photonic band gap crystals can be used in many applications such as omni-directional reflectors [5], tunable optical filters [6] and optical sensors [7]. 1
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Highly sensitive optical elements for detection of biological objects are of central significance not only for biological research but also for pharmaceutical manufacturing, public health and environmental observing [8]. Diverse sensing schemes based on mechanical [9], electrochemical [10] and optical [11-20] interactions have been proposed for a wide variety of applications. One of the most significant techniques is based on measuring the change in the index of refraction that analyte molecules experience upon binding. To attain this, physicists have used resonance shifts in surface plasmons [21-27], microresonators [28] or nanoparticle plasmons [29,30]. Each technique depends on the field confinement in a waveguide configuration that acts as a large optical label with characteristic resonance. Moreover, new artificial materials with simultaneously negative electric permittivity and magnetic permeability have been proposed for different applications [31-40]. These materials were referred to as left-handed materials due to the left-handed behavior they exhibit. Among these applications of left-handed materials is slab waveguide sensing [41-47] in which minute changes in the index of refraction of an aqueous cladding can be detected with high sensitivity due to the enhancement of evanescent field attained by left-handed materials. In this paper, one dimensional ternary photonic crystal is studied as an optical sensor for detection small changes in the refractive index of an analyte. The transmission of an incident wave from a ternary photonic crystal is studied with the angle of incidence and wavelength of incident wave. The effects of number of layers and thickness of the layers on the sensor performance are also investigated. 2. Reflection and transmission from ternary photonic crystal Figure 1 shows a schematic diagram of 1-D ternary photonic crystal having three layers of refractive indices 𝑛1, 𝑛2 and 𝑛3 with thicknesses 𝑑1, 𝑑2 and 𝑑3, respectively. The period of the lattice is given by d=d1+d2+d3. x air
n1
n2
n3
....
Өo d1
d2
d1
d2
d3
d3
z
Fig. 1. Structure of one- dimensional ternary photonic crystal sensing element. We consider TE mode in which the characteristic matrix of one period is given by [7,48,49]
M[d] =
∏𝑙 𝑖=1
[
cos 𝛽𝑖 ‒ 𝑖𝑝𝑖sin 𝛽𝑖
2
‒ 𝑖sin 𝛽𝑖
][
𝑀11 = 𝑀 21 cos 𝛽𝑖 𝑝𝑖
𝑀12 𝑀22 ,
]
(1)
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where l=3, β1=
2𝜋𝑛1𝑑1cos 𝜃1 𝜆0 𝜀3
𝑐𝑜𝑠𝜃2 and 𝑝3 =
, β2 =
µ3𝑐𝑜𝑠𝜃3.
2𝜋𝑛2𝑑2cos 𝜃2 𝜆0
, β3 =
2𝜋(𝑛3)𝑑3cos 𝜃3 𝜆0
, 𝑝1 =
𝜀1
𝜀2
1
µ2
µ 𝑐𝑜𝑠𝜃1, 𝑝2 =
and 𝜆𝑜 is the free space wavelength.
Here θ1, θ2 and θ3 are the ray angles inside the layers 1, 2 and 3, respectively which related to the angle of incidence θ0 by
[
cos 𝜃1 = 1 ‒
𝑛20𝑠𝑖𝑛2𝜃0 𝑛21
1 2
] ,cos 𝜃 = [1 ‒
𝑛20𝑠𝑖𝑛2𝜃0
2
1 2
] and cos 𝜃 = [1 ‒ 3
𝑛22
𝑛20𝑠𝑖𝑛2𝜃0 (𝑛3)2
]
1 2
The elements of the matrix M[d] are given by
(
𝑀11 = cos 𝛽1cos 𝛽2cos 𝛽3 ‒
𝑝2sin 𝛽1sin 𝛽2cos 𝛽3 𝑝1
‒
𝑝3cos 𝛽1sin 𝛽2sin 𝛽3 𝑝2
‒
𝑝3sin 𝛽1cos 𝛽2sin 𝛽3 𝑝1
) (2)
𝑀12 = ‒𝑖
(
sin 𝛽1cos 𝛽2cos 𝛽3 𝑝1
+
cos 𝛽1sin 𝛽2cos 𝛽3 𝑝2
+
cos 𝛽1cos 𝛽2sin 𝛽3 𝑝3
‒
𝑝2sin 𝛽1sin 𝛽2sin 𝛽3 𝑝1𝑝2
)
(3)
𝑀21 =‒ 𝑖
(𝑝 sin 𝛽 cos 𝛽 cos 𝛽 1
1
2
𝑝1𝑝3sin 𝛽1sin 𝛽2sin 𝛽3
3 + 𝑝2cos 𝛽1sin 𝛽2cos 𝛽3 + 𝑝3cos 𝛽1cos 𝛽2sin 𝛽3 ‒
𝑝2
(4)
,
(
𝑀22 = cos 𝛽1cos 𝛽2cos 𝛽3 ‒
𝑝1sin 𝛽1sin 𝛽2cos 𝛽3 𝑝2
‒
𝑝2cos 𝛽1sin 𝛽2sin 𝛽3 𝑝3
‒
𝑝1sin 𝛽1cos 𝛽2sin 𝛽3 𝑝3
). (5)
The matrix M[d] in Eq. (1) is unimodular as |𝑀[𝑑]| = 1. For an N period structure, the characteristic matrix of the matrix is given by [7] [𝑀(𝑑)]𝑁 =
[𝑀
11𝑇𝑁 ‒ 1(𝑎) ‒ 𝑇𝑁 ‒ 2(𝑎)
𝑀21𝑇𝑁 ‒ 1(𝑎)
𝑀12𝑇𝑁 ‒ 1(𝑎) 𝑠11 𝑀22𝑇𝑁 ‒ 1(𝑎) ‒ 𝑇𝑁 ‒ 2(𝑎) = 𝑠21
] [
𝑠12 𝑠22
]
(6)
where TN are the Chebyshev polynomials of the second kind 𝑇𝑁(𝑎) =
sin [(𝑁 + 1)𝑐𝑜𝑠 ‒ 1𝑎] 1
[1 ‒ 𝑎 2 ]2
,
(7)
where 1
𝑎 = 2[𝑀11 + 𝑀22]
(8)
3
)
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The transmission coefficient of the multilayer is given by 𝑡 = (𝑆
2𝑝0
11 + 𝑆12𝑝0
)𝑝0 + (𝑆21 + 𝑆22𝑝0).
(9)
where 𝑝𝑜 = 𝑛0cos 𝜃0 = cos 𝜃0.
(10)
with 𝑛0 = 1. The transmissivity for this structure can be written in terms of the transmission coefficients as 𝑇 = |𝑡|2 ,
(11)
3. Numerical Results To study the transmission profile with the angle of incidence, we consider onedimensional ternary photonic crystal with the parameters n1=1.00, n2 =4.00, n3=1.57, d1=150 nm, d2= 160 nm, d3=100 nm and λ=632.8 nm. The layer of index of refraction 𝑛1 is considered the analyte material. The transmission profile is explored in Fig. 2 in which the transmission spectrum for different values of the repetitive periodic layers at normal incidence of light is shown. As can be seen, there exists a photonic band gap. The most remarkable feature of this figure is the appearance of precursors of the band structure characteristic of a multilayer configuration and it arises even for a number of repetitive periodic layers of 4. In each panel of the figure there exist photonic band gaps in which the transmission approaches zero and allowed energy bands in which the transmission is close to unity. Each allowed energy band contains a number of ripples which totally dependent on the number of repetitive period layers. As can be seen from the figure, there is N-1 ripples for N periods. Considering the second valley in which T approaches zero, it gets flatter and deeper as N increases. Considering regions of T=1 (allowed energy bands), the width of the band increases which increasing the wavelength. Taking the forth panel of N = 10, the first allowed band has a width of 43 nm and the last allowed band has a width of 160 nm. The variation of the transmissivity with the angle of incidence for different values of periods is shown in Fig. 3. The angle of incidence was varied from 0o to 90o in a step of 0.01o. The calculations were conducted for periods of N = 4, 6, 8 and 10. The transmission is zero until an angle of incidence of 40o then it increases toward unity. When it reaches unity it shows a number of ripples which depend on the number of periods. When the number of repetitive period layers is 4, the number of ripples is 1. When the number of periods increases to 6, the number of ripples is 2. When the number of repetitive periodic layers becomes 8 and 10, the number of ripples becomes 3 and 4, 4
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respectively. We conclude when the number of periods is N, the number of ripples is ‒ 1.
𝑁 2
We now turn our attention to the use of the proposed structure as a refractometric sensor. We assume two analytes: air (n1=1) and water (n1=1.33). The transmission versus angle of incidence for the proposed one-dimensional ternary photonic crystal is illustrated in Fig. 4 for Δn = 0.02 for both water and air analytes. It is known that the one-dimensional ternary photonic crystal exhibits transmission peaks in the transmission-angle of incidence curve which shift for any variation in the analyte index. To show this transmission peak shift, we plot the transmission curve for pure analyte (solid lines) of indices n1=1.33 (water) and n1=1.00 (air). We also show the transmission peaks at refractive index difference of 0.02 (dotted lines). There is a distinct shift in the transmission peaks due to the analyte index variation. For water as an analyte this shift is Δθ = 1.6o and for air it is Δθ = 1o. Assuming a measurable angle resolution of 0.01o, the resolution of the proposed one-dimensional ternary photonic crystal is ultra-high. The transmission profile is explored further in Fig. 5 which shows the transmission spectrum for different values of n1 with water and air as analyte layers. As is clearly seen there is a peak shift towards larger wavelength as the refractive index of the analyte layer changes. For water analyte (upper panel) the wavelength shift is Δλ = 2.6 nm for Δn = 0.02 and for air analyte this shift is Δλ = 2.2 nm for Δn = 0.02 where these transmission peak shifts are very easily detectable with existing optoelectronic devices. For refractometric applications, as the peak angular shift (Δθ) due to a given refractive index change of the analyte increases, the sensitivity of the sensor to detect the changes in the index of refraction increases. We now optimize d1, d2, and d3 that correspond to the highest Δθ for Δn1 = 0.02 for air analyte. Figure 6 shows the variation of the peak angular shift (Δθ) with d1 (upper left panel), d2 (upper right panel) and d3 (lower panel) for Δn1 = 0.02. As observed from the figure, the angular shift increases monotonically with d1 until it reaches 7.05o for d1 = 500 nm whereas it varies with d2 to reach 2o at d2=145nm. The lower panel shows that Δθ can reach 7.05o for d3=100 nm. Finally, we optimize n2 and n3 that correspond to the highest Δθ for Δn1 = 0.02 for air analyte. Figure 7 shows the variation of the peak angular shift (Δθ) with n2 (left panel) and n3 (right panel) for Δn1 = 0.02. As clearly seen from the figure, the angular shift reaches 1.85o for n2 = 3.55 whereas it reaches 1.00o at n3 = 1.20. It is worth mentioning that the peak disappears for n2 < 3.55 and n2 > 4.10. It also disappears when n3 < 1.20 and n2 > 1.57.
5
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N=4
Transmission
1.0 0.5
0.5
0.0
0.0
400
600
Transmission
800
N=8
1.0
400
0.5
0.0
0.0 600
600
800
800
N=10
1.0
0.5
400
N=6
1.0
400
600
(nm)
800
(nm)
Fig. 2. Transmission from one dimensional ternary photonic crystal with wavelength. Parameters are the same as in Fig. 2.
N=4
Transmission
1.0 0.5
0.5
0.0
0.0 0
20
40
80
N=8
1.0
Transmission
60
0
0.5
0.0
0.0 20
40
60
20
40
80
0
(deg.)
60
80
N=10
1.0
0.5
0
N=6
1.0
20
40
60
80
(deg.)
Fig. 3. Transmission from one dimensional ternary photonic crystal with the angle of incidence with N = 4, 6, 8, and 10 for d1=150 nm, d2 = 160 nm, d3 = 100 nm, λ = 632.8 nm, n1 = 1.00, n2 = 4.00, and n3 = 1.57.
6
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Transmission
1.0
n1 = 1.33
0.8
N=6
n1 = 1.35
0.6 0.4 40
45
50
55
60
65
70
75
80
Transmission
1.0
N=6
0.8 0.6 0.4
n1 = 1.00
0.2 0.0
n1 = 1.02 40
45
50
55
60
65
70
75
80
(deg.)
Transmission
Fig. 4. Transmission from one dimensional ternary photonic crystal with the angle of incidence for analyte of water of n1 = 1.33 (upper panel) and air of n1 = 1.00 (lower panel).
1.0
n1 = 1.33
0.8
n1 = 1.35
0.6
600 Transmission
N=4
0.4
0.9 0.6
650
700
800
n1 = 1.00 n1 = 1.02
0.3 0.0 600
750
N=4
650
700
750
800
(nm)
Fig. 5. Transmission from one dimensional ternary photonic crystal with wavelength for analyte of water of n1 = 1.33 (upper panel) and air of n1 = 1.00 (lower panel).
7
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7
1.75
5
6
2.00
d2=145 nm d3=100 nm
1.50
4 3
1.25
2
d1=500 nm d3=100 nm
200
300
400
d1(nm) 7
500
120
d1=500 nm d2=145 nm
6
1.00 100
140
160
d2(nm)
5 4 3 0
20
40
60
d3(nm)
80
100
Fig. 6. The angular shift for one dimensional ternary photonic crystal versus d1, d2, and d3 respectively, with N = 4 for λ = 632.8 nm, n1 = 1.00, n2 = 4.00, and n3 = 1.57.
2.0 1.0
1.5
0.9
1.0
0.8
3.6
3.8
4.0
1.2
n2
1.3
1.4
1.5
n3
Fig. 7. The angular shift for one dimensional ternary photonic crystal versus n2 and n3 for N = 4 and λ = 632.8 nm. Conclusion In this work, one dimensional ternary photonic crystal was investigated as a refractometric sensor to detect very small changes in the refractive index of an analyte 8
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material homogeneously distributed in one of the photonic crystal layers. The principle of operation of photonic crystal as a refractometric sensor in reflection mode was explained. The transmission of an incident electromagnetic wave was studied in details using Chebyshev polynomials of the second kind. The variation of the transmissivity with the incidence angle and wavelength of the incident electromagnetic wave was studied for different values of number of periods. The transmissivity shows a number of ripples 𝑁 when plotted with the angle of incidence. The number of ripples is 2 ‒ 1 where N is the number of periods. When the transmissivity is plotted with the light wavelength, the signs of the band structure characteristic of a multilayer configuration appear. For any number of periods, allowed energy bands and stop bands in which the transmission is zero appear. In each allowed energy band, there is a number of ripples (N-1) for N periods. The valley gets flatter and deeper as N increases. When used as an optical sensor for refractometric purposes, the proposed ternary photonic crystal shows an ultra-high sensitivity. For water as an analyte, the peak angular shift is Δθ = 1.6o and for air it is Δθ = 1o for Δn = 0.02. For the same change in the index of refraction of an analyte, the transmission spectrum shows a peak shift towards larger wavelength of Δλ = 2.6 nm for water and Δλ = 2.2 nm for air. Finally, we tried to optimize the peak angular shift (Δθ) to obtain the highest sensitivity. The shift can be reach up to 7.05o for d1 = 500 nm, d2=145nm and d3=100 nm. References [1] S. Sandhu, S. Fan, M. Yanik, M. Povinelli, Advances in theory of photonic crystal, Journal of light wave technology. 24 (2006) 4493-4501. [2] I. Kriegel, F. Scotognella, Disordered one--‐dimensional photonic structures composed by more than two materials with the same optical thickness, Optics Communications. 338 (2015) 523-527. [3] Z. Zare , A. Gharaati, Investigation of band gap width in ternary 1D photonic Crystal with Left-Handed Layer, Acta Physica Polonica A. 125 (2014) 36-38. [4] C. J. Wu, Y. H. Chung, T.J. Yang, B. J. Syu, Band Gap Extension in A OneDimensional Ternary Metal-Dielectric Photonic Crystal, Progress In Electromagnetics Researc. 102 (201081-93). [5] S. K Awasthi, U. Malaviya, S. P. Ojha, Enhancement of omnidirectional totalreflection wavelength range by using one dimensional ternary photonic band gap material, J. Opt. Soc. Am. B. 23 (2006) 2566–2571. [6] S. K Awasthi, S. P. Ojha, Design of a tunable optical filter by using one-dimensional ternary photonic band gap material, Progress In Electromagnetic Research M. 4 (2008) 117–132. [7] A. Banerjee, Enhanced refractometric optical sensing by using one-dimensional ternary photonic crystals, Progress In Electromagnetic Research, PIER. 89 (2009) 11–22. [8] A.P.F. Turner, Biosensors: sense and sensibility, Chem. Soc. Rev. 42 (2013) 3184. [9] J.L. Arlett, E.B. Myers, M.L. Roukes, Comparative advantages of mechanical biosensors, Nature Nanotechnology. 6 (2011) 203–215. [10] N.J. Ronkainen, H.B. Halsall, W.R. Heineman, Electrochemical biosensors, Chem. Soc. Rev. 39 (2010) 1747- 1763. [11] S. A. Taya, T. M. El-Agez, Comparing optical sensing using slab waveguides and total internal reflection ellipsometry, Turk. J. Phys. 35 (2011) 31-36. 9
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