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Progress In Electromagnetics Research B, Vol. 1, 197-208 2008

ENHANCEMENT OF OMNIDIRECTIONAL REFLECTION IN PHOTONIC CRYSTAL HETEROSTRUCTURES R. Srivastava and ShyamPati Department of Physics UdaiPratap Autonomous College Varanasi 221007, India S. P. Ojha Chaudhary Charan Singh University Meerut 200005,.India . Abstract—In this paper we have theoretically studied the omnidirec­ tional total reflection frequency range of a multilayered dielectric het­ erostructures. Three structures of NaaAlFg/Ge multilayer have been studied. The thickness of the two layers of the first and second struc­ ture is differing from each other and the third photonic structure is the combination of first and second structures. Using the Transfer Matrix Method (TMM) and the Bloch theorem, the reflectivity of one dimen­ sional periodic structure for TE- and TM-modes at different angles of incidence is calculated. From the analysis it is found that the proposed structure has very wide range of omnidirectional total frequency bands for both polarizations.

1. INTRODUCTION During the last 15 years, photonic crystals (PCs) have drawn much attention as a new kind of optical materials. . These materials are based on the interaction between an optical field and materials exhibiting periodicity on the scale of wave length [1-5]. Photonic crystals are composite structures with a periodic arrangement of refractive index in one-dimension (ID), two-dimensions (2D), or three-dimensions (3D) [6,7]. The main feature of photonic crystals is that they can prohibit the. propagation of electromagnetic waves within a certain frequency range called photonic band gap (PBG), which is analogous i

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Srivastava, ShyamPati, and Ojha

to the electronic band gap in ordinary materials. The materials containing PBG have many potential applications in optoelectronics and optical communication [8-12]. Ojha etal. [13] observed filtering properties in PBG materials and extended the idea for constructing monochromators [14]. Reflectors are one of the most widely used optical devices. There are two types of reflectors, the metallic reflectors and multilayer dielectric reflectors. Metallic reflector can reflect , light over a wide range of frequencies for arbitrary incident angles. At infrared, optical or higher frequencies, there is a considerable power loss owing to absorption. However, Multilayer dielectric reflectors can have an extremely low loss, means they have a high reflectivity over a broad range of frequencies at all incident angles, i.e., an omnidirectional total reflection, [15-29] if the refractive index and thickness of the constituent dielectric layers are properly chosen. This kind of omnidirectional dielectric reflectors can have potential applications in many ways, for instance, as micro-cavities, antenna substrates, or coaxial waveguides, etc. [30]. For the optical range within which the main application is expected, most of experimental effort has been concentrated on 2D and 3D photonic crystals. However,, due to technological problem and high cost, the application of 2D and 3D photonic crystals is limited. The ID photonic crystals are attractive since their production is more feasible at any wavelength scale and their analytical and numerical calculations are simpler. These crystals can exhibit the property of omnidirectional reflection [16,17,31]. A one-dimensional photonic crystal is a periodic multilayer structure consisting two type of layer which differs in the dielectric constant. The index of refraction is periodic in the y coordinate and consists of an endlessly repeating stack of dielectric slabs, which alternate in thickness from d\ to d,2 and. in index of refraction from n\ to n%. Incident light can.be either s-polarized or p-polarized [32,33]. In 1998 Fink etal. [16] first reported that one dimensional dielectric lattice displays total .omnidirection reflection for incident light under certain conditions, Chigrin et al. [19] described the effect at optical frequencies (604.3-638.4 nm) using 19 layers of NagAlFe/ZnSe. ' Large band gap is one aim in the study of photonic crystals. Large refractive, index contrast and specific structures are required to obtain a wide band gap. Disordered ID photonic crystal has been studied for band gap extension [34,35]. Combinations of two or more photonic crystals have been used to enlarge the frequency range of reflection [32,33]. Photonic crystals are used in manufacturing optical devices and to

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increase the capability of optoelectronic circuits. Many optical devices are designed using the property of high reflectance, attributed to high dielectric contrast at the NasAlFg/Ge interface (An ss 2) [34,36], These structures are suitable for antireflection coating and design the interference filter and thin films near infrared-visible region [37]. Such a structure can be used as an useful optical devices in the optical industry. In this article, we study theoretically the reflection properties of NaaAlFe/Ge ID photonic crystals. Two structures of same period and different layer thicknesses are studied and then these; two structures are combined to form a single PC. We show that it is possible to enhance the total reflection frequency range of omnidirectional reflector in this combined PC. The condition, for obtaining this large omnidirectional PBG is that the directional PBGs of constituent ID PCs should overlap each other. 2. THEORY The propagation of electromagnetic radiation in a simple periodic layered medium, consist of alternating layer of transparent materials with different refractive indices, we take x axis along the direction normal to the layer and assume that the materials are nonmagnetic. The index of refraction is

{rji

o <x
•q2 b< x < A with

■q{x) = rj(x + A)

(2)

. To solve propagation of electromagnetic radiation in this medium we use transfer matrix formulation [38,39].- The electric field is given as

E = E(x)-'e^4-M .

.

(3)

where /? is the z component of the wave vector. The electric field distribution E{x) within each homogenous layer can be expressed as. the sum of an incident plane wave and a reflected wave. The complex amplitudes of these two waves constitute the component of a column vector. The electric field of nth cell can be written as

{an- e~vkllT (x~nA)+bn ■ el'kll! (x~nA) ‘

'

nA—a<x< nA ....

Cn- e~ik2ir {x-nA+a)+
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Srivastava, ShyamPati, and Ojha

with fax = |(??1

Vl -u

- P2

C

fax = [(^2 • ^)

- P2

• cos 0i (5)

t]2 ' W

: COS 02

where 0i and 02 are the ray angles in the layer. Using matrix method e(ifex-6) . ^ +

bn—l)

2

blj

e-(ife2x-6) .

_

fax fak

e(i-k2x-b) ■ (l —

e-(i-k2x-b) . ^ _|_ fax fa k

(6)

and similarly ,(i-kix-a) .

dn)

2 ,{i-kix-a) .

klx\

(l + %T \

fak

(*

fax

g—(i-fclx-o) .

_

fax fak

e-(i-kix-a) . (

\~fak

1 4.

fax fak

(7)

. By eliminating (Cn, the matrix equation dn—l\ _ (A bn-1 ~ 1(7

B\ fOjj, D '{bn

(8).

is obtained a

, 1. (fa-x

fa x

2 \fa.%

fa-3

Ate = ei,klx'a cos(fa.x -b) + -i

Bte = e~iklx'a

sin(fc2.x - b) (9)

Progress In Electromagnetics Research B, Vol. 1, 2008 -i-k\.x-a

Dte

cos(k2.x • b) - -i

k2-x k\.x

fel-'a k2.x

sin(/52.a. • b)

This is for s polarization of electromagnetic wave. polarization Atm Btu Ctm

0i'k\.X'a

3 i-ki-x'a

ji'k

PtM = 6-^^

cos(b.«.6) + ii 1. (h-xvi

+

201

Similarly for p

I Mh»-b)

k2.xri\ . „

1, fkhxril k2.xnl\ sin(fc2.x • b) 2 fel-aT?2/ n u\ i-fkixV2 faxVi\ ■ a,

(10)

•

According to the Block equation Ek(x + A) = Ek(x)

(11)

and Block wave function K(/3 ■ oj) = — cos 1

(12)

Regimes 1/2[A + D] < 1 corresponds to real, and thus to propagating Block waves; when 1/2[A + D] > 1 however = rmr/A + i ■ Ki which has an imaginary parts K{ so that the Block wave, is evanescent. These are so called forbidden bands of the periodic medium he band edges are the regimes where 1/2 [A + D] = 1 the band structure for a typical periodic layered is obtained for TE and TM waves. The TM is forbidden band shrink to zero when (3 — f]2u>/c • sin &b with 6b the Brewster angle. /? = 0, for normal incident (TE). The dispersion relation u> verse K for normal incident can be written as cos (AA). = cos(fcx • a) cos (k2 ■ b)

~ f— + —) sin(fci ■ oj sin(fe2 • b) 2 \m mj (13)

where ' *i =

m -u c

and

k2

rj2-w =------c

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Srivastava, ShyamPati, and Ojha

3. RESULT AND DISCUSSION We have studied two ID photonic crystals PCI and PC2 consisting NasAlFg/Ge periodic multilayer structure of the same period, but with different layer thicknesses: PCI has 10 pairs of Na3AIF6/Ge as two dielectric layers with and layer thicknesses d\ — 0.75d and d% = 0.25d respectively and PC2 has 10 pairs of two dielectric layers of NasAlFe/Ge with layer thicknesses d\ = 0.50d and d2 = 0.50d respectively, where d is the stack thickness. The refractive indices of the crystals are ni = 1.34 and n
TE

TM

0

0.161-0.323

0.161-0.323.

45

0.164-0.360

0.186-0.337

85 ODR = 0.232-0.323

0.167-0.407

0.232-0.352

Table 2. For PC2. Angle(deg) 0 45

TE

TM

. 0.131-0.208

0:131-0.208

0.301-0.415

0.301-0.415

0.132-0.216

6.148-0.208

0.307-0.438

.

, 0.323-0.425

0.134-0.226

0.178-0.204

0.314-0:460 ODR = 0.204-0.131 and 0.358-0.415

0.358-0.433

85

It is clear from the Table 1 that the ODR range for PCI is 0.2320.323 and it is shown in Fig. 4(a). The ODR range for PC2 is 0.2040.131 and 0.358-0.415 and it is shown in Fig. 4(b). The ODR range

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203

Table 3. For PC1/PC2. Angle(deg)

TE

TM

0

0.129-0,417

0.129-0.417

45 .

0.131-0.438

0.146-0.427

85

0.134-0.462

0.172-0.434

Figure 1. Calculated reflectance spectra of PCI and PC2, and PC1/PC2 at the incident angle 0°. TM and TE polarizations are plotted as dotted and solid lines respectively. for PC1/PC2 is 0.172-0.417 and it is shown in Fig. 5. From the Tables 1, 2 and 3, we see that, although omnidirectional photonic band gaps for the two photonic crystals PCI and PC2 do not overlap, the directional photonic band gaps for PCI and PC2 overlap each other at different angles and for both the polarizations, TE and . TM, except at 85°, where there is some non-overlapping region, which appears as transmission peaks in the reflectance spectra of PC1/PC2.

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Figure 3. Calculated reflectance spectra of PCI and PC2 and PC1/PC2 at the incident angle 85°. TM and TE polarizations are plotted as dotted and solid lines respectively.

205

Figure 4. Omnidirection reflection band for (a) PCI and (b) PC2; TM and TE polarizations are plotted as dotted and solid lines respectively.

Figure 5. Omnidirection reflection band for PC1/PC2; TM and TE polarizations are plotted as dotted and solid lines respectively. 4. CONCLUSION Prom, the above study, we see that the 100% (approx.) reflectionrange becomes larger as compared to PCI and PC2, for both TE and TM polarization and for all incident angles. Thus, the omnidirectional reflection ranges are enhanced. Therefore, large, omnidirectional reflection range can be obtained for NasAlFg/Ge structure by combining the two photonic crystals as considered here. For obtaining this large frequency range, the directional photonic band gaps of two constituent PCs should overlap each other.

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REFERENCES 1. Pendry, J., “Photonic band structure,” J. Mod. Opt., Vol. 41, 209-229, 1994. 2. Jonopoulos, J, D., P. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature, Vol'.386, 143-149, 1997. 3. Noda, S., A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature, Vol. 407, 608-610, 2000. 4. Jonopoulos, J. D., R. D. Meade, and J. N. E. Yablonovitch, “Photonic crystals,” J. Mod. Opt., Vol. 41, 173-194, 1994. 5. Winn, J. N., Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. ■ 6. Yablonovitch, E., “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett., Vol. 58, 2059-2062, 1987. 7. John, S., “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett., Vol. 58, 2486-2489, , 1987. ' 8. Chen, K. M., A. W. Sparks, H. C. Luan, K. Wada, and L. C. Kimerling, “Si02/Ti02 omnidirectional reflector and . microcavity resonator via sol gel method,” Appl. Phys. Lett., Vol. 75, 3805-3807, 1999. .. 9. Temelkuran, B. and E. Ozbay, “Experimental demonstration of . photonic crystal based waveguides,” Appl. Phys. Lett., Vol. 74, 486-488, 1999. 10. Scalora, M., J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and. switching of ultrashort pulse in nonlinear photonic band gap materials,” Phys. Rev. Lett., Vol. 73, 13681371, 1994: 11. Zheng, Q. R., Y. Q. Fu, and N. C. Yuan, “Characteristics of planar PBG structures with a cover layer,” Journal of Electromagnetic Waves and Applications, Vol. 20, 1439-|-1453, 2006. 12. Ozbay, E., B. Temelkuran, and M. Bayinder, “Microwave application of photonic crystals”, Progress In Electromagnetics : Research, PIER 41, 185-209, 2003. 13. Ojha, S. P., P. K. Chaudhary, P. Khastgir, and O. N. Singh, “Operating characteristics of an optical filter with a linearly periodic refractive index pattern in the filter material,” Jpn. J. Appl. Phys., Vol. 31, 281-285, 1992. i4- Srivastava, S. K. and S. P. Ojha, “Operating characteristics of an

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15. 16. 17. 18. 19.

20. 21. 22.

23. 24. 25. .26.

27.

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optical filter in metallic photonic band gap materials,” Microwave Opt. Technol. Lett., Vol. 35, 68-71, 2002. Winn, J. N., Y. Fink, J. N. Winn, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection froni a one-dimensional photonic crystal,” Opt. Lett., Vol. 23,1573-1575, .1998. Fink, Y., J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopou­ los, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science, Vol. 282, 1679-1682, 1998. Dowling, J. P., “Mirror on the wall: You’re omnidirectional after all?” Science, Vol. 282, 1841-1842, 1998. . Yablonovitch, E., “Engineered omnidirectional externalreflectivity spectra from one-dimensional layered interference filters,” Opt. Lett., Vol. 23, .1648—1649, 1998. Chigrin, D. N., A.. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A: Mater. Sci. Process., Vol. 68, 25-28, 1999. Southwell, W. H., “Omnidirectional mirror design with quarterwave dielectric stacks,” Appl. Opt., Vol'. 38, 5464—5467, 1999. Russell, P. St. J., S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in ID multilayer dielectric structures,” Opt. Comrriun., Vol. 160, 66—71, 1999. Chigrin, D. N.., A. V. Larinenko, . D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightwave Technol., Vol. 17, 2018-2024, 1999. Abdulhalim, I., “Reflective phase-only .modulationusing one­ dimensional photonic crystals,” J. Opt. A: Pure Appl. Opt., Vol. 2, 9-11, 2000. Lekrier, J., “Omnidirectional reflection by multilayer dielectric mirrors,” . J. Opt. A: Pure Appl. Opt., Vol. 2, 349-352, 2000. Lusk, D., I. Abdulhalim, and F. Platido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun., Vol. 198, 273-279, 2001. Singh, S. K;, J. P. Pandey, K. B. Thapa, and S. P. Ojha, “Structural parameters in the formation of omnidirectional high reflectors,” Progress In Electromagnetics Research, PIER 70, 5378, 2007. Zandi O., Z, Atlasbaf, and K. Forooraghi, “Flat multilayer dielectric reflector antennas,” Progress In Electromagnetics

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28.

29.

30. 31. 32. 33. 34.

35. 36.

37. 38. 39.

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Research, PIER 72, 1-19, 2007. Srivastava, S. K. and S. P. Ojha, “Omnidirectional reflection bands in one-dimensional photonic crystal structure using fullerence films,” Progress In Electromagnetics Research, PIER 74, 181-194, 2007. Srivastava, S. K. and S. P. Ojha, “Omnidirectional reflection bands in one-dimensional photonic crystals! with left-handed materials,” Progress In Electromagnetics Research, PIER 68, 91111,2007. ; Ibanescu, M., Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science, Vol. 289, 415-419, 2000. . Deopura, M., C. K. Ullal, B. Temelkuran, and Y. Fink, “Dielectric omnidirectional visible reflector,” Optics Letters, Vol. 26, 11971199,: 2001. Li, H., H. Chen, and X. Qiu, “Band-gap extension of disordered ID binary photonic crystals,” Physica B, Vpl.. 279, 164-167, 2000. Zi, J., J. Wan, and C. Zhang, “Large frequency range of negligible transmission in one-dimensional photonic quantum well structures,” Appl. Phys. Lett., Vol. 73, 2084-2086, 1998. Wang, X., X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi,’ “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,” Appl. Phys. Lett., Vol. 80, 4291-4293, 2002. Guida, G., “Numerical studies of disordered photonic crystals,” Progress In Electromagnetics Research, PIER 41, 107-131, 2003. Singh, S. K., K. B. Thapa and S. P. Ojha, “Large frequency range of omnidirectional reflection in Si-based one-dimensional photonic crystals, International Journal of Microwave and Optical Technology, Vol. 1, 686-690, 2006. Mohamed, S. H., M. M. Wakkad, A. M. Ahmed and A. K. Diab, “Structural and optical properties of Ge-As-Te thin films,” Eur. Phys. J. Appl. Phys., Vol. 34, 165-171, 2006. Yeh, P., Optical Waves in Layered Media, John Wiley & Sons, New York, 1988. Born, M. and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1998.

Progress In Electromagnetics Research, PIER 81, 225-235, 2008

DESIGN OF PHOTONIC BAND GAP FILTER R. Srivastava Department of Physics Udai Pratap Autonomous College Varanasi 221007, India K. B. Thapa • Department of Physics U. I. E. T., C. S. J. M. University Kanpur-208024, India S. Pati Department of Physics Udai Pratap Autonomous College Varanasi 221007, India S. P. Ojha Choudhary Charaii Singh University Meerut 200005, India. Abstract—In this paper a new type of optical filter using photonic band gap materials has been suggested. iA detailed mathematical analysis is presented to predict allowed and forbidden bands of wavelengths ■ with variation of angle of incidence. It is'possible to get desired ranges of the electromagnetic spectrum, filtered with this structure by changing the incidence angle of light.

1. INTRODUCTION Photonic crystals PCs have drawn much attention as a new kind of optical materials [1-6]. PCs made of periodic dielectric materials in one, two or three spatial directions that exhibit electromagnetic stop bands or photonic band gaps (PBGs). PBGs have been investigated intensively relating to their ability of controlling the propagation of

226

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light [6-9]. The absence of electromagnetic wave or light wave inside PBGs will lead to some unusual properties, which can be used for.Bragg mirrors or narrow-band filters [10-16]. An optical filter is a device, which has the property of adding or dropping a particular wavelength channels from the multi wavelength network. . A great deal of work has been done by technologists for the development of methods for designing multi-layer films with prescribed characteristics [17-21], Tunable optical filters have received much attention due to their application in fibre optic communications and other optical fields. Several configurations have been proposed, including tunable multiple electrode asymmetric directional couplers [23], tunable Mach Zehnder interferometers [24,25], fibre Fabry-Perot filters. [26,27], .tunable waveguide arrays [28,29], liquid crystal Fabry-Perot filters [30,31], tunable multi grating filters [32], and acousto-optic tunable filters. [33]. Another class of most popular filter based on the phenomenon of multi-beam interference and based on waveguides [34-37]. Fabrication of optical filters in the near infrared region of the wavelength was suggested by Ojha et al. [38] in 1992. Chen et al. [39] in 1996 suggested the design of optical filters, by photonic band gap air bridges and calculated the important, results and some aspects of. filter properties. Recently D’Orazio et al. [40] have fabricated the photonic band gap filter for wavelength division multiplexing. In another investigation Villar et. al. [41] have analyzed the one­ dimensional photonic band gap structures with a liquid crystal defect for the development of fiber-optic tunable wavelength filters. In this paper a new type of optical filter using photonic band gap materials has been suggested. The working principle as well as the theoretical analysis of this filter is based on the Kronig and Penney model. 2. THEORY

It is well known that when electrons move through a periodic lattice, allowed and forbidden energy bands, are obtained. The same idea may be applicable to the case of optical radiation if the electron waves are replaced by optical waves, and the lattice periodicity structure is replaced by a periodic refractive index pattern. One expects allowed and forbidden bands of frequencies instead of energies. By choosing a linearly periodic refractive index profile in the filter material one. obtains a given set of wavelength ranges that are allowed or forbidden to pass through the filter material. Selecting a particular *-axis through the material, we shall assume a periodic step function for

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227

the index of the form [22,42]

n

ni,

0 < x < a;

Ti2,

—b<x<0\

(1)

where n(x) = n (x + md) and m is the translation factor, which takes the values m = 0, ±1, ±2, ±3, ..and d = a + b is the period of the lattice with a and b being the width of the two.regions having refractive indices (ni) and (n.2) respectively. The refractive index profile of the materials in the form of rectangular symmetry is shown in the Figure 1.

tti

| ( '•

-b

.0

,a

v'

'x

Figure 1. Periodic refractive index profile of material. If 6 is the angle of incident on this periodic structure the onedimensional wave equation for the spatial part of 'the electromagnetic eigen mode (x) is given by

(g) . w2(a)cos2 0-“k:, {t) _ 0 dx2 C2 Vk(X)~ u,

(2)

where n(x) is given by Equation (1). Therefore, Equation (2) for wave equation may be written as 0<x
(3a)

—b<x< 0

(3b)

where 0\ and Q% .are ray angle in the layer of refractive index n\ and fi2 respectively. . The periodic nature of the problem allows the application of Bloch’s theorem which solution can be written as ipK — uk (x) elKx where K is known as Bloch wave number and uk (x) is the value of the eigen function. Thus using this, theorem Equations (3a) and (3b) can be written as

-~j~Y +

+ (a2 - Ku\ = 0;

0<x< a

(4a)

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Srivastava et al.

d?U2 + 2iK^ + (f - K2) u2

dx2

dx

0;

—b<x<0

(4b)

where a = (^cos0i), /3 = cos62), 0\ — cos_1[l d2 = cos_1[l — ~~]1/2 and u\ represents the value of uk(x) in the interval (0, a) and u2 in the interval (—6, 0) respectively. The solution of differential Equations (4a) and (4b) can be written as ui= Aei{-a~K rel="nofollow"> + Be~i{-a+K^

(5a)

u2 = CeW~K> + De~W+K>

(5b)

Now applying the boundary conditions as given below «i(®)U=o = U2(x)\x=o u'i{x)\x=0 = it2.(*)U=o .

.

(6a) (6b)

’ «i(*)U=a = U2{x)\x=-b

(6c)

«l(®)|x=a. ='U2{x)\x=-b

(6d)

we get four equations having four unknown constants. To obtain a non­ trivial solution for the equations, the determinant of the coefficients of the unknown constants must be'zero, which is given as A\\ A21

A12 A13 Au A22 A23 A24

^■31

^.32

^33

A41

A42

>143 A44

^.34

_n

/7\ ’

where

An=Ai2 = Aw = A14 = 1; A2i=i{a - K), A22 = ~i(a + K), A22=i(P - K), A24 = —i(P + K)\ Azl=eia{~a-K\ A32 = e~ia{a+K\ A33 = e~ib^~K\ ^34 = ei6^+*>; Aii=i (a - K) eia{o~K\ A42 =-i (a + K) e~ia^+K\ Ai3=i (p - K) e~ib^~K\ A44 - t (/3 + K) eib^+K\ On solving Equation (7) we obtain cos (if d) ■

—

cos (aa) • cos (/36) 1 / ni • cos 0 ri2 • cos 02 •sin (aa) ■ sin (/3b) (8) 2 \7l2 • cos 02 n\ • cos 0i \

Now, abbreviating the L.H.S. as L\, Equation (8) may be written as L\ = cos (K ■ d)

(9)

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229

3. RESULT AND DISCUSSION For the proposed filter, we have chosen the dielectric materials as NagAlFe/ZnS for ultraviolet region with low and high index contrast. The refractive index for NasAlFg is n\ — 1.34 and for ZnS is «2 = 2.2. The thickness of the layers is a and 6 respectively. Taking the values of the a and b as Yablonovite structure a = 85% of d and b — 15% of d' where d = a + b. Using these values, Equation (9) is plotted against the wavelength A and the curves are depicted in the Figures 2 to 5 respectively. The photonic bands obtained in this manner are shown in the Tables 1 to 4 respectively. Because of the existence of the cosine function on the right-hand side of the Equation (9), the upper and lower limiting values will obviously be +1 and —1 respectively. The

Figure 2, Variation of L(A) with wavelength (A).for n\ = 1.34,. n2 = 2.2, d = 500nm, a = 0.85d and b = 0.15d and 0 = 0°.

Figure 3. Variation of L (A) with wavelength (A) for nx = 1.34, n2 = 2.2, d = 500 nm, a = 0.85d and b — 0.15d and 9 = 30°.

Srivastava et al.

230

«2

= 2.2, d — 500nm, a — 0.85d and b = 0.15d and 6 = 45°.

wavelength (A) .■

Figure. 5. Variation of L (A) with wavelength (A) for ni = 1.34, n2 = 2.2, d = 500 nm, a = 0.85d and b — 0.15d and 9 — 60°. Table 1. Photonic bands for (ni = 1.34, a = 0.85d and b = 0.15d and 0 = 0°. Allowed Bands ■ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Allowed Ranges (in A0) 1000-1044 1055-1127 ■ ‘ 1133-1210 1237-131-8 1355-1455 1487-1631 1634-1813 1855-2054 2143-2397 2509-2910 2978-3500 ■

— 2.2, d — 500 nm,

Band Width (in A0) ' 44 72. , '77 81 100 144 179 199 254 401 522

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231

Table 2. Photonic bands ,for.(ni = 1.34, »2 = 2.2, d — 500nm, a = 0.85d and b — 0.15d and 6 = 30°. Allowed Bands 1.

.

2. 3. 4. ' .5.

. '

Allowed Ranges, (in A0)

Band Width (in A0)

1000-1059

59

1062-1137

'75

1157-1233

76

1271-1359

88

1401-1524

123

6.

1542-1706

164

7.

1736-1923 '

187

8.

2011-2239

228

9.

2363-2715

352

10.

2810-3413

603

Table 3. Photonic bands for {ri\ = 1.34, a = 0.85d and b = 0.15d and 9 = 45°. Allowed Bands

Allowed Ranges (in A0)

712

,

= 2.2, d — 500 nm,

Band Width (in A0)

1.

1000-1062

62 .

2.

1070-1145

75

3. 4.

1179-1257

78

1304-1407

103

5.

1442-1594

152

6.

1604-1787.

•183

7.

1863-2070

234

8.

2202-2506

304

9.

2631-3190

559

portion of the curve lying between these limiting values will yield the allowed ranges of A and those outsides will show the forbidden ranges of transmission. Prom the. study of these figures it is found that the width of the allowed photonic bands increases as the wavelength increases, for a fixed values of a, b, n\ and n2. Actually these allowed bands give the different ranges of wavelengths that can be transmitted through the filter structure. The ranges of transmission depend on the values of controlling parameters a, b, n\ and n2. So by choosing suitable values

232

Srivastava et al.

Table 4. Photonic bands for (rai = 1.34, hi = 2.2, d = 500 nm, a = 0.85d and b = 0.15d and 6 = 60°. Allowed Bands

Allowed Ranges, (in A0)

Band Width (in A°j

1.

1000-1055

55

2.

1075-1148

73

3.

1196-1279

83

4.

. 1332-1459

127

5.

1476-1643

6. 7.

1700-1887 2023-2275.

8.

2400-2921

.

■

167 187 .

252 531

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