ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)
Vol. 03, Issue 04,April 2018 Available Online: www.ajeee.co.in/index.php/AJEEE
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THEORETICAL ANALYSIS OF SPHERICAL DIELECTRIC RESONATORS WITH RADIALLY VARYING DIELECTRIC CONSTANT
Sujit Kumar Verma
Assistant Prof. and Head, Department of Physics, SMM Town PG College, BALLIA (UP) Abstract:- Dielectric resonators are of considerable interest in the field of microwave technique due to their applications as filters, oscillators etc at high frequencies. At frequencies in the optical range the dimensions of the dielectric resonators are small which puts limitation in fabrication. Electromagnetic analysis of spherical dielectric resonators suitable for the microwave region has been made by a numbers of workers [1-4].However no such study appears to have been made for the optical frequency range. In the present paper the electromagnetic field analysis starting from the Maxwell’s Equations is reported for spherical dielectric resonators with radially varying dielectric constant.
Keywords: Spherical Resonators, Inhomogeneous, resonance frequency, confinement.
1. INTRODUCTION
Optical communication refers to the transmission of speech, data, picture or other information by light. Optical frequencies are some five order of magnitude higher than, say microwave frequencies. Therefore larger volume of information can be transmitted through fiber cable compared to that through copper coaxial cable of similar size. The reductions of transmission loss in optical fibers and development in the area of the light sources and detector have brought about a phenomenal growth of the fiber optic industry. The optical fiber is essentially a dielectric wave guide [9]. Dielectric resonators using the material with high dielectric constant, lower losses and better temperature coefficient have been studied in literature. Although cylindrical resonators are commonly preferred the spherical resonators have also been considered. A dielectric sphere with a given dielectric constant and radius possess natural modes of oscillation having characteristics frequencies [4]. Such oscillations/resonance have been studies both theoretically and experimentally in the microwave regions [6,7] and more recently in the optical region of the electromagnetic spectrum [11]. Ideally even if the material loses or neglected the theoretical analysis for a dielectric resonator shows that the characteristics frequencies are complex and therefore their exist leaky modes. These radiation losses can be minimized using material of high dielectric constant. The resonant frequency and quality factor for free and shielded spherical dielectric resonators has also been calculated for millimeter applications [7]. In order to investigate the possibility of the confinement of EM field inside the resonator, a theoretical analysis of spherical dielectric resonators is presented there. In this we propose to find field expression, characteristics equation and calculate resonance frequency for different mode. In the present paper the electromagnetic field analysis starting from Maxwell’s equations is reported for spherical dielectric resonators with radially varying dielectric constant.
2. THEORY
The spherical dielectric resonator is equivalent to a spherical hole in a perfect conductor filled with dielectric material. Waldron [1] has presented the analysis for a spherical cavity in a perfect conductor. In the present work theoretical electromagnetic field analysis of spherical resonator for eigenmodes is completed using straightforward method. In a source free homogeneous and dielectric medium the four Maxwell’s equations are given by
𝛁. 𝐃 = 0 … … … (1)
𝛁. 𝐁 = 0………..(2)
𝛁 × 𝐄 = −∂𝐁∂t………(3)
𝛁 × 𝐇 =∂𝐃∂t……….(4) As we know relation for B and D
B=μ𝐇 = μ0μrH D=ε𝐄= ε0εrE
ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)
Vol. 03, Issue 04,April 2018 Available Online: www.ajeee.co.in/index.php/AJEEE
2
Where μ0 and ε0 are permeability and permittivity of free space. So that the velocity of light in vacuum c2 =( μ0ε0)-1.For non-magnetic dielectric μr=1.In order to derive wave equations for E taking curl of equation (3) and for H taking curl of equation (4).[11,15]. Since we are interested in the case of dielectric constant which is a function of radial coordinate only, further simplification can be achieved by separating the solution of vector wave equation into two parts: transverse electric and transverse magnetic. The condition satisfied by the fields for TE and TM modes are respectively given by
r.E=0 r.H=0 Then we get
∇2E -ε
C2
∂2𝐄
∂t2=0 for TE
∇2H -Cε2∂∂t2𝐇2=0 for TM
Assuming e−jωt time dependence for all field components the problem essentially simplifies to the solution of scaler equation
∇2Ψ + ε r k2Ψ =0………..(5)
The field components are obtained using equation (3) and (4).Thus for TE mode we use H= 1
jωμ0(∇ × 𝐄) and for TM modes E= - jωε1
0(∇ × 𝐇) The solution of TE and TM modes are given by
E= r× 𝛁Ψ𝐞−𝐣𝛚𝐭 and H= r× 𝛁Ψ𝐞−𝐣𝛚𝐭The general solution can now be expressed as the sum of two types of field.
Solution of scalar equation: The equation ∇2Ψ + ε r k2Ψ =0
Where ε r is dielectric constant and depends on r this equation in spherical polar coordinates takes the form 1
r2sin θ[sin θ∂r∂(r2 ∂∂rΨ)+∂
∂θ(sin θ∂Ψ∂θ)+ 1
sin θ
∂2𝚿
∂ϕ2=- ε r k2Ψ
Now we suppose the solution of this equation Ψ(r, θ, ϕ)=R(r)Θ(θ)Φ(ϕ) To split it into a set of ordinary differential equation. Using standard differential equation we get
r2 ddr2R2+2rdR
dr+{ ε r k2r2 –l(l+1)}R=0
For uniform Dielectric Constant: Here ε r =ε,the solution of equation (9) can be expressed in terms of spherical Bessel Equation. If we Substitute ε1/2 kr=e Then using spherical Bessel function we can write the solution as R={A[(ε12kr)J1(ε12kr)] + B[(ε12kr)n1(ε12kr)}
For Inhomogeneous Dielectric constant: In Model –I , take the form ε r =εm-β2r2
The solution of equation (9) can be obtained in terms of generalized laguerre polynomial.
The solution is u(ξ)=e−ξ22ξl+1L2l+1n (ξ2) Here L2l+1n (ξ2) is generalized laguerre polynomial of order (n+l+1).
In Model -II, take the form ε r =εr01- α2
Again, The solution of equation (9) can be obtained in terms of generalized laguerre polynomial. The solution is u(η)=e−η2ηl+1L2l+1n (η) Here L2l+1n (η) is generalized laguerre polynomial of order (n+l+1).
Calculation of Electromagnetic fields: The complete expression for Ψ
Is thus obtained in which radial part Rnl is Characterized by two indices (n,l)and depending upon the dielectric constant formε(r).The electric field components for TE modes are given by
Er =0
ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING
Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767, (ISSN NO. 2456-1037)
Vol. 03, Issue 04,April 2018 Available Online: www.ajeee.co.in/index.php/AJEEE
3 Eθ=Rnl(r)[- 1
sin θ
dYlm(θ,ϕ) dϕ ] Eϕ=Rnl(r)[dYlm(θ,ϕ)
dθ ]
We observe that TE and TM modes are degenerate. The lowest value of l is 1since all components will vanish for l=0.
Calculation of resonant frequencies: The dielectric function given by ε r =εm-β2r2 and ε r =εr01- α2 is varying with value of r. Since the dielectric constant have singularity at r=0, so for the practical case we assume that it is given by two segments ε = εm for (0<r<qa) and ε r =εr01- α2 for (qa<r<a) where ‘a’ is the radius of the sphere and ‘q‘ is the function of radius in which ε is constant. In order to compare our calculated result with the known result I have tabulated the values obtained by Julein and Guillion[12] .
3 RESONANT FREQUENCY FOR SOME RADII OF SPHERICAL RESONATOR
Taking a=1.56 mm and 0.72 mm and value of maximum dielectric constant εm =9.7 to 36, I have calculated the resonant frequency for metallic resonator and for both modes of dielectric resonator. The value of dielectric constant is adjusted such that at r=a, ε r =1, i.e free space value [12,14]. The result shown in table indicates that the resonant frequency for inhomogeneous resonators is higher than the homogeneous ones.
REFERENCES:
1. A. Julien & P. Guillion, IEEE Trans. Microwave Th. Tech vol-MTT-15, pp-723-729 June 1986.
2. T. Yamasaki, K. Sumioka, T.Tsutsai, Appl. Phy. Lett.7 6 (2000)1243.
3. W.Y.Zhang, X. Y. Lei, Z. L. Wang, et. al, Phy. Rev. Lett. 84(2000) 2853.
4. K.A. Zaki and A.E. Atia, “Modes in dielectric – loaded waveguide and resonant, IEEE Trans. MTT-31 No.7, pp .815-824 july 1986.
5. D. Brady, G. Papen, J. E. Sipe, J. opt. soc. Am. B10 (1993)644.
6. K. G. Sullivan, D. G. Hall, Phy. Rev. A50(1994) 2701.
7. M.Gastine, L. Courtoiis and J. Dormann, IEEE Trans. Microwave Th. Tech vol-MTT-304, pp-694-7700, 1967.
8. P. Debye, Ann. Phys.30, 57(1909).
9. R A Waldron, Theory of guided electromagnetic waves (Van Nostrand Reinhold Company, London, 1969).
10. D G Blair and S K Jonnes, J. Phys.D20, 1559(1987).
11. J D Jackson, Classical electrodynamics, third edition (John Wiley and Sons, Inc. New York, 1998).
12. R A Yadav and I D Singh ,Pramana –Journal of Physics ,vol.62, No. 6 June 2004 ,pp 1255-1271.
13. C.C. Chen, IEEE Trans. Antenns propagate vol.-46-pp-1704-1803, July 1998.
14. P Affolter and B Eliaasson, IEEE Trans. MTT-21, 573 (1973).
15. J A Stratton, Electromagnetic theory (McGraw-Hill, New York, 1950).
16. N. Hodgson, H. Weber, optical resonator, London, Berlin Springer 1997.
Radius
(mm) Types of Resonators Dielectric Constant
Parameters Frequency (GHz)
0.72
Metallic
Unshielded uniform dielectric
ε =36.0 33.9 48.8
ε =9.7 63.1 -
Non uniform
Dielectric(Model-1)
εm =36.0,β=8.217mm-1
εm =9.7,β=4.097mm-1 54.48
100.82 54.48 100.82 No uniform Dielectric
(Model-2)
ε01=2.80,α2=2.889, εm=36.0
ε01=1.566,α2=1.175, εm=9.7 115.88
132.18 115.88 132.18 1.56 Metallic
Unshielded uniform dielectric
ε =36.0 15.67 22.6
ε =9.7 29.7 -
Non-uniform Dielectric(Model-1)
εm =36.0,β=3.92mm-1
εm =9.7,β=1.891mm-1 25.14
46.50 25.14 46.50
Non uniform
Dielectric(Model-2)
ε01=6.067,α2=2.889, εm=36.0
ε01=3.93,α2=1.175, εm=9.7 53.47
60.98 53.47 60.98
