J. of Electromagn. Waves and Appl., Vol. 26, 1007–1017, 2012

ANALYSIS OF ELECTROMAGNETIC FIELD TRANS-MITTED BY A UNIAXIAL CHIRAL LENS

A. Ghaffar1, *, M. H. Shahzad2, M. Y. Naz1, S. Ahmed3,M. Sharif2, Q. A. Naqvi3, and A. A. Syed3

1Department of Physics, University of Agriculture, Faisalabad,Pakistan2Department of Physics, GC University, Faisalabad, Pakistan3Department of Electronics, Quaid-i-Azam University Islamabad,Pakistan

Abstract—In this paper, the transmission of the electromagneticwaves through a plano convex lens of lossless uniaxial chiral mediumis derived using dyadic transmission coefficient. Derived geometricaloptics field is valid everywhere except at focal point where it givesinfinite value.Singularity of the field at focal point is addressed usingMaslov’s method. The effects of different constitutive parameters onthe refracted field are also discussed.

1. INTRODUCTION

During last few decades, considerable effort has been put to thestudy of electromagnetic fields in complex artificial materials, i.e.,materials composed of chiral objects [1–10]. It is known that randomarrangement of chiral objects (e.g., wire spirals) give rise to isotropicchiral material, which is not straightforward and also costly toproduce [11]. If all the spirals in the lens have the same direction,the effective material of that lens becomes uniaxial like sculpturedthin films [12]. Uniaxial chiral materials may be prepared for opticalpurposes which is non homogeneous in the thickness direction buttransversely homogeneous. Anisotropic property is an additionalfeature of these materials. These features give rise to a mathematicalframework for enabling precise and predictable engineering of thepolarization state of light, in addition to the widely practiced spectrum

Received 24 December 2011, Accepted 14 March 2012, Scheduled 5 April 2012* Corresponding author: Abdul Ghaffar (chabdulghaffar@yahoo.com).

1008 Ghaffar et al.

engineering and bandwidth control [13]. This is the basic reason for ourmotivation to consider the problem of transmission of electromagneticwaves through a plano convex lens of a lossless uniaxial chiral mediu.Reflection and transmission through slabs of such materials have beenstudied for the case when the axis of uniaxial medium is transverse tothe interfaces of slab [14, 15].

In the uniaxial chiral medium, the electromagnetic fields satisfythe following constitutive relations

D = ¯ε · E − j√

μoεo ¯γ · H (1)B = ¯μ · H + j

√μoεo ¯γ · E (2)

with the medium parameter dyadics¯ε = εxaxax + εt(ayay + azaz) (3)¯μ = μxaxax + μt(ayay + azaz) (4)¯γ = γaxax (5)

where γ is the chirality parameter. In above equations ax, and ay arethe axes transverse to z-axis with ax = az ×ay. These are chosen suchthat they form natural coordinates for the geometrical optics.

In this study, transmission of plane wave through a uniaxialchiral plano convex lens placed in free space is considered. Dyadictransmission coefficient for uniaxial chiral plano convex lens is derived.Field expressions are written for three regions of the geometry, andboundary conditions are employed at two interfaces. The analysis isrestricted to the incidence normal to the lossless uniaxial chiral planoconvex lens and propagation is transverse to the axis of the uniaxiallens. The field transmitted by lossless uniaxial lens converges at focalpoint of the lens. The field propagating away from the lens is obtainedby using geometrical optics (GO) technique which gives singularityat focal point. To overcome this disadvantage of geometrical optics,Maslov’s method is used to obtained uniform field around the causticregion. According to Maslov’s method, the field expression near thecaustic can be constructed by using the geometrical optics information,though we must perform the integration in the spectrum domain inorder to predict the field in the space domain [16]. This method hasbeen successfully applied to predict the field in the focal region offocusing systems by many researchers [17–23].

2. FORMULATIONS

Consider a uniaxial plano convex lens with aperture diameter a andfocal length f is placed in free space (ε0,μ0) as shown in Figure 1.

Electromagnetic field transmitted by uniaxial chiral lens 1009

Figure 1. Normal incidence on uniaxial chiral lens.

The surface of the plano convex lens is described by followingequation [9]

ζ = g (ξ) =n

n + 1f − 1√

n2 − 1

√ξ2 +

n − 1n + 1

f2 (6)

where (ξ, ζ) are Cartesian co-ordinates, n the refractive index of chirallens, and f the focal length of the lens. Surface of the uniaxial chirallens at z = 0 is excited by plane wave described by the followingexpression

Ei (r) = Eie−jk0z (7)

Hi (r) = Hie−jk0z (8)

Hi =1η0

az × Ei.

Then the reflected field may be written as [16]

Er (r) = Erejk0z (9)

Hr (r) = Hrejk0z (10)

Hr = − 1η0

az × Er

where k0 = ω√

μ0ε0 and η0 =√

μ0/ε0.Inside the uniaxial chiral lens, the total field is assumed as

combination of four waves, two propagating toward the interface z = ζand two propagating toward the interface z = 0 as shown in Figure 1.

1010 Ghaffar et al.

The electric and magnetic fields corresponding to waves propagatingtoward the interface z = ζ may be written as

E+c (r) = E+

1 e−jk+z + E+2 e−jk−z (11)

H+c (r) = H+

1 e−jk+z + H+2 e−jk−z (12)

H+1,2 =

k±ktηt

az × E+1,2

where k± = kt

√A±, kt = ω

√μtεt, ηt =

√μt/εt E+

1,2 = E+1,2e±,

e± = s±ax + jay and A± = 12

(μz

μt+ εz

εt

)±√

14

(μz

μt− εz

εt

)2+ γ2μ0ε0

μtεt

and s± = kt

√A±

γk0

(A± − εz

εt

).

Above expression may be simplified using following notations

E =εz

εt, M =

μz

μt, N =

γk0

ktand X =

√ME − N2

where parameters εz , μz, εt and μt are assumed to be real.These quantities are assumed to be positive, and A± may be

expressed as

A± =12

[M + E ±

√(M + E)2 − 4X2

](13)

The electric and magnetic fields corresponding to waves propagatingtowards the interface z = 0 may be written as

E− (r) = E−1 ejk+(z−ζ) + E−

2 ejk−(z−ζ) (14)

H− (r) = H−1 ejk+(z−ζ) + H−

2 ejk−(z−ζ) (15)

H−1,2 = − ¯Y±E−

1,2, E−1 = ¯R1E

+

1 e−jk+ζ E−2 = ¯R2E

+

2 e−jk−ζ

On the other side of the lens, the total transmitted field may be writtenas

Et (r) = Et+e−jk0(z−ζ) (16)

Ht (r) = Ht+e−jk0(z−ζ) (17)

Ht+ =

1η0

az ×Et+

The unknown dyadic coefficients may be obtained using the boundaryconditions at z = 0 and z = ζ as

Ei (0) +Er (0) = E+ (0) +E− (0)Hi (0) +Hr (0) = H+ (0) +H− (0)E+ (ζ)+E− (ζ) = Et (ζ)H+ (ζ)+H− (ζ) = Ht (ζ)

Electromagnetic field transmitted by uniaxial chiral lens 1011

If we define the dyadic transmission coefficients through this expressionEt = ¯T · Ei and following the same procedure as given in [15, 16] thetransmission dyadic may be given as

¯T =(

¯A + ¯B)·(

¯R+¯It)

(18)

where

¯A =[¯It + ·

(¯Y1

+ + ¯It)−1

·(

¯Y1+−¯It

)]e−jk+ζ

¯B =[¯It+ ·

(¯Y1−+¯It

)−1 ·(

¯Y1−−¯It

)]e−jk−ζ

¯It = axax + ayay = e+e+ + e−e− = a+a+ + a−a−¯Y1± = η0

(az × ¯Y±

)¯Y = a+e+ + a−e− = (e+a+ + e−a−)−1

a± =η0

ηt

(s±√A±

ax + j√

A±az

)

The admittance dyadic may be expanded in Cartesian coordinates

¯Y =η0

ηt

[√M + E + 2X

M + X(axax + Xazaz) +

jN

M + X(axaz + azax)

](19)

The reflection dyadic is given by [15]¯R = Rxxaxax + Rxyaxay + Ryxayax + Ryyayay (20)

where

Rxx =η2

t (M + X) − η20 (E + X) + ηtη0 (X − 1)

√M + E + 2X

η2t (M + X) + η2

0 (E + X) + ηtη0 (X + 1)√

M + E + 2X

Rxy = Ryx =−2jNηtη0

η2t (M + X) + η2

0 (E + X) + ηtη0(X + 1)√

M + E + 2X

Ryy =η2

t (M + X) − η20 (E + X) + ηtη0(X − 1)

√M + E + 2X

η2t (M + X) + η2

0 (E + X) + ηtη0(X + 1)√

M + E + 2X

The transmission dyadic in simplified form is written as

¯T =[Υ+ (Rxx + 1) axax + RxyΥ+axay + RyxAt1ayax

+At1(Ryy + 1) ayay

]e−jk+ζ +

[Υ− (Rxx + 1) axax

+RxyΥ−axay + RyxAt2ayax + At2(Ryy+1) ayay

]e−jk−ζ (21)

1012 Ghaffar et al.

where

Υ+ =η0

ηt

√A+

+ 1, Υ− =η0

ηt

√A−

+ 1, At1 =η0

ηt

√A+ + 1

and At2 =η0

ηt

√A− + 1.

The wave vector corresponding to the wave refracted from the rearside of the uniaxial lens in free space is [9]

q = npi +√

1 − n2 + n2 (pi ·N)N −(piN

)N (22)

where pi = az and N is unit normal

N = − g′(ξ)√

1 + (g′(ξ))

2ax +

1√1 + (g

′(ξ))

2az

Thus q can be written as

q =K (α) sin αax + (n + K (α) cos α) az = qxax + qzaz

whereK (α) =

√1 − n2 sin2 α − n cos α.

The GO transmitted electric field can be obtained using following [20]

Et = Et

[D(τ)D(0)

]−12

ejkt(S0+τ) (23)

where τ represents the parameter along the ray, S0 = nζ shows theinitial value of the function. D(τ) represents the Jacobian of thetransformation from Cartesian coordinates to ray coordinates. TheHamilton’s equations for refracted field are

dx

dτ=qx,

dz

dτ=qz,

dqxdτ

= 0,dqzdτ

= 0

For x = ξ z = ζ, qx = qx0 and qz = qz0, we obtain initial values for(x, y) and (qx, qz).

x = ξ + qxτ, z = ζ + qzτ

The Jacobian is evaluated as

D (τ) =∂(x, z)∂(ξ, T )

= n +K(α)cos α

+(

qz∂qx

∂ξ− qx

∂qz

∂ξ

)τ

J (τ) =∂(τ)∂(0)

= 1 +cos α

ncosα + K(α)

(qz

∂qx

∂ξ− qx

∂qz

∂ξ

)τ

(24)

Electromagnetic field transmitted by uniaxial chiral lens 1013

where

∂qx

∂ξ=

∂qx

∂α· ∂α

∂ξ=

(cos α

(1 − 2n2 sin2 α

)√

1 − n2 sin2 α− n cos 2α

)(n − 1)

32√

n + 1f2

∂qz

∂ξ=

∂qz

∂α

∂α

∂ξ= −

(sinα

(1 + n2 cos 2α

)√

1 − n2 sin2 α+ n sin 2α

)(n − 1)

32√

n + 1f2

It is obvious from above expression that due to the presence ofsingularity, the GO method fails in the vicinity of the focal point.Thus we apply the Maslov’s method and according to this method,the transmitted electric field expression is obtained as [17–20]:

Et(r)=

√K(α)2πj

τ/2∫−τ/2

Et

√n cos α + K(α)

qz cos α

∂qx

∂x

ejkt(K(α)ρ sinα sinφ−K(α)ξ sinα+(n+K(α) cos α)ρ cos φ−K(α)ζcos α)dα (25)

The electric field expression in component form may be expressedas

Ex(r)E0

=

√K(α)2πj

τ/2∫−τ/2

(Rxx+1)(Υ+e−jk+ζ+Υ−e−jk−ζ

)[n cos α+K(α)qz cos α

∂qx

∂x

]12

×ejkt(K(α)ρ sinα sinφ−K(α)ξsinα+(n+K(α) cosα)ρ cosφ−K(α)ζcosα)dα (26)

Ex (r)E0

=

√k

2πj

τ/2∫−τ/2

Ryx

((η0

ηt

√A++1

)e−jk+ζ +

(η0

ηt

√A−+1

)e−jk−ζ

)[n cos α+K(α)

qz cos α

∂qx

∂x

]12

×ejkt(K(α)ρ sinα sinφ−K(α)ξ sinα+(n+K(α)cosα)ρ cosφ−K(α)ζ cosα)dα (27)

3. RESULTS AND DISCUSSION

In this section, analytical expressions (26) and (27) are used forcalculating the refracted electromagnetic field of a cylindrical uniaxialchiral lens. For numerical computation Mathcad software is employed.Both equations give similar results, so only results obtained fromEquations (26) are presented. Present problem is extension of ourrecent work [9] on isotropic chiral plano convex lens. The field of acylindrical uniaxial plano convex lens at the caustic point is obtained

1014 Ghaffar et al.

using Maslov’s method. The effects of the chirality, the permittivityand permeability of the uniaxial chiral lens on the transmitted fieldare observed.

The refractive index n = 2.5, focal length f = 175λ and radiusa = 200λ have been assumed in the simulation. Figures 2 show the fieldintensity distributions along z-axis for different values of the chiralityparameter γ. The solid line represents the fields for γ = 0, whichcorresponds to a dielectric lens. The other curves show the results forγ = 1.5, γ = 2.5, γ = 3.2, γ = 4.2 and γ = 5.2 respectively. FromFigure 2, it is observed that the intensity of the field at the causticpoint decreases with increase in the value of chirality parameter.

Figure 3 shows the field intensity distributions along z-axis fordifferent values of the transverse permeability The solid line representsthe fields for μ = 0.1μ0. These results have been compared with othertechnique which exactly matches [6, 14]. Other curves show the resultsfor μ = 0.3μ0, μ = 0.6μ0, μ = 0.9μ0, ε = 1.12μ0 and μ = 1.15μ0

respectively. It is observed that intensity of the field at caustic pointdecreases as value of transverse permeability of uniaxial lens increases.Figure 4 shows the field intensity distributions along z-axis for severaldifferent values of the transverse permittivity. The solid line representsthe fields for ε = 0.1ε0. The other curves show the results for ε = 0.3ε0,ε = 0.6ε0, ε = 0.9ε0, ε = 1.12ε0 and ε = 1.15ε0 respectively. FromFigure 4, it is observed that the field intensity of the field at the causticpoint is also decreased as value of transverse permittivity of uniaxiallens is increased.

Figure 2. Variation of normal-ized field intensity around focalpoint along z-axis due to chiralityparameter.

Figure 3. Variation of normal-ized field intensity around focalpoint along z-axis due to trans-verse permeability.

Electromagnetic field transmitted by uniaxial chiral lens 1015

Figure 4. Variation of normal-ized field intensity around focalpoint along z-axis due to trans-verse permittivity.

Figure 5. Variation of normal-ized field intensity around focalpoint along z-axis due to axialpermittivity and permeability.

Figure 5 shows the field intensity distributions along z-axis fordifferent values of the axial permittivity and permeability. The solidline represents the field intensity for εz = 0.3ε0, μz = 0.3μ0. Theother curves show the results for εz = 0.6ε0, μz = 0.6μ0, εz =0.9ε0, μz = 0.1μ0, εz = 1.2ε0, μz = 1.2μ0 and εz = 1.5ε0, μz = 1.5μ0

respectively. From Figure 5, it is observed that the field intensity of thefield at the caustic point is increased as value of axial permittivity andpermeability of uniaxial lens is increased. The maximum field intensityis found around the caustic point located at kz = 250 (caustic point)of the lens. The caustic point has been obtained according to GOanalysis. In all Figures kz is normalized quantity.

4. CONCLUSION

In this paper, the transmitted field from a uniaxial chiral cylindricalplano convex lens is analyzed using Maslov’s method. The problem isformulated only for normal incidence. Dyadic transmission coefficientfor uniaxial chiral dielectric interface is derived analytically. Numericalresults for the field distribution are presented for different parametersso that the focusing behaviors for the uniaxial chiral lens can beclarified. It is found that in analysis with the increase of chiralitypermittivity and permeability may be used to increase the fieldintensity. This analysis may be helpful for study and design ofmicrowave and optical filters. As a special application, polarizer maybe suggested by choosing suitable values for the medium parametersand for the thickness of the lens.

1016 Ghaffar et al.

ACKNOWLEDGMENT

Authors would like to thank Higher Education Commission (HEC) forstartup research grant under IPFP.

REFERENCES

1. Kong, J. A., Electromagnetic Wave Theory, Wiley, New York,1986.

2. Lakhtakia, A., Beltrami Fields in Chiral Media, World Scientific,Singapore, 1994.

3. Jarchi, S., J. R. Mohassel, and R. F. Dana, “Analysis of microstripdipole antennas on a layered metamaterial substrate,” Journal ofElectromagnetic Waves and Applications, Vol. 24, No. 5–6, 755–764, 2010.

4. Tang, M. C., S.-Q. Xiao, T. Deng, D. Wang, and B.-Z. Wang, “A dual-band epsilon negative material design usingfolded-wirestructures,” Journal of Electromagnetic Waves andApplications, Vol. 25, No. 2–3, 327–337, 2011.

5. Zhao, X., L. Zhao, K. Huang, and C. Liu, “A circularly polarizedarray composed of linear polarized microstrip patches fed bymetamaterial transmission line,” Journal of ElectromagneticWaves and Applications, Vol. 25, No. 11–12, 1545–1553, 2011.

6. Manapati, M. B. and R. S. Kshetrimayum, “SAR reductionin human head from mobile phone radiation using singlenegative metamaterials,” Journal of Electromagnetic Waves andApplications, Vol. 23, No. 10, 1385–1395, 2009.

7. Naqvi, A., S. Ahmed, and Q. A. Naqvi, “Perfect electromagneticconductor and fractional dual interface placed in a chiral nihilitymedium,” Journal of Electromagnetic Waves and Applications,Vol. 24, No. 14–15, 1991–1999, 2010.

8. Rahim, T., M. J. Mughal, Q. A. Naqvi, and M. Faryad, “Focalregion field of a paraboloidal reflector coated with isotropic chiralmedium,” Progress In Electromagnetics Research, Vol. 94, 351–366, 2009.

9. Ghafiar, A., “Focusing by a cylindrical plano convex lens of chiralmedium,” Journal of Electromagnetic Waves and Applications,Vol. 25, No. 5–6, 833–844, 2011.

10. Dong, J.-F., J. Li, and F.-Q. Yang, “Guided modes in the four-layer slab waveguid containing chiral nihility core,” Progress InElectromagnetics Research, Vol. 112, 241–255, 2011.

Electromagnetic field transmitted by uniaxial chiral lens 1017

11. Semchenko, I. V., S. A. Khakhomov, S. A. Tretyakov, andA. H. Sihvola, “Electromagnetic waves in artificial chiral struc-tures with dielectric and magnetic properties,” Electromagnetics,Vol. 21, 401–414, 2001.

12. Semchenko, I. V. and S. A. Khakhomov, “Artificial uniaxialbianisotropic media at oblique incidence of electromagneticwaves,” Electromagnetics, Vol. 22, 71–84, 2002.

13. Lakhtakia, A. and R. Messier, Sculptured Thin Films: Nanoengi-neered Morphology and Optics, SPIE Press Book, 2005.

14. Viitanen, A. J. and I. V. Lindell, “Uniaxial chiral quarter-wavepolarization transformer,” Electron. Lett., Vol. 29, 1074–1075,1993.

15. Lindell, I. V. and A. H. Sihvola, “Plane wave reflection fromuniaxial chiral interface and its application to polarizationtransformation,” IEEE Transaction on Antenna and Propagation,Vol. 43, 1397–1403, 1995.

16. Uckun, S., “Plane wave propagation through a uniaxial chiral slaband transmission coefficient,” Microwave and Optical TechnologyLetters, Vol. 18, 171–174, 1998.

17. Ji, Y. and K. Hongo, “Analysis of electromagnetic waves refractedby a spherical dielectric interface,” J. Opt. Soc. Am. A, Vol. 8,541–548, 1991.

18. Hongo, K. and H. Kobayashi, “Radiation characteristics of a planoconvex lens antenna,” Radio Sci., Vol. 31, 1025, 1987.

19. Ghaffar, A., A. Hussain, Q. A. Naqvi, and K. Hongo, “Radiationcharacteristics of an inhomogeneous slab using Maslov’s method,”Journal of Electromagnetic Waves and Applications, Vol. 22,No. 2–3, 301–312, 2008.

20. Ghaffar, A., Q. A. Naqvi, and K. Hongo, “Study of focusing offield refracted by a cylindrical plano-convex lens into a uniaxialcrystal by using Maslov’s method,” Journal of ElectromagneticWaves and Applications, Vol. 22, No. 5–6, 665–679, 2008.

21. Gorman, A. D., “Vector field near caustics,” J. Math. Phys., Vol.26, 1404–1407, 1985.

22. Ziolkowski, R. W. and G. A. Deschamps, “Asymptotic evaluationof high frequency field near a caustic: An introduction to Maslov’smethod,” Radio Sci., Vol. 19, 1001–1025, 1984.

23. Kravtsov, Y. A., “Two new methods in the theory of wavepropagation in inhomogeneous media (review),” Sov. Phys.Acoust., Vol. 14, No. 1, 1–17, 1968.