Introduction

This device can be used in microwave engineering, in high frequency technology and it has also applications in optics. Chiral materials (optically active materials) have applications as shielding and absorbing materials [1, 2], in antenna radomes [3] and in waveguide structures [4]. It has been shown that a slab made of uni¬ axial chiral material changes the polarisation of propagating electromagnetic plane wave. By choosing the thickness of the slab and the material parameters properly, the slab which transforms linearly polarised field to elliptically polarised field is achieved. The ellipticity of the field depends on the angle ( in Figure 1) between the polarisation plane of the incident wave and the axis of the material. Because chi¬ ral material is reciprocal, the device works also reversely, it transforms elliptically polarised field to linearly polarised field. The advantage compared to previously published solutions [5, 6, 7] , the reflection from the slab is eliminated so that the polarisation transforming effect is remained. By choosing the thichness of the slab as a quarter of a polarisation wave length, I = λ _p /4, and the material parameters P _z l 'μ _t i ^e _z / ^e _t , ^{κ ar} td η _t as given in Tables 1 and 2, a slab is achieved, which transforms the linearly polarised incident electric field to elliptically polarised field without re¬ flection. All possible polarisation states (right hand circular, right hand elliptical, linear, left hand elliptical, left hand circular ) are achieved by rotating the slab.

Description

Theory of the uniaxial chiral media can be found in literature [6, 8, 9, 10] and the polarisation transforming properties are studied in [5, 6]. The uniaxial chiral material is achieved by putting small metal helicies of same handedness oriented in same direction, parallel to z axis as illustrated in Figure 1. Macroscopically uniaxial chiral media is described by the constititive equations

D = _ E + ? H, B = H + C E (1) where _ _

_ = e _t + e ₂ u ₂ u_, = μ _t + μ _z u _z u _z ,

C = jκ M- ^e ° _u_, ; = -j ^' /--//-__oU_u ₂ . The propagation factors propagating in perpendicular to the z direction are [5, 6] = ω /μ _{t t} sJ A ₊ , __ (2) where the parameters A± are

(3)

The electric and magnetic fields are written with the eigenvectors. The eigenvectors for the electric field are: e± = JJ/±Uα u ₂

__! (4) and eigenvectors for the magnetic field are: j 1 h± = — u _x 7τ=Uz, (5) y± ± where y± = (A± — ^)/ ^κ _\ r^ • The parameters y± and -/_f± depend on the material parameters and are real numbers. The total electric field consists of the sum of the field components propagating in positive and negative y direction:

E(y) = E ₊ e- ^jk+y e+ - E ₊ ' e ^jk+y e ₊ + E ^jk~y e_ - E_' e ^jk - ^y e ^* _ . (6)

The total magnetic field is

H(y) = [_E_ _- ^J' *- ^, 'h_ - E_' e ^jk - ^{y *} _}. (7)

The coefficients E± present the part which is propagating in positive y direction, and the coefficients E± the part which is propagating in negative y direction of the total field. The complex conjugate of the vector is denoted by . . It can be shown that by choosing the thickness of the slab as a quarter of the polarisation wave length λ _p

the polarisation of the incident linearly polarised field is transformed maximally to elliptically polarised field [5, 6].

Next it is required that there is no reflected field from a slab whose thickness is I. This requirement binds the material parameters μ-//_t, t _z jt _t , K and the impedance level η _t in a certain way. The calculation of reflection coefficients from the uniaxial chiral slab of thickness . is straitforward, but the results are quite complicated in analytical form. In the following the procedure for eliminating the reflected fields, and the conditions which follow for the material parameters from that procedure are presented.

In Figure 1, at the interface y = I the continuity of the tangential fields ( E and H ) couples the coefficients E± and E±' through the equation

-u _y x E(f) = H(f) (9) as

E ₊ ' = r ₊₊ e- ^{j k+ l} E ₊ + r ₊ _e-' ^{(fc+ +fc} ->'£_ , (10)

E'_ = r _→ e- ^{j(k+ +k} - ^)l E ₊ + r __ e- ^32k - ^l E- , (11)

where η ₀ = Jμ ₀ j(. ₀ is the impedance of the medium outside the slab and the real coupling coefficients are:

After that the fields at the interface y = 0 are written with the equations (6) and

(7):

E(0) = E+[e+ - r ₊ e- ^j2k+l e ^* ₊ - r _ _+e - * ⁺⁺ ^ V ]

+ £_[e_ - r__e- ^i2fc - - r ₊ _e- ^j(fc+ +*->'_ ], (16)

H(0) = ^-E ₊ [y±( ₊ - r _++e -' ^Λ+ 'h ) - y_r_ ₊ e-Λ* ^++fc - ⁾ 'h ^* ] ._

+ ^£_[y_(h_ - r__e-^-'h ^* _) - y ₊ r ₊ _e- ^fc++fc - ⁾ 'h ]. (17)

»7_

The reflected field, E _re/i (0) = ϊ_ • E(0), is written with the reflection dyadic as [11]

_. = [ ₀ + ]- ¹ -[ ₀ - ], where Y ₀ = — u _υ x I. The admittance dyadic Y binds the electric and magnetic fields to each other as

F-E(0) = H(0). (18)

By eliminating from the equations (16) and (17) the coefficients E± and E- the components of the admittance dyadic Y can be calculated. These components are presented in detail in Appendix A.

By requiring that the reflected field vanishes at the interface y 0, the condition between the fields is obtained:

After writing this equation in component form

(-_u_ - jBu _z )E± + (Cu _x - jDn _z )E- = 0 (20) which is in matrix form as

A C E±

(21)

-jB -jD _._ 0,

where the coefficients A, _?, C and D are complex numbers:

__ = 1 - + (22) o Z

Det = 0. This leads to the following equation:

- ₃ 2k±l

(e + e -.2J__. - e - ₃ 2(k+ +k-)l

) χ

₊ ( _e -j2(fc ₊ +-_)i _ 9--j(- 4-fc- )' ) ^{y+ y} - [1 _ ( _--) ² ] ² __ ₊ ___ 77„

0 (26)

This is a complex valued equation which is fulfilled separately for real and imaginary parts, or separately for amplitude and phase parts. Because all the terms inside the

square brackets are real numbers the above characteristic equation is fulfilled when the phase factor terms are real numbers. This condition is fullfilled when

(_+ + __). = nπ, (27)

2k±l = {n ± l)τr, (28) where n is an integer. The thickness of the polarisation transformer was chosen according to (8) so that

(k± — fc_)Z = _ (29)

From these conditions it follows that the parameters A± are:

(30)

Because from the equation (3) it also follows that A± -+ ^■ Λ_ = - ^tά , the parameters A+ can also be written in the form:

and the chirality parameter (K is here taken as a positive number)

The square bracket part of the determinant equation have two different solutions depending if n is odd or even integer. If n is odd, the determinant equation reduces to the form

This solution, η _t /η ₀ = 1, is a solution, when the plane wave is propagating through the slab without reflection but no polarization transforming effect is occured. If n is even, the determinant equation reduces to the form

_____ ( ₁ + 5)(ι _

This is the same equation as r±±r = 0. From this equation two different conditions for the impedance level η _t / o are obtained: y+ y- Vt 1

+ -^=)(^) ² + (y ₊ - y_)(l - y/AZZ y/X± ^~ η, ^ > _ - ⁽ TIM . _ ^{) = O (36)}

or

The equation (36) corresponds to the case r = 0 and the equation (37) corresponds to the case r = 0. This is all information obtained from the condition of the reflected fields.

The other condition concerns the polarisation requirements of the transmitted field. The polarisation must be transformed so that all the polarisation states are achieved by rotating the slab when the incident field is linearly polarized. Let us next consider the transmitted field.

Let us take first the case r±± = 0, i.e., ηt/Vo is determined by the equation (36). The other case can be considered in the same way.

The electric field at the interface y __ 0 is

E(0) = E±[e± - r_±e'_] + _E _ [e_ + r__e_. - r±_e ^* ₊ ] (38) and at the interface y _= _

E(Z) = jE+ [e± - r_+e ^* _] - _. ^' __ _ [__ - . __ _ ^* _ - . +_ _;]. (39)

The incident field is assumed to be linearly polarised, ( in Figure 1 in direction )

E(0) = __ ₀ (sin αu. -f cos α_ι ₂ ). (40)

After eliminating the coefficients E± and _._ the transmitted field is obtained which is written with real and imaginary parts as:

E(Z) = E _r + jEi = — (-[o cos αu, + _>sin u_] - j ^' fc sin αu- — c. cos αu.]) , (41)

where the coefficients a, b, c, d and q are real numbers: α = 2(y ₊ + y-r_±)(y_ + y±r±_ ), (42)

^{c = (} ^7 ⁺ ^τ ^{)(1 + r} -- - ^r+ - ^r - ⁺⁾ ' ^{(44) i = (} ^ ⁺ ^r ⁾⁽¹ - ^r -- ^{^+} - ^r - ^{+) (45)}

, ( β)

The transmitted field is elliptically polarized and the ellipse is located on xz plane as shown in Figure 1. Now, it is required that all polarisation states are achieved when rotating the slab. The polarisation of the transmitted field is studied with the polarisation vector p which is a real vector and the value of which is between [—1, 4-1]. When the value of the polarisation vector with respect to the propagation direction is 4-1, the field is right handed circularly polarised, and, when the value of the polarisation vector is —1, the field is left handed circularly polarised. When p = 0 the field is linearly polarised. The polarisation vector is defined as [10, 11]

2E _t x E _r 2(α_ cos ² α 4- - _ sin ² α)

^{P =} |E _r | ² + |Ei| ^{2 =} (α ² 4- _ ² ) cos ² + (. 4 c ² ) sin ² "* ^' ^'

All polarization states are obtained when — 1 < p < 1. This condition is fulfilled when

_ = c, and _ = —a, (48) or b = —c, and d = α, (49)

In fact the two conditions in (48) describes the same equation. Also the two con¬ ditions in (49) describes the same equation, so, it is necessary only to study one of those conditions, for example, b — c = 0 oτ b + c = 0. Let us consider first the equation (48) which is b — c = 0 :

τfc ^{)(1 +} '-- ^" '-'-> - " ^{■ (50} »

Then the polarisation vector is b " ^?2 s: in' — a ,2" c _.os' 2 a u, (51) b ² sin Q - - cos ² α If especially a ² = b\ (52) the polarisation vector reduces to a very simple form p = — cos 2 u _y , (53) from which it is seen that the value of p varies between —1 < p < 1, when 0 < < π/2. The special values for material parameters which fullfill the above conditions are given in Table 1. These values are in the region μ _z /μt > ε _* / _ • Other solutions which give positive real values for material parameters are not found.

Let us next consider the case r = 0, thus, the impedance level Vt/Vo is calculated from the equation (37). This leads to similar expression for the transmitted field as previously when r is changed to r±±. In this case with the condition (49)

² < ^{~ '} 7 ⁾⁽ 7T. - ⁽⁵⁴⁾

positive real values for material parameters in region μ _z /μ _t < _ _z / _. are found. The polarisation vector is in this case α ² cos ² a — b ² sin ²

P ⁼ 1 a ² cos 2 ² 41- o r ₂ si • 2 ^u v (55) '

If the condition (52) is also valid the polarisation vector can be written in simple form p = cos 2 u _v , (56) the value of which varies between 1 > p > — 1, when 0 < a < π/2, thus, values of opposite sign compared to previous case. This means that the handedness of the polarisation is changed. These special values for the material parameters are given in Table 2.

As a conclusion the uniaxial chiral slab works as a polarisation transformer when the thickness of the polarisation transformer is

( A _t is the wave length in transverse plane) and the two equations (36) and (48), or, (37) and (49) with the condition (52) are fulfilled. These conditions determine the values for the material parameters. In addition the values for the material parameters __//_t, e _z /e _t , K and η _t /Vo must be positive real numbers. Some values for material parameters which fullfill these conditions are given in Tables 1 and 2. In addition, it can be shown that the main axis of the ellipse of the transmitted field are at 45° angle in x and z axis. The direction of the main axis of the transmitted field is obtained from the equation [11]

Because with the conditions given above the transmitted field is

E(.) = — - ( — [cos αu- 4- sin _u _z ] ± [sin u _x 4- cos u.]) , (59) where 4- sign refers to the case r±± = 0 and — sign to the case r__ = 0, the direction of q _P is u _x 4- u _z when 0 < a < π/2 and u _x — u ₂ , when π/2 < a < π. The chirality parameter n is here assumed to be a positive number. If the chirality parameter is negative, K < 0, this means that the handedness is changed, otherwise the same analysis is valid.

Table 1: Some values of the material parameters and the corresponding values for the thickness of the slab for which the value for the polarization vector is p — — cos 2a.

Table 2: Some values of the material parameters and the corresponding values for the thickness of the slab for which the value for the polarization vector is p = cos 2a.

Example

The reflection coefficients is calculated using material parameters given in Tables. Let us take the values corresponding to n = 6 in Table 1: μ _z /μ _t = 1.300 , e _z /e _t = 0.810 , η _t /η ₀ = 0.790 , = 0.233 ,

and the thickness I _= 1.481 λ _t . Then KJ μ _{0 0} / μ _t e. _t _- 0.239 and other parameters

A± = 1.397 , ___ = 0.713 , y± = 2.458 , y_ = -0.407 , r±± = -1.150 ^■ 10 ^~8 , r +_ = -3.969 • 10 ^-2 , r_ ₊ = -1.713 • 10 ^-1 , r __ = 1.015 ■ 10 ^"3 , 2π . 2 2ππ ^■■ 11,, 448811 . . 2τr , 2π ^■ 1.481 _, . = / = = 2.506_ , fc_ Z = rrΛ - — . = 3.508π,

" ^{+ "} λ _t /X+ ^" _\ _ L397 ^r AZ y/όJϊS and the phase factors j2k ₊ l _{= e} -i5.012 _ _ _ _{1 {} ?} --J2--J __ _e - 7.016 _W _ _ _Q ^ _e -i(fc ₊ +-_ )! ₌ -j6.014 _ 1.0.

The expression for the transmitted field is

E(.) - E ₀ ([0.706 cos αu, - 0.708 sin αu-] 4- [0.707 si u _x - 0.706 cos u,]) , and the corresponding polarisation vector is

2(-0, 706 ² cos ² a 4- 0, 708 • 0, 706 sin ² α)

P = u,_ — cos 2αu„

(0, 706 ² 4- 0, 706 ² ) cos ² a 4- (0, 708 ² 4- 0, 706 ² ) sin ² α ^y

Especially, if the incident field is linearly polarised parallel to x axis (_ = π/2), the transmitted field is right handed circularly polarised. If the incident field is linearly polarised parallel to z axis the transmitted field is left handed circularly polarised. The polarisation of incident field which is linearly polarised with a = π/4 remains unchanged.

The polarisation transformer works also in reverse direction. In this example, the incident field is right handed circularly polarised

E(0) = ^ (u _x 4- _ u_), (60)

which is a combination of two linearly polarised field which are in 90° phase. The transmitted field is obtained by considering i and _ components separately. The total field is a sum of these two partial fields which are tranformed separately

E(_) = M§ [(- ^u _* + J ^'u _* ) + - ( ^u χ - J ^'u _* )] = jV2E _{0 x} (61)

thus, the transmitted field is linearly polarised in x direction. Similarly, if the incident field is left handed circularly polarised, the transmitted field is linearly polarised in z direction.

The reflected field is written with the reflection dyadic:

E _re /.(0) = [_ff _xx u _x u _x 4- __ _x _u u- 4- i_ _2Z u-u _x 4- R _zz u _z u _z ] ■ E(0).

The values for the reflection coefficient are calculated numerically. The values for parameters are calculated according to Appendix A: αj = 2.528 , ₂ = -5.039 , α ₃ = 1.049 , α ₄ = 1.219 , fei - 0.829 , - ₂ = 0.959 , 6 ₃ = 1-997 , δ ₄ = -0.399 , after that the components of the admittance dyadic are calculated:

_Vt Y _xx = 1.120 • 10 ^~3 , η _t Y _xz = 0.787 , η _t Y„ = -0.791 , η _t Y _zz = -1.120 • 10 ^~3 .

Values for the reflection coefficients are obtained:

R _xx - -3.002 • 10 ^-4 , R _xz = J7.235 ^■ 10 ^-4 , __ _« = J7.235 • 10 ^"4 , R _zz = 1.744 • 10 ^"3

The reflection from the slab is extremely low. Also by using other values for material parameters given in Tables the reflection is practically zero.

Figure 1. Uniaxial chiral slab. In this Figure is illustrated the direction of the polarisation plane (the angle a) of the incident linearly polarised electric field E(0), elliptically polarised transmitted field E(/), the orientation of the ellipse in xz plane and the polarisation vector p of the transmitted field. In uniaxial chiral material the metal helices of same handedness are parallel to z axis.

References

Varadan et al, "Electromagnetic shielding and absorptive materials," United States Patent 4948922, Aug. 14, 1990.

Jaggard et al, "Novel shielding, reflection and scattering control using chiral materials," United States Patent 5099242, Mar. 24, 1992.

Engheta et al, "Novel radomes using chiral materials," WO92/ 12549, PCT/US92/00050, July 23, 1992.

Engheta et al, "Waveguides using chiral materials," United States Patent 5165059, Nov. 17, 1992.

A.J. Viitanen, I.V. Lindell, "Uniaxial chiral quarter-wave polarization trans¬ former," Electronics Letters, Vol. 29, No. 12, pp. 1074-1075, June 1993.

A.J. Viitanen, I.V. Lindell, "Plane wave propagation in a uniaxial bianisotropic medium with application to a polarization transformer," Int. J. of Infrared and Millimeter Waves, Vol. 14, No. 10, pp. 1993-2010, 1993.

A.J. Viitanen, "Plane wave reflection from a uniaxial chiral quarter-wave slab," Proc. of CHIRAL '94, 3rd Int. Workshop on Chiral, Bi-isotropic and Bi-anisotropic Media, pp. 379-384, Perigueux, France, May 18-20, 1994.

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

I.V. Lindell, A.J. Viitanen, "Plane wave propagation in a uniaxial bianisotropic medium," Electronics Letters, Vol. 29, No. 2, pp. 150-152, January 1993.

I.V. Lindell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Norwood, MA: Artech House, 1994.

I.V. Lindell, Methods for Electromagnetic Field Analysis, Oxford: Clarendon Press, 1992.

Appendix A

In the equation (16) the x and z components of the electric field are

E(0) • u = ja _x E± + ja ₂ E_ (62)

E(0) • u _z - -a ₃ E± - α ₄ £_, (63) where the real coefficients are

<_ =y+(l -r ₊₊ )4-y_τ-_ ₊ , α ₂ =y_(l-.__)4-y+-+_ , (64)

° ³⁼ - ^{(1 + r++)} -^' ^α<= 3r ^(1+r -- ⁾ -^ ^{' (65)}

The amplitudes E± and E- are solved:

E± = -j ^ E(0) • u + - E(0) • u _z , (66) θια — α _{2 3} αια — a _{2 3}

___ - j E(0) • u ₂ . (67)

In addition, the equation (17) for the magnetic field in component form is

7? _t H(0) = -[_ _! __+ 4- b ₂ E_}u _x - j[b ₃ E± - _._-_ ., (68) where the real coefficients are

_ _α = 1 - r±± 4- r _→ , b ₂ = 1 - r__ 4- r ₊ _, (69) _{fc3 =} -J ^< ±-. _(1+r++) _-^- ₊ , ^-^.(l + ,__)- - _,-_. ₍₇₀ )

By substituting the expressions of E± into this equation, the magnetic field can be written in the form

H(0) == F • E(0), (71)

Y = _. 3_χU _x u _x 4- F_ u u ₂ 4- __χu _z u _x 4- jY _zz u _z u _z (72) where the real valued components for the admittance dyadic are:

_ 1 (-i- ₄ - b ₂ a ₃ ) _v _ 1 (b ₂ -ι - _ια ₂ )

^■ *xx \ ) XZ i -.1 ( t )

Vt ( _, αι<- ₄ — a ₂ a ₃ ) η _t α^ — α ₂ α ₃ )

_ __ ____________________). - _____________________ , ₇ ,N η _t (aιa ₄ - _. ₂ -_) ??. ( _α α ₄ - α ₂ α ₃ )

Moreover

F ₀ = [u _x u ₂ - u_u _x ] (75) and the reflection dyadic

I-[F ₀ 4-F]- ^α -[F ₀ -F]. (76)

The components of the reflection dyadic can be calculated in matrix form from the equation (76)

__χχ tt-XZ jYxx (r + ^Y **) -jY: (_1o:-n_) (77)

Rzx Rzz jYz ^■ ( „ + ^y« ) -JYz

The expressions for the reflection dyadic are:

(78) n, ^y _M + ( -r. ( -r_,) ' ^Λ " d + Yχ)(i-Y _X )-YχχYzz- ^{{ )}