Progress In Electromagnetics Research B, Vol. 7, 53–73, 2008

THREE-DIMENSIONAL GAUSSIAN BEAMSCATTERING FROM A PERIODIC SEQUENCE OFBI-ISOTROPIC AND MATERIAL LAYERS

V. Tuz

Department of Theoretical Radio PhysicsKharkov National UniversitySvobody Sq. 4, Kharkov, 61077, Ukraine

Abstract—The three-dimensional Gaussian beam scattering fromthe bounded periodic sequence of one-to-one composed isotropicmagnetodielectric and bi-isotropic layers are investigated. The beamfield is represented by an angular continuous spectrum of plane wave.The problem of the partial plane wave diffraction on the structure issolved using the circuit theory and the transfer matrix methods. Itis found that after reflection from the structure, the circular Gaussianbeam becomes, in general, an elliptical Gaussian beam, in addition toa displacement of the beam axis from the position predicted by rayoptics.

1. INTRODUCTION

For solving many diffraction problems the simple plane waves areemployed. In practice, a simple plane wave is not an option, but mustapproximated by a beam of radiation, which can be modeled as it isin the present paper by a Gaussian beam [1, 2].

The problem of reflection and transmission of a wave beam fromsingle isotropic and anisotropic layers, metal-dielectric heterogeneitiesand their periodical sequences has been the subject of numerousresearch papers [1–12]. Several beam-wave phenomena such as lateralshift, focal shift, angular shift, beam splitting that are not found in thereflection of plane wave are the major features for investigation. Papers[13–20] are devoted to investigate the wave beam scattering on thestructures with the spatial dispersion that include single layers of thenatural and artificial reciprocal (chiral) and nonreciprocal bi-isotropic,bi-anisotropic medium, gyrotropic crystals, etc. Most of these studieswere based on a two-dimensional beam-wave structure. The reason for

54 Tuz

this choice was perhaps the confidence that essential insights would notbe lost while mathematical complexities could be reduced. However,it was shown in [5, 20] that this confidence may not be warranted, inparticular, the two-dimensional model is not to take into considerationthe polarization effects and not able to predict the change in ellipticityof the scattered beam.

In the present work the scattered fields of a tree-dimensionalGaussian beam on a bounded periodical sequence of one-to-onecomposed isotropic magnetodielectric and bi-isotropic layers areinvestigated. The scattering coefficients of the plane monochromaticwaves are determined using the circuit theory and the transfer matrixmethods [21–25] and the beam field is represented by the angularcontinuous spectrum of the plane wave [1, 2, 9, 20].

The aim of this work is the performance improvement andfunctionality expansion for the electromagnetic field systems based onthe exploitation of the media chirality [26–37].

2. PROBLEM FORMULATION

A periodic in the z-axis direction, with period L, structure of Nidentical basic elements (periods) is investigated (Fig. 1). Each ofperiods includes a homogeneous magnetodielectric and bi-isotropiclayers with thicknesses d1 and d2 (L = d1 + d2) that are defined withthe material parameters ε1, µ1 and ε2, µ2, ξ, ζ, respectively (ξ, ζ are themagnetoelectric interaction parameters). In general, the parametersε1, µ1 and ε2, µ2, ξ, ζ, can be frequency dependent and complex for lossmedia. The outer half-spaces z ≤ 0 and z ≥ NL are homogeneous,isotropic and have permittivities ε0, µ0 and ε3, µ3, respectively.

Note that if ξ = ζ = 0 (i.e., the both of layers are conventionalmaterials), then the structure in Fig. 1 is known as a distributed Braggreflector (DBR) [23].

The auxiliary coordinate system xin, yin, zin is introduced for anincident beam field description [9]. In it, the incident field Ein,H in

is written as the continued sum of the plane waves with the spectralparameter κin (it has a sense of the transverse wave vector of thepartial plane wave):

Ein = ein

∫∫ ∞

−∞U(κin) exp(iκin · (rin+ain)+iγin(zin+a3))dκin,

H in = hin

∫∫ ∞

−∞U(κin) exp(iκin · (rin+ain)+iγin(zin+a3))dκin,

(1)

Progress In Electromagnetics Research B, Vol. 7, 2008 55

Figure 1. The periodical sequence of bi-isotropic and material layers.

In the equation (1) the vectors are introduced

ein = PVp − bin × PVs, hin = PVs + bin × PVp. (2)

Here the vector P describes the field polarization,

P = z0 × n, (3)

and in the structure coordinates x, y, z, the vector n is characterized viathe components (cos θin cosϕin, cos θin sinϕin, 0), θin = 90◦ − αin, z0

is the basis vector of z-axis, and the vector bin describes the directionof the incident wave beam propagation

bin =(

cos θin cosϕin, cos θin sinϕin,−√ε0µ0 − cos2 θin

),

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U(κin) is the spectral density of the beam in the plane zin = 0, γin =√k2

0 − κin · κin, 0 < arg(√k2

0 − κin · κin) < π and ain = (a1, a2).The transformation from the coordinate system x, y, z to

xin, yin, zin can be realize via three possibility: rotating on the angleϕin around the z-axis; rotating on the angle αin around the x-axis;shifting the point of origin to point (a1, a2, a3). The transition matrixfor the first and the second transformations is written in standard view:x0

in

y0in

z0in

=

cosϕin sinϕin 0− cosαin sinϕin cosαin cosϕin sinαin

sinαin sinϕin − sinαin cosϕin cosαin

·

x0

y0z0

.(4)

From relations (4) it is continue, that the wave vector components inthe mentioned coordinate systems are related in the following way:κx

κy

γ

=

cosϕin − cosαin sinϕin sinαin sinϕin

sinϕin cosαin cosϕin − sinαin cosϕin

0 sinαin cosαin

·

κxin

κyin

γin

(5)

or,κx

κy

γ

=

cosϕin − sin θin sinϕin cos θin sinϕin

sinϕin sin θin cosϕin − cos θin cosϕin

0 cos θin sin θin

·

κxin

κyin

γin

,(6)

where γ =√k2

0 − κ · κ, 0 < arg(√k2

0 − κ · κ) < π. Taking intoaccount (4) and (5) for the reflected field we have

Eref = U eeref + Uhe

ref

= PVe

∫∫ ∞

−∞U(κin)Ree(κ) exp(iκ · r − iγz)dκin

−bref × PVh

∫∫ ∞

−∞U(κin)Rhe(κ) exp(iκ · r − iγz)dκin,

Href = Uhhref + U eh

ref

= PVh

∫∫ ∞

−∞U(κin)Rhh(κ) exp(iκ · r − iγz)dκin

+bref × PVe

∫∫ ∞

−∞U(κin)Reh(κ) exp(iκ · r − iγz)dκin,

(7)

Progress In Electromagnetics Research B, Vol. 7, 2008 57

and for the transmitted field

Etr = U eetr + Uhe

tr

= PVe

∫∫ ∞

−∞U(κin)τ ee(κ) exp(iκ · r + iγ(z −NL)dκin

−btr × PVh

∫∫ ∞

−∞U(κin)τhe(κ) exp(iκ · r + iγ(z −NL)dκin,

H tr = Uhhtr + U eh

tr

= PVh

∫∫ ∞

−∞U(κin)τhh(κ) exp(iκ · r + iγ(z −NL)dκin

+bref × PVe

∫∫ ∞

−∞U(κin)τ eh(κ) exp(iκ · r + iγ(z −NL)dκin,

(8)In (7) and (8) the reflection Rss′ and transmission τ ss′ complexcoefficients (s, s′ = e, h) of the partial plane electromagnetic wavesfrom the structure were introduced. They depend from the frequencyof the incident field, angles (αin, ϕin) and the other electromagneticand geometrical parameters of the structure. The coefficients withcoincident indexes (ss) describe the transformation of the incident waveof the perpendicular (s = e) or the parallel (s = h) polarization intothe co-polar wave, and the coefficients with indexes (ss′) describe thetransformation into the cross-polar wave. The left index corresponds tothe incident wave and the right index — to the reflected or transmittedwave. The scattering coefficients Rss′ and τ ss′ are determined troughthe solution of the plane monochromatic wave diffraction problem onthe structure under study.

3. TRANSFER MATRIX OF PERIOD. SCATTERINGCOEFFICIENTS OF PLANE MONOCHROMATICWAVES

According to the phenomenological description [26, 27], the spatialdispersion effects of the first order are defined via the constitutiveequations where the electric and the magnetic inductions arerelated with the electromagnetic field intensity via the permittivity,permeability and magnetoelectric interaction parameters (the gyrationparameters in terms of crystalloptics)

D = ε2E + ξH, B = µ2H + ζE, (9)

where ξ = χ+ iρ, ζ = χ− iρ, χ is a parameter of the nonreciprocalitydegree of the medium, ρ is the chiral parameter. In a particular caseof ρ �= 0, χ = 0 the medium is chiral and reciprocal (the Pasteur

58 Tuz

medium), when ρ = 0, χ �= 0 is the nonchiral, nonreciprocal medium(the Tellegen medium).

In the homogeneous along the x-direction bi-isotropic layers, theelectromagnetic field is governed by the coupled differential equations

∆⊥Ex + k20

(n2

2 + ζ2)Ex − 2ik2

0ρµ2Hx = 0,∆⊥Hx + k2

0

(n2

2 + ξ2)Hx + 2ik2

0ρε2Ex = 0,(10)

where n2 =√ε2µ2 is the medium refractive index, and ∆⊥ = ∂2/∂y2+

∂2/∂z2.The waves of the perpendicular (Ee‖x0) and parallel (Hh‖x0)

linear polarizations can be presented as the superposition of the wavesof the right (Q+

s ) and left (Q−s ) circular polarizations [26]:

Eex = Q+

e +Q−e , He

x = −i(

1η+2

Q+e − 1

η−2Q−

e

),

Ehx = i

(η+2 Q

+h − η−2 Q−

h

), Hh

x = Q+h +Q−

h ,

(11)

where η±2 =√µ±2 /ε

±2 is the wave impedances of a bi-isotropic medium,

ε±2 = ε2 ∓ iξη−12 exp(∓iυ), µ±2 = µ2 ± iζη2 exp(±iυ), sin υ = (ξ +

ζ)/2n2, η2 =√µ2/ε2. Such substitution transforms (10) to two

independent Helmholtz equations:

∆⊥Q+s + (γ+)2Q+

s = 0, ∆⊥Q−s + (γ−)2Q−

s = 0. (12)

Here s = e, h; γ± = k0

√ε±2 µ

±2 = k0n

±2 is the propagation constants of

the right- (γ+) (RCP) and left- (γ−) (LCP) circularly polarized planewave in the unbounded bi-isotropic medium. Their general solutionsfor the RCP and LCP waves in a bounded bi-isotropic layer can bewritten as

Q±e =

1

2√Y e±

2

{Ae± exp

[i(γ±y y+γ

±z z

)]+Be± exp

[i(γ±y y+γ

±z z

)]},

Q±h =

√Y h±

2

2

{Ah± exp

[i(γ±y y+γ

±z z

)]+Bh± exp

[i(γ±y y−γ±z z

)]},

(13)where As±, Bs± are the wave amplitudes, Y e±

2 = cosα±2 /η

±2 , Y

h±2 =

(η±2 cosα±2 )−1 are the wave admittances, γ±y = γ± sinα±

2 , γ±z =

γ± cosα±2 , sinα±

2 = sinα0n0/n±2 , α0 are the incidence angle of the

partial plane wave on the z = 0 structure boundary, α±2 are the

Progress In Electromagnetics Research B, Vol. 7, 2008 59

refraction angles in the bi-isotropic medium. The substitution (13)to (11) gives the field components of the E- and H-polarizations.

Due to the scattering of the given polarization plane electromag-netic wave (s) by the bi-isotropic layers, the cross-polar components(s′) appear in the secondary field. Denote their amplitudes as As′ andBs′ for the transmitted and reflected waves, respectively. The fieldcomponents in the m-th period are shown in the Appendix A.

The structure under study can be considered as a consecutiveconnection of the eight-poles which are equivalent to the illuminatedboundary (T 01), repeated heterogeneity (T = T 1T 2) and thelast element which is loaded on the waveguide channel havingthe admittance Y s

3 (T ′) (Fig. 1). The equations coupling thefield amplitudes at the structure input (As

0, Bs0, B

s′0 ) and output

(AsN+1, A

s′N+1) for the incident fields of E-type (Ah

0 = 0) and H-type(Ae

0 = 0) are obtained as:

As

0

Bs0

0Bs′

0

= T 01T

N−1T ′

As

N+1

0As′

N+1

0

,

As

m

Bsm

As′m

Bs′m

= T 1

A+

m

B+m

A−m

B−m

,

A+

m

B+m

A−m

B−m

= T 2

As

m+1

Bsm+1

As′m+1

Bs′m+1

,

As

m

Bsm

As′m

Bs′m

= T 1T 2

As

m+1

Bsm+1

As′m+1

Bs′m+1

, (14)

In [25] it has been shown that T 01TN−1T ′ = T 01T

NT 13, and in theblock representation (2 × 2) the transfer matrices are

T pv =

((T s

pv) 0

0 (T s′pv)

), T 1 =

((T ss

1+) (T ss1−)

(T ss′1+) (T ss′

1−)

), T 2 =

((T ss

2+) (T ss′2+)

(T ss2−) (T ss′

2−)

)

(15)where T pv corresponds to the matrices T 01 and T 13. The blocks of thequasi-diagonal matrices are

T spv =

1

2√Y s

p Ysv

(Y s

p + Y sv ±(Y s

p − Y sv )

±(Y sp − Y s

v ) Y sp + Y s

v

),

where the upper sign relates to s = h, and the lower sign relates tos = e in terms of the wave types.

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The elements of the transfer matrices T 1 and T 2 are determinedfrom solving the boundary-value problem and are shown in theAppendix B.

The reflection, transmission and transformation coefficients of theplane monochromatic wave of the reflected (z ≤ 0) and transmitted(z ≥ NL) fields are determined by the expressions Rss = Bs

0/As0, τ

ss =As

N+1/As0, R

ss′ = Bs′0 /A

s0 and τ ss′ = As′

N+1/As0.

The parametrical (including angular) dependences of thescattering coefficients of plane monochromatic waves from thestructure with a large number of elements have interleaved areas withthe high (the quasi-stop bands) and low (the quasi-pass bands) averagelevel of the reflection (Fig. 2). The interference of the reflected wavefrom outside boundaries of layers gives N − 1 small-scale oscillationsin the quasi-pass bands. The maximum of the reflection coefficientmagnitude for the cross-polar wave corresponds to the minimum ofthe reflection coefficient magnitude for the co-polar wave. Whenthe structure is backed by a metal ground plane (the reflectingregime) (Fig. 2(a)), the high-Q resonances of the reflection coefficientmagnitude in the quasi-stop bands appear. In the presence of chirality(ρ �= 0) and the dissipation losses (Im (ε2, ε1) �= 0), the magnitude ofthe small-scale oscillations of the reflection coefficient decreases, andthe full-resonant transparency vanishes. More detailed analysis of thescattered fields of the partial plane wave on the chiral and materiallayer sequence is given in [28].

(a) (b)

Figure 2. The angular dependences of the reflection coefficientmagnitude of the partial plane wave for the sequence of N = 5 bi-isotropic and material layers in reflecting (a) and passing (b) regimes:ε0 = ε1 = µj = 1, j = 0 ÷ 3, k0L = 10, d1/L = d2/L = 0.5, ρ = χ =0.2; (a) ε2 = 4 + 0.02i, ε3 → ∞; (b) ε2 = 4, ε3 = 1.

Progress In Electromagnetics Research B, Vol. 7, 2008 61

Figure 3. The distribution of the absolute value of the incident beamfield in the z = 0 plane: k0wx = k0wy = 10, ϕin = 0◦.

4. ANALYSIS OF GAUSSIAN BEAM SCATTERING

Let’s consider the scattering of a Gaussian beam with the spectraldensity U(κin) that assigns due to law

U(κin) = exp(−(w · κin)2/16

)Hn

(kxinwx/

√2)Hm

(kyinwy/

√2)

(16)where w = (wx, wy), wx and wy are beam widths along xin andyin axis, respectively, Hn(·) is the Hermit polynomial of n-th order.Restrict oneself to case of the zeroth-order (n = m = 0) beam (Fig. 3).

Some effects, like the beam form distortion, ellipticity change,beam splitting, lateral shift were discovered in the scattered beam(Figs. 4–6). Those effects are connected to the angular dependence ofthe phase and magnitude of the co-polar |Rss| and the cross-polar |Rss′ |component of the partial plane wave reflection coefficients (Fig. 2).

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(a) (b)

Figure 4. The distribution of the absolute value of the field (a) andthe lateral shift value of its maximum (b) of the reflected wave beamin the z = 0 plane for the structure that includes single isotropic andsingle bi-isotropic layers (N = 1): ε0 = ε1 = µj = 1, j = 0 ÷ 3, ε2 =4+0.02i, ε3 → ∞, k0L = k0w = k0b = 10, ρ = χ = 0.2, ϕin = 0◦; (a)d1/L = 0.2, d2/L = 0.8.

Progress In Electromagnetics Research B, Vol. 7, 2008 63

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Figure 5. The distribution of the absolute value of the field |Uref | ofthe reflected wave beam in the z = 0 plane for the sequence of N = 5isotropic and bi-isotropic layers (the passing regime): ε0 = ε1 = µj =1, j = 0÷3, ε2 = 4, k0L = k0w = k0b = 10, d1/L = d2/L = 0.5, ρ =χ = 0.2.

If the structure includes the single basic element (N = 1) and isbacked by a metal ground plane, the reflection coefficient magnitudeof the co-polar wave |Rss| is weakly depend from the falling angle andis nearly per unit (Fig. 2a). When ϕin = 0, for the two-dimensionalbeam, during [1], the field components U ee

ref and Uhhref of the reflected

beam have the next view

U ssref (y, z) = exp {iΦss(κ0)}Rss(κ0)Vs

×∫ ∞

−∞U(κin) exp

{i[(κin − κ0)(y + ∆s

y) − iγz]}dκin,

(17)

where κ0 = k sinαin and Φss(κ0) is the phase of the reflection

Progress In Electromagnetics Research B, Vol. 7, 2008 65

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Figure 6. The distribution of the absolute value of the field |Uref |of the reflected wave beam in the z = 0 plane for the sequenceof N = 5 isotropic and bi-isotropic layers (the reflecting regime):ε0 = ε1 = µj = 1, j = 0 ÷ 3, ε2 = 4 + 0.02j, ε3 → ∞, k0L =k0w = k0b = 10, d1/L = d2/L = 0.5, ρ = χ = 0.2, αin = 30◦.

Progress In Electromagnetics Research B, Vol. 7, 2008 67

coefficient of the partial wave which falls under an angle αin. Thescattered beam shift along the illuminated boundary of the structureis defined from the condition ∆s

y = −(∂Φss/∂κ)κ=κ0 . It is sizeablewhen the phase of the partial plane wave scattering coefficient rapidlychanges with the angle. When the angle of the falling beam is nearly tothe quasi-Brewster angle or greater then it, the splitting of the reflectedbeam into two beams with different intensity value appears (Fig. 4).

When N > 1, the shift of the reflected beam along the illuminatedboundary of the structure increases due to multiple reflections of wavesfrom the boundaries of the layers (Figs. 5, 6). The beam splitting isobserved on the angles that lie nearly of the totally transmission anglesof the partial plane wave.

Increasing the chiral parameter ρ raises the structure reflection,changes the ellipticity and slightly raises the lateral shift of thereflected beam. The maximum of the intensity for the cross-polarwave corresponds to the minimum of the intensity for the co-polarwave. There are the areas of the practically totally transformation ofthe co-polar wave into the cross-polar wave. To the presence of themedia nonreciprocity (χ �= 0) is |U eh| �= |Uhe| and the sizeable cross-polar component in the scattered field appears even for case of thestraight incident wave beam (αin = 0◦).

We would like to emphasize an important peculiarity of the systemunder study regarding its application in the design of high-precisionmatched loads and layered absorbing coatings (Fig. 6) [24, 25, 29]. Thispeculiarity consists in reducing the co-polar reflection due to wavetransformation into the cross-polar reflection.

5. CONCLUSION

In this paper, we have investigated the tree-dimensional Gaussian beamscattering for the DBR-like bounded periodic sequence of pairs of bi-isotropic and magnetodielectric layers. The lateral shift, ellipticitychange, beam splitting is studied. The revealed effects allows us torecommend the application of the studied structure in the design ofcascaded high-Q and stop-band frequency filters, wave transformers,angular discriminators, absorbers, etc.

APPENDIX A.

With zm = mL, zm1 = mL+d1, notations, the field components in them-th period of the structure zm ≤ z ≤ zm1 and zm1 ≤ z ≤ (m + 1)L

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are written as follows (the factor exp[−i(ωt− kyy)] is omitted):

{Ee

x1

Ehy1

}= ±

1/√Y e

1

i/√Y h

1

×({Ae

m

Ahm

}exp[ikz1(z − zm)] ±

{Be

m

Bhm

}exp[−ikz1(z − zm)]

),

{He

y1

Hhx1

}=

√Y e

1

i√Y h

1

×({Ae

m

Ahm

}exp[ikz1(z − zm)] ∓

{Be

m

Bhm

}exp[−ikz1(z − zm)]

),

{Ex2

Ey2

}=±

1/2√Y e+

2

i/2√Y h+

2

(A+

m exp[iγ+z (z−zm1)]±B+

m exp[−iγ+z (z−zm1)]

)

+

1/2√Y e−

1

i/2√Y h−

1

(A−

m exp[iγ−z (z−zm1)]±B−m exp[−iγ−z (z−zm1)]

),

{Hy2

Hx2

}=

√Y e+

2 /2

i√Y h+

2 /2

(A+

m exp[iγ+z (z−zm1)]∓B+

m exp[−iγ+z (z−zm1)]

)

±

√Y e−

2 /2

i√Y h−

2 /2

(A−

m exp[iγ−z (z−zm1)]∓B−m exp[−iγ−z (z−zm1)]

),

The field components at the structure output are

{Ee

x3

Ehy3

}= ±

(1/

√Y e

3

)Ae

N+1(i/

√Y h

3

)Ah

N+1

exp[ikz3(z −NL)],

{He

y3

Hhx3

}= ±

√Y e

3 AeN+1

i√Y h

3 AhN+1

exp[ikz3(z −NL)].

Here kzj = kj cosαj , kyj = kj sinαj , kj = k0nj , nj = √εjµj , Y

ej =

η−1j cosαj , Y

hj = (ηj cosαj)−1, ηj =

√µj/εj , sinαj = sinα0n0/nj

and j �= 2.

Progress In Electromagnetics Research B, Vol. 7, 2008 69

APPENDIX B.

The elements of the transfer matrix T = T 1T 2 are:

T ee1± =

1

2√Y e

1 Ye±2

(Y e

1 + Y e±2 Y e

1 − Y e±2

Y e1 − Y e±

2 Y e1 + Y e±

2

)E1,

T eh1± = ± 1

2√Y h

1 Yh±2

(Y h±

2 + Y h1 Y h±

2 − Y h1

Y h±2 − Y h

1 Y h±2 + Y h

1

)E1,

T hh1± =

1

2√Y h

1 Yh±2

(Y h±

2 + Y h1 Y h±

2 − Y h1

Y h±2 − Y h

1 Y h±2 + Y h

1

)E1,

T he1± = ∓ 1

2√Y e

1 Ye±2

(Y e

1 + Y e±2 Y e

1 − Y e±2

Y e1 − Y e±

2 Y e1 + Y e±

2

)E1,

T ee2± =

1

4Y e∓2

√Y e

1 Ye±2

×

(Y e∓

2 + Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 − Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 − Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 + Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 − Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 + Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 + Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 − Y e1

) (Y e∓

2 − Y e±2

) E±

2 ,

T eh2± = ∓ 1

4Y h∓2

√Y h

1 Yh±2

×

(Y h∓

2 + Y h1

) (Y h∓

2 + Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 − Y h±2

)(Y h∓

2 + Y h1

) (Y h∓

2 − Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 + Y h±2

)(Y h∓

2 + Y h1

) (Y h∓

2 − Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 + Y h±2

)(Y h∓

2 + Y h1

) (Y h∓

2 + Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 − Y h±2

) E±

2 ,

T hh2± =

1

4Y h∓2

√Y h

1 Yh±2

×

(Y h∓

2 + Y h1

) (Y h∓

2 + Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 − Y h±2

)(Y h∓

2 + Y h1

) (Y h∓

2 − Y h±2

)−

(Y h∓

2 − Y h1

) (Y h∓

2 + Y h±2

)

70 Tuz(Y h∓

2 +Y h1

) (Y h∓

2 −Y h±2

)−

(Y h∓

2 − Y h1

)(Y h∓

2 + Y h±2

)(Y h∓

2 +Y h1

) (Y h∓

2 +Y h±2

)−

(Y h∓

2 − Y h1

)(Y h∓

2 − Y h±2

)E±

2 ,

T he2± = ∓ 1

4Y e∓2

√Y e

1 Ye±2

×

(Y e∓

2 + Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 − Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 − Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 + Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 − Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 + Y e1

) (Y e∓

2 − Y e±2

)(Y e∓

2 + Y e1

) (Y e∓

2 + Y e±2

)−

(Y e∓

2 − Y e1

) (Y e∓

2 − Y e±2

) E±

2 ,

where

E1 = Diag(exp(−ikz1d1) exp(ikz1d1)),E±

2 = Diag(exp(−iγ±z d2) exp(iγ±z d2)).

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