Propagation of Gaussian beam through a uniaxial anisotropic slab

Optics Communications 380 (2016) 336–341

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Propagation of Gaussian beam through a uniaxial anisotropic slab Zhixiang Huang, Feng Xu, Benxuan Wang, Minquan Li, Huayong Zhang n Key Lab of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University, Hefei, Anhui 230039, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 26 April 2016 Received in revised form 11 June 2016 Accepted 14 June 2016

An exact analytical solution to the reflection and transmission of an incident Gaussian beam by a uniaxial anisotropic slab is obtained in terms of the rectangular vector wave function expansion form. In the uniaxial anisotropic slab, the optical axis is parallel to the interface. For a localized beam model, numerical results of the normalized field intensity distributions are presented, and the propagation characteristics are analyzed concisely. & 2016 Elsevier B.V. All rights reserved.

Keywords: Gaussian beam reflection and transmission uniaxial anisotropic slab

1. Introduction The interaction between electromagnetic (EM) waves and anisotropic media has been studied extensively over the years, for a variety of applications in optical signal processing, optimum design of optical fibers, radar cross section controlling, and microwave device fabrication, etc. It is of fundamental importance to analyze the reflection and transmission of EM waves at a plane interface separating an isotropic and anisotropic medium. With the wave splitting technique, the reflection and transmission properties have been discussed for an EM plane wave normally incident on a stratified bianisotropic slab [1]. Graham et al. investigated the reflection and transmission of an EM plane wave striking a biaxially anisotropic–isotropic interface [2]. For an incident shaped beam, Stamnes et al. presented the formulations and numerical results of focused paraxial field intensities inside a uniaxial and biaxial crystal [3–6]. In one previous paper, we have studied the reflection and transmission of an incident Gaussian beam (focused TEM00 mode laser beam) by a uniaxial anisotropic slab with the optical axis perpendicular to the interface by using the cylindrical vector wave functions (CVWFs) [7]. This paper, based on the rectangular vector wave function (RVWF) expansion form, is devoted to the presentation of an exact analytical solution to the case of the optical axis parallel to the interface. The body of this paper is organized as follows. Section 2 provides the theoretical procedure for the determination of the reflected, internal and transmitted fields for a Gaussian beam incident on a uniaxial anisotropic slab. Numerical results of the n

Corresponding author. E-mail address: [email protected] (H. Zhang).

http://dx.doi.org/10.1016/j.optcom.2016.06.042 0030-4018/& 2016 Elsevier B.V. All rights reserved.

normalized field intensity distributions are given in Section 3. The work is summarized in Section 4.

2. Formulation 2.1. Expansions of Gaussian beam, reflected beam, transmitted and internal beams in terms of the rectangular vector waves As shown in Fig. 1, an incident Gaussian beam propagates from free space to an infinite uniaxial anisotropic slab of thickness d , with its propagation direction O′z′ having the polar coordinates ζ , η with respect to the Cartesian coordinate system Oxyz and its beam waist middle located at origin λ on the axis O′z′. Origin O has a coordinate z0 on the axis O′z′, and the planes z = 0 and z = d are the interfaces between free space and the uniaxial anisotropic slab. In this paper, a time dependence of the form exp( − iωt ) is assumed and suppressed for the EM fields. In Appendix A, an expansion for the EM fields of an incident Gaussian beam (focused TEM00 mode laser beam) in terms of the RVWFs with respect to the system Oxyz is obtained, as follows:

Ei = E1i + Ei2

(1)

where the electric field E1i is described by

E1i = E0 E2i

∫0

2π

π

dβ

∫0 2 [ITEmk (α, β ) + ITMnk (α, β )]dα 0

0

E1i

(2)

and by the same expression as but integrated over α from π/2 to π . For a TE-polarized mode, the Gaussian beam shape coefficients ITE and ITM are

Z. Huang et al. / Optics Communications 380 (2016) 336–341

337

Fig. 1. A Gaussian beam striking a uniaxial anisotropic slab. (a) The coordinate system, (b) the configuration.

ITE

Et = E0

i 4π k 0

=

+ m2

Pnm(cos ζ ) Pnm(cos α ) ⎤ ⎥ sin ζ sin α ⎦

i 4π k 0

∑ ∑

gn

n = 1 m =−n

+

2n + 1 (n − m)! im(β − η) ⎡ dPnm(cos ζ ) Pnm(cos α ) e m⎢ n(n + 1) (n + m)! ⎣ dζ sin α

Pnm(cos ζ ) dPnm(cos α ) ⎤ ⎥ dα ⎦ sin ζ

2 q=1

(4)

⎡ −s 2(n + 1/2)2 ⎤ 1 ⎥ exp(ik 0z 0)exp⎢ 1 + 2isz 0/w0 ⎣ 1 + 2isz 0/w0 ⎦

2

2

is from 0 to π/2 for E1i . Following Eq. (2), the reflected beam and transmitted beam can be expanded as

dβ

∫0 2 [a(α, β )mk (π − α, β ) + b(α, β )nk (π − α, β )] 0

0

∫0 2 fq (α, β )F eq(α, β ) (8)

2

∑∫ q=1

0

π

2π

dβ

∫0 2 gq(α, β )Geq(α, β )

exp[i(k 0x sin α cos β + k 0y sin α sin β − kqzz )]dα

(9)

For the sake of brevity, only the expansions of the electric fields are presented, and the magnetic fields can be expanded correspondingly with the following relations

H= =

1 ∇ × E, ⎡⎣ m k 0(α, β ) n k 0(α, β )⎤⎦ iωμ 0 1 ∇ × ⎡⎣ n k 0(α, β ) m k 0(α, β )⎤⎦ k0

(10)

2.2. Gaussian beam propagation through a uniaxial anisotropic slab The unknown expansion coefficients a(α , β ), b(α , β ) in Eq. (6), c (α , β ), d(α , β ) in Eq. (7), as well as fq (α , β ), gq(α , β ) (q = 1, 2) in Eqs.

π

2π

dβ

E2w

(5)

uration in Fig. 1 we can see that only E1i represents those rectangular vector waves that are incident on the interface z = 0, due to the fact that α , made by the propagation vector k 0 and the axis Oz ,

∫0

π

2π

exp[i(k 0x sin α cos β + k 0y sin α sin β + kqzz )]dα

coordinates α and β in the system Oxyz . Then, from the config-

dα

(7)

= E0

0

Er = E 0

0

ability μ0 . From Appendix B, we obtain the eigen plane wave spectrum representations of the internal beams that propagate towards the interfaces z = d and z = 0, respectively described by E1w and E2w as

∑∫

where s = 1/(k0w0), and w0 is the beam waist radius. The corresponding expansions for a TM-polarized Gaussian beam can also be obtained by replacing ITE in Eq. (2) with iITM , and ITM with iITE . Eq. (1) can be interpreted that an incident Gaussian beam is expanded into a continuous spectrum of rectangular vector waves, with each rectangular vector wave having a propagation vector k = k (sin α cos βx^ + sin α sin βy^ + cos αz^ ) defined by the polar 0

0

= E0

When the Davis–Barton model of the Gaussian beam is used [8], μr can be computed by the localized approximation as [9,10]

gn =

∫0 2 [c(α, β )mk (α, β ) + d(α, β )nk (α, β )]dα

E1w n

∞

π

dβ

1

(3)

ITM =

2π

We consider that the uniaxial anisotropic medium of the slab has an optical axis parallel to the interfaces z = 0 and z = d , and that its constitutive relations are expressed by a permittivity ten^^ε in the system Oxyz and a scalar perme^^ε + yy ^ ^ ε + zz sor ε¯ = xx

⎡ dP m dP m ∑ ∑ gn 2n + 1 (n − m)! eim(β − η)⎢ n (cos ζ ) n (cos α ) n(n + 1) (n + m)! ⎣ dζ dα n = 1 m =−n n

∞

∫0

0

(6)

(8) and (9) can be determined by using the boundary conditions respectively at z = 0 and z = d , as follows:

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Z. Huang et al. / Optics Communications 380 (2016) 336–341

E1i x + Exr = E1wx + E2wx H1i x + Hxr = H1wx + H2wx

E1wx + E2wx = Ext H1wx

+

H2wx

=

Hxt

E1i y + Eyr = E1wy + E2wy ⎫ ⎪ ⎬ i r w w⎪ H1y + H y = H1y + H2y ⎭

E1wy + E2wy = Eyt ⎫ ⎪ ⎬ w w t⎪ H1y + H2y = H y ⎭

at

at

z=0 (11)

⎤ ⎡k k ⎤⎡ i⎢ 0 sin2αcos2β − 1 ⎥⎢ f1(α, β )exp(ik1zd) + g1(α, β )exp( − ik1zd)⎥ k k ⎣ 1 ⎦ 0 ⎦⎣ ⎤ ⎡ = ⎢ − c (α, β )cos α cos β + id(α, β )sin β⎥k 0 sin α ⎦ ⎣ exp(ik 0d cos α )

z=d

(12)

where the subscripts x and y denote the x and y components of the EM fields. The boundary conditions in Eqs. (11) and (12) are valid for each value of α and β . Then, by virtue of the above field expansions, Eq. (11) can be written as

i

⎤ ⎡ k0 2 sin α sin β cos β⎢ f1(α, β )exp(ik1zd) + g1(α, β )exp( − ik1zd)⎥ ⎥⎦ ⎢⎣ k1 ⎤ ⎡ − i⎢ f2 (α, β )exp(ik2zd) − g2(α, β )exp( − ik2zd)⎥ ⎥⎦ ⎢⎣

[iITE sin β − ITM cos α cos β + ia(α, β )sin β + b(α, β )cos α cos β ] k 0 sin α = − [f2 (α, β ) + g2(α, β )]

1 2 (a2 − k 02sin2αcos2β ) k22

(13)

k 02 k22

k12

k22

k22

k 02

⎤ ⎡ − sin2α = ⎢ − c (α, β )cos α sin β − id(α, β )cos β⎥k 0 ⎥⎦ ⎢⎣ (20)

From the system consisting of Eqs. (13)–(20) the expansion coefficients a(α , β ), b(α , β ), c (α , β ), d(α , β ), fq (α , β ) and gq(α , β )

(q = 1, 2) can be determined. By substituting them into Eqs. (6)– (9), the reflected, transmitted and internal beams are obtained.

sin β ]k 0 sin α = [f1(α, β ) − g1(α, β )] k 02

a22

sin αexp(idk 0 cos α )

[ − iITE cos β − ITM cos α sin β − ia(α, β )cos β + b(α, β )cos α

1−

(19)

sin2α + [f2 (α, β ) + g2(α, β )] 3. Numerical results

2

sin α sin β cos β

(14)

2

[ − ITE cos α cos β + iITM sin β + a(α, β )cos α cos β + ib(α, β ) sin β ]k 0 sin α = i[f1(α, β ) + g1(α, β )][

k0 2 k sin αcos2β − 1 ] k1 k0

In this paper, the normalized field intensity distributions (NFIDs) will be calculated, which are respectively defined by

(Ei + Er )/E0 =

2 2 2 2⎞ ⎛ i ⎜ E + Er + Eyi + Eyr + Ezi + Ezr ⎟/ E0 x x ⎝ ⎠

(15) 2

2

2

2

(E1w + E2w)/E0 = ( E1wx + E2wx + E1wy + E2wy + E1wz + E2wz )/ E0 [ − ITE cos α sin β − iITM cos β + a(α, β )cos α sin β − ib(α, β ) cos β ]k 0 sin α = i[f1(α, β ) + g1(α, β )]

k22 k 02

2

Et /E0 =

− sin2α (16)

and Eq. (12) as

−[f2 (α, β )exp(ik2zd) + g2(α, β )exp( − ik2zd)] 1 2 (a2 − k 02sin2αcos2β ) = [ic (α, β )sin β − d(α, β )cos α cos β ] k22 k 0 sin αexp(ik 0d cos α )

(22)

(

2

2

Ext + Eyt + Ezt

2

)

2

/ E0

(23)

By inserting the x , y and z components of the electric fields of the incident Gaussian beam, reflected beam, internal beam as well as transmitted beam into Eqs. (21)–(23), the explicit expressions of the NFIDs are obtained. We take the reflected electric field in Eq. (6) as an example, and write its x , y and z components as

Exr

= E0

(17)

2π

π

∫0 ∫0 2 [ik0a(α, β )sin α sin β + k0b(α, β )sin α cos α cos β ]

⎤ ⎡ 2 ⎢ f (α, β )exp(ik d) − g (α, β )exp( − ik d)⎥ 1 − k 0 sin2α 1 1 z z 1 ⎥⎦ ⎢⎣ 1 k12

× exp[ik 0(x sin α cos β + y sin α sin β − z cos α )]dαdβ

⎤ 2 ⎡ k + ⎢ f2 (α, β )exp(ik2zd) + g2(α, β )exp( − ik2zd)⎥ 02 sin2α sin β ⎥⎦ k2 ⎢⎣

(24)

Eyr

= E0

⎤ ⎡ cos β = ⎢ − ic (α, β )cos β − d(α, β )cos α sin β⎥k 0 sin α ⎥⎦ ⎢⎣ exp(ik 0d cos α )

2

and

k0 2 sin α sin β k1

cos β − i[f2 (α, β ) − g2(α, β )] a22 k22

(21)

2π

π

∫0 ∫0 2 [ − ik0a(α, β )sin α cos β + k0b(α, β )sin α cos α (18)

sin β ] × exp[ik 0(x sin α cos β + y sin α sin β − z cos α )] dα dβ

(25)

Z. Huang et al. / Optics Communications 380 (2016) 336–341

Fig. 2. NFIDs in the plane xOz for a uniaxial anisotropic slab (a12 = 3k 02, a22 = 2k02) illuminated by a TE-polarized Gaussian beam with ζ = π /4 and η = 0 .

Ezr = E0

2π

339

Fig. 4. NFIDs in the plane xOz for a uniaxial anisotropic slab (a12 = (2 + 0.1i )k02, a22 = 4k02) illuminated by a TE-polarized Gaussian beam with ζ = π /3 and η = 0 .

π

∫0 ∫0 2 k0b(α, β )sin2αexp[ik0(x sin α cos β + y sin α

sin β − z cos α )]dαdβ

(26)

In the following calculations, the thickness of the slab d and the beam waist radius w0 are assumed to be fifteen and three times the wavelength of the incident Gaussian beam, and z0 = 0. Figs. 2 and 3 show the NFIDs respectively in the planes xOz and yOz , coinciding with the plane of incidence (the plane containing the axes O′z′ and Oz ), for a uniaxial anisotropic slab illuminated by a TE-polarized Gaussian beam. From Figs. 2 and 3 we can see that the differences in the NFIDs are quite obvious in the different planes of incidence, which dose not appear for a dielectric isotropic slab or a uniaxial anisotropic slab with the optical axis perpendicular to the interface [7]. In addition, due to the different refractive index in the different plane of incidence, the middle of the transmitted beam in Fig. 2 is at the right of that in Fig. 3, showing that the latter is refracted more strongly. The NFIDs in the xOz plane are shown in Figs. 4 and 5 for an absorbing uniaxial anisotropic slab, respectively illuminated by a TE and TM-polarized Gaussian beam. It is demonstrated that, compared with the case of a TE-polarized Gaussian beam, an incident TM-polarized Gaussian beam is reflected less and

Fig. 5. NFIDs for the same model as in Fig. 4 but illuminated by a TM-polarized Gaussian beam with ζ = π /3 and η = 0 .

attenuated more quickly when transmitted into the slab.

4. Conclusion By using the field expansions in terms of the rectangular vector waves, an approach is presented to compute the reflection and transmission of an incident Gaussian beam by a uniaxial anisotropic slab with the optical axis parallel to the interface. The numerical results show that both the incident direction and polarized mode of an incident Gaussian beam have a great impact on its propagation characteristics. Furthermore, this study can also be extended to other anisotropic cases such as uniaxial chiral, biaxial and gyrotropic slabs, which we hope may offer the reference to research on the EM properties of anisotropic media.

Acknowledgments Fig. 3. NFIDs in the plane yOz for the same model as in Fig. 2 illuminated by a TEpolarized Gaussian beam with ζ = π /4 and η = π /2 .

The authors would acknowledge the supports by the NSFC (Nos. 51477001, 61471001) and AHNSF (No. 1408085MF123).

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Z. Huang et al. / Optics Communications 380 (2016) 336–341

Appendix A

k1 = a2

In [11], we have expanded an incident Gaussian beam as in Fig. 1 in terms of the spherical vector wave functions (SVWFs) natural to the system Oxyz , as follows (TE mode) [12]:

k2 = a1a2[a22cos2θk + sin2θk(a12cos2ϕk + a22sin2ϕk )]− 2

n

∞

Ei = E0 ∑

∑

⎡ iG m m (1) (k r , θ , ϕ) + G m n r (1)(k r , θ , ϕ)⎤ ⎣ n, TE mn 0 ⎦ n, TM mn 0

(A1)

n = 1 m =−n

Gnm, TM

Gnm, TE

where and are the Gaussian beam shape coefficients in spherical coordinates

Gnm, TE = gni n + 1

Gnm, TM

= igni

m −imη dPn (cos

2n + 1 (n − m)! e n(n + 1) (n + m)!

n+ 1

ζ)

dζ

(A2)

2n + 1 (n − m)! −imη Pnm(cos ζ ) e m n(n + 1) (n + m)! sin ζ

(A3)

The RVWFs are defined by [13]

m k 0(α, β )

{

=∇×

n k 0(α, β ) =

z^exp⎡⎣ ik 0(sin α cos βx + sin α sin βy + cos αz )⎤⎦

}

1 ∇ × m k 0(α, β ) k0

(1)

( ) ∫0

π

mn

i−(n + 1) = 4π k 0

∫0

2π

⎡ dP m(cos α ) m ⎢ n n dα ⎣

e

imβ

1

e E˜ (k) = f˜q (θk, φk )F˜ q(θk, φk ) = f˜q (θk, φk ) e e e (F˜qxx^ + F˜qyy^ + F˜qzz^) (q = 1, 2)

e e e F˜1x = 0, F˜1y = cos θk, F˜1z = − sin θk sin ϕk

(B10)

e F˜2y = sin2θk sin ϕk cos ϕk

(B11)

e F˜2z = sin θk cos θk cos ϕk

(B12)

The solution to Eq. (B1) is then given by

E(r) = E0 ∑

( )

( )

2π

∫0 ∫ f˜q (θk, ϕk)F˜ eq(θk, ϕk)eik ⋅ rkq2 sin θkdθkdϕk q

(B13)

Owing to the phase continuity over the interfaces z = 0 and z = d when applying the boundary conditions, we have kq sin θk = k0 sin α , ϕk = β . Then, Eq. (B13) is transformed into Eq. (8), where e f (α , β )dαdβ = f˜ (θ , ϕ )k 2 sin θ dθ dϕ , F e(α , β ) =F˜ (θ , ϕ ) , and

dβ

⎤ (α, β )⎥dα ⎦ k0

(B9)

a2 e F˜2x = − 22 (sin2θksin2ϕk + cos2θk ) a1

q=1

P m(cos α ) n (α, β ) + m n m sin α k0

(B8)

where f˜q (θk, ϕk ) is the unknown angular spectrum amplitude, and

2

(A5)

(B7)

So the eigenvectors E˜ (k) can be obtained from Eq. (B2) as

(A4)

The SVWFs are expanded in terms of the RVWFs as

m n

(B6)

q

(A6)

By substituting Eq. (A6) into Eq. (A1) and then by performing some algebra as in [14], we can obtain Eq. (1).

q

k

q

k

k

k1 = a2, k1z =

k12 − k 02sin2α

F1ex = 0, F1ey =

1−

k 02 k12

k

q

q

k

k

k

(B14)

sin2α , F1ez = −

k0 sin α sin β k1

(B15)

1

APPENDIX B The E-field vector wave equation in the uniaxial anisotropic medium can be readily derived from the source-free Maxwell's equations, which is of the form

∇ × ∇ × E(r) − ω2μ 0 ε¯⋅E(r) = 0

−∞

∞

−∞

∞

∫−∞ dkx ∫−∞ dky ∫−∞

−∞

K¯ (k)⋅E˜ (k)eik ⋅ r dk z = 0

k2z =

(B16)

1 1 2 2 [a1 a2 − k 02sin2α(a12cos2β + a22sin2β )]2 a2

(B17)

(B1)

The solution to Eq. (B1) can be obtained by following the theoretical procedure as in [15]. A substitution of the Fourier trans∞ ∞ ∞ form E(r) = ∫ dk x ∫ dk y ∫ E˜ (k)eik ⋅ rdkz into Eq. (B1) leads to ∞

⎡ ⎤2 ⎛ a2 ⎞ k2 = ⎢ a12 + k 02⎜⎜ 1 − 12 ⎟⎟sin2αcos2β⎥ ⎢⎣ ⎥⎦ a2 ⎠ ⎝

F2ex = −

F2ey = (B2)

where

F2ez =

⎛ k 2 − k2 + a 2 ⎞ k xk y k xk z 1 ⎜ x ⎟ ⎟ K¯ (k) = ⎜ k yk x k y2 − k 2 + a22 k yk z ⎜ ⎟ 2 2 2⎟ ⎜ k zk x k zk y k z − k + a2 ⎠ ⎝

(B3)

a12 = ω2μ 0 ε1, a22 = ω2μ 0 ε2

(B4)

1 2 (a2 − k 02sin2αcos2β ) k22

k 02 k22

(B18)

sin2α sin β cos β

(B19)

k0 k2 sin α cos β 1 − 02 sin2α k2 k2

(B20)

Let θk = π − ψk in Eq. (B13). Then, by following the same theoretical procedure as in obtaining Eq. (8), i.e., having kq sin ψk = k0 sin α , ϕk = β , Eq. (B13) becomes Eq. (9), where g (α , β )dαdβ = f˜ (π − ψ , ϕ )k 2 sin ψ dψ dϕ , G e = F e , G e = − F e , q

q

k

k

q

k

k

k

1x

1x

1y

1y

G1ez = F1ez , G2ex = F2ex , G2ey = F2ey , and G2ez = − F2ez .

For nontrivial solutions of E˜ (k) the following characteristic equation has to be satisfied

Det⎡⎣ K¯ (k)⎤⎦ = 0

References

(B5)

From Eq. (B5) we can have its two roots of kq(q = 1, 2), as follows:

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