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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3705–3711
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Reflection and transmission of Gaussian beam by a uniaxial anisotropic slab

Mingjun Wang, Huayong Zhang, Guosheng Liu, Yingle Li, and Qunfeng Dong  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3705-3711 (2014)
http://dx.doi.org/10.1364/OE.22.003705


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Abstract

The reflection and transmission of an incident Gaussian beam by a uniaxial anisotropic slab are investigated, by expanding the incident Gaussian beam, reflected beam, internal beam as well as transmitted beam in terms of cylindrical vector wave functions. The unknown expansion coefficients are determined by virtue of the boundary conditions. For a localized beam model, numerical results are provided for the normalized field intensity distributions, and the propagation characteristics are discussed concisely.

© 2014 Optical Society of America

1. Introduction

The body of this paper proceeds as follows. Section 2 provides the theoretical procedure for the determination of the reflected, internal as well as transmitted fields for a Gaussian beam incident on a uniaxial anisotropic slab. In Section 3, numerical results of the normalized field intensity distributions are displayed. Section 4 is the conclusion.

2. Formulation

As shown in Fig. 1
Fig. 1 Geometry of an incident Gaussian beam from free space on a uniaxial anisotropic slab.
, an incident Gaussian beam propagates in free space and along the positive z axis in the plane xOz of the Cartesian coordinate system Oxyz, with the middle of its beam waist located at origin O on the axis Oz. The planes z=0 and z=d are the interfaces between free space and an infinite uniaxial anisotropic slab of thickness d. Origin O has a coordinate z0 on the axis Oz, and the angle of incidence or the angle made by the axis Oz with the axis Oz is β. In this paper, the time-dependent part of the electromagnetic fields is assumed to be exp(iωt).

In [8

8. H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]

], an expansion has been obtained of the electromagnetic fields of an incident Gaussian beam (focused TEM00 mode laser beam) in terms of the CVWFs with respect to the system Oxyz, which, for the sake of subsequent applications of boundary conditions, can be written as
Ei=E1i+E2i
(1)
where the electric field E1i is described by
E1i=E0m=0π2[Im,TE(ζ)mmλ(1)(h)+Im,TM(ζ)nmλ(1)(h)]exp(ihz)dζ
(2)
and E2i by the same expression as E1i but integrated over ζ from π/2 to π.

In Eq. (2) λ=k0sinζ, h=k0cosζ, k0 is the free-space wavenumber, and In,TEm, In,TMm are the Gaussian beam shape coefficients. For a TE polarized Gaussian beam, the coefficients In,TEm, In,TMm are
Im,TE=(i)m+1k0n=|m|(nm)!(n+m)!2n+12n(n+1)gn[m2Pnm(cosβ)sinβPnm(cosζ)sinζ+dPnm(cosβ)dβdPnm(cosζ)dζ]
(3)
Im,TM=(i)m+1k0mn=|m|(nm)!(n+m)!2n+12n(n+1)gn[Pnm(cosβ)sinβdPnm(cosζ)dζ+dPnm(cosβ)dβPnm(cosζ)sinζ]
(4)
where gn (Gaussian beam shape coefficients in spherical coordinates), when the Davis-Barton model of the Gaussian beam is used [9

9. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

], can be computed by the localized approximation as [10

10. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]

, 11

11. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112(1), 1–27 (2011). [CrossRef]

]
gn=11+2isz0/w0exp(ik0z0)exp[s2(n+1/2)21+2isz0/w0]
(5)
where s=1/(k0w0), and w0 is the beam waist radius.

For a TM polarized Gaussian beam, the corresponding expansions can be obtained only by replacing In,TEm by iIn,TMm, and In,TMm by iIn,TEm.

Equation (1) can be interpreted as an incident Gaussian beam being expanded into a continuous spectrum of cylindrical vector waves, with each cylindrical vector wave having a propagation vector k0=λr^+hz^. The angle between k0 and the positive z axis is ζ, so that only E1i represents the cylindrical vector waves that are incident on the interface z=0.

By following Eq. (2), the reflected beam is expanded as

Er=E0m=0π2[amζ)mmλ(1)(h)+bm(ζ)nmλ(1)(h)]exp(ihz)dζ
(6)

Let’s consider that a uniaxial anisotropic medium is characterized by a permittivity tensor ε¯=x^x^εt+y^y^εt+z^z^εz in the system Oxyz and a scalar permeability μ0.

In [12

12. H. Y. Zhang, Z. X. Huang, and Y. Shi, “Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam,” Opt. Express 21(13), 15645–15653 (2013). [CrossRef] [PubMed]

], based on the eigen plane wave spectrum representation of the fields and the Fourier expansion for the unknown angular spectrum amplitude, the eigenfunction representations of the fields in uniaxial anisotropic media are developed in terms of the CVWFs, as follows:
Ew=E0q=12m=Gmq(θk)[Aqe(θk)mmλq(1)+Bqe(θk)nmλq(1)+Cqe(θk)lmλq(1)]eihqzdθk
(7)
where

a12=ω2εtμ0,a22=ω2εzμ0
(8)
λq=kqsinθk,hq=kqcosθk
(9)
k1=a1,k2=a1a21a12sin2θk+a22cos2θk
(10)
A1e(θk)=1,B1e(θk)=C1e(θk)=A2e(θk)=0
(11)
B2e(θk)=ia12sin2θk+a22cos2θka12sinθk
(12)
C2e(θk)=a12a22a12sinθkcosθk
(13)

Due to the phase continuity over the interfaces z=0 and z=d when using the boundary conditions, we have λq=kqsinθk=λ=k0sinζ (q=1,2 and ζ is from 0 to π/2). Then, Eq. (7) is transformed into
E1w=E0q=12m=0π2Emq(ζ)[αqe(ζ)mmλ(1)(hq)+βqe(ζ)nmλ(1)(hq)+γqe(ζ)lmλ(1)(hq)]eihqzdζ
(14)
where
h1=a12λ2,h2=a1a2a22λ2
(15)
k1=a1,k2=1a2a12a22(a12a22)λ2
(16)
α1e(ζ)=1,β1e(ζ)=γ1e(ζ)=α2e(ζ)=0
(17)
β2e(ζ)=ia23λ1a12a22(a12a22)λ2
(18)
γ2e(ζ)=(a12a22)a2a1λa22λ2a12a22(a12a22)λ2
(19)
and Emq(ζ)dζ=Gmq(θk)dθk.

Let θk=πψk in Eq. (7). Then, by following the same theoretical procedure as in obtaining Eq. (14), i.e., having λq=kqsinψk=λ=k0sinζ (q=1,2 and ζ is from 0 to π/2), Eq. (7) becomes
E2w=E0q=12m=0π2Fmq(ζ)[αqe(ζ)mmλ(1)(hq)+βqe(ζ)nmλ(1)(hq)γqe(ζ)lmλ(1)(hq)]eihqzdζ
(20)
where Fmq(ζ)dζ=Gmq(πψk)dψk

It should be pointed out that E1w represents the electromagnetic waves within the uniaxial anisotropic slab (internal beam) that propagate towards the interface z=d, and that E2w towards the interface z=0.

An appropriate expansion of the transmitted beam in terms of the CVWFs can be given by

Et=E0m=0π2[cm(ζ)mmλ(1)(h)+dm(ζ)nmλ(1)(h)]exp(ihz)dζ
(21)

For the sake of brevity, only the expansions of the electric fields are written, and the corresponding expansions of the magnetic fields can be obtained with the following relations

H=1iωμ0×E,[mmλeihzmmλeihz]=1k×[nmλeihzmmλeihz]
(22)

The unknown expansion coefficients am(ζ), bm(ζ) in Eq. (6), Emq(ζ), Fmq(ζ) in Eqs. (14) and (20) as well as cm(ζ), dm(ζ) in Eq. (21) can be determined by using the following boundary conditions
E1ri+Err=E1rw+E2rwE1ϕi+Eϕr=E1ϕw+E2ϕwH1ri+Hrr=H1rw+H2rwH1ϕi+Hϕr=H1ϕw+H2ϕw}atz=0
(23)
E1rw+E2rw=ErtE1ϕw+E2ϕw=EϕtH1rw+H2rw=HrtH1ϕw+H2ϕw=Hϕt}atz=d
(24)
where the subscripts r and ϕ respectively denote the r and ϕ components of the electromagnetic fields.

By virtue of the fields expansions, the boundary conditions in Eq. (23) can be written as
Im,TE(ζ)+am(ζ)=Em1(ζ)+Fm1(ζ)
(25)
Im,TM(ζ)hk0bm(ζ)hk0=Em2(ζ)[β2e(ζ)h2k2iγ2e(ζ)]Fm2(ζ)[β2e(ζ)h2k2iγ2e(ζ)]
(26)
Im,TE(ζ)hk0am(ζ)hk0=h1k0Em1(ζ)h1k0Fm1(ζ)
(27)
Im,TM(ζ)+bm(ζ)=Em2(ζ)k2k0β2e(ζ)+Fm2(ζ)k2k0β2e(ζ)
(28)
and the boundary conditions in Eq. (24) as

Em1(ζ)eih1d+Fm1(ζ)eih1d=cm(ζ)eihd
(29)
Em2(ζ)[β2e(ζ)h2k2iγ2e(ζ)]eih2dFm2(ζ)[β2e(ζ)h2k2iγ2e(ζ)]eih2d=dm(ζ)hk0eihd
(30)
Em1(ζ)eih1dFm1(ζ)eih1d=cm(ζ)hh1eihd
(31)
Em2(ζ)β2e(ζ)eih2d+Fm2(ζ)β2e(ζ)eih2d=dm(ζ)k0k2eihd
(32)

The expansion coefficients am(ζ), cm(ζ), Em1(ζ) and Fm1(ζ) can be determined from Eqs. (25), (27), (29) and (31), and bm(ζ), dm(ζ), Em2(ζ) and Fm2(ζ) from Eqs. (26), (28), (30) and (32). By substituting them into Eqs. (6), (14), (20) and (21), the reflected, internal and transmitted beams are obtained.

3. Numerical results

In this paper, we will focus our attention on the normalized field intensity distributions, which are defined, respectively, by
|(Ei+Er)/E0|2=(|Eri+Err|2+|Eϕi+Eϕr|2+|Ezi+Ezr|2)/|E0|2
(33)
|(E1w+E2w)/E0|2=(|E1rw+E2rw|2+|E1ϕw+E2ϕw|2+|E1zw+E2zw|2)/|E0|2
(34)
and

|Et/E0|2=(|Ert|2+|Eφt|2+|Ezt|2)/|E0|2
(35)

By substituting the r, ϕ and z components of the CVWFs into Eqs. (1), (6), (14), (20) and (21), the r, ϕ and z components of the electric fields of the incident Gaussian beam, reflected beam, internal beam as well as transmitted beam can be obtained [8

8. H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]

, 12

12. H. Y. Zhang, Z. X. Huang, and Y. Shi, “Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam,” Opt. Express 21(13), 15645–15653 (2013). [CrossRef] [PubMed]

], and then the explicit expressions of Eqs. (33)(35).

Figures 2
Fig. 2 |(Ei+Er)/E0|2, |(E1w+E2w)/E0|2 and |Et/E0|2 for a uniaxial anisotropic slab illuminated by a TM polarized Gaussian beam.
and 3
Fig. 3 Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.
, sharing the same colorbar, show the normalized field intensity distributions in the xOz plane for a uniaxial anisotropic slab, illuminated by a TM and TE polarized Gaussian beam, respectively. The thickness of the slab d and the beam waist radius w0 are assumed to be ten and two times the wavelength of the incident Gaussian beam, and a1=3k0, a2=2k0, β=π/3, z0=0.

From Figs. 2 and 3 we can see that the middle of the transmitted beam for a TM polarized Gaussian beam is at the left of that for a TE polarized one, showing that the former is refracted more strongly, which does not appear for the case of a dielectric isotropic slab. Another noticeable difference lies in the fact that the normalized field intensity of the reflected beam for a TM polarized Gaussian beam becomes negligibly small, as a result of the Brewster angle phenomenon (Brewster's angle of a dielectric medium with a relative refractive index 3 is π/3).

4. Conclusion

An approach to compute the reflection and transmission of an incident Gaussian beam by a uniaxial anisotropic slab is given. Numerical results of the normalized field intensity distributions demonstrate that, compared with a TE polarized Gaussian beam, a TM polarized one is reflected less and refracted more strongly. As a result, this study provides an exact analytical model for interpretation of Gaussian beam propagation phenomena through a uniaxial anisotropic slab, and can also be extended to other cases such as chiral, biaxial and gyrotropic anisotropic slabs.

Acknowledgments

This works is partly supported by the National Natural Science Foundation of China (Grant No. 61271110), New Scientific and Technological Star of Shaanxi Province funded project (Grant No. 2011KJXX39) and the Natural Science Foundation of Shaanxi Province education office, China (Grant No. 12Jk0955).

References and links

1.

J. W. Graham and J. K. Lee, “Reflection and transmission from biaxially anisotropic-isotropic interfaces,” Prog. Electromagn. Res. 136, 681–702 (2013). [CrossRef]

2.

J. F. Dong and J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Prog. Electromagn. Res. 127, 389–404 (2012). [CrossRef]

3.

J. J. Stamnes and V. Dhayalan, “Double refraction of a Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 29(4), 486–497 (2012). [CrossRef] [PubMed]

4.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, D. Velauthapillai, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26(3), 691–698 (2009). [CrossRef] [PubMed]

5.

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250(4–6), 231–240 (2005). [CrossRef]

6.

G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209(1–3), 55–67 (2002). [CrossRef]

7.

G. S. Sithambaranathan and J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18(7), 1670–1677 (2001). [CrossRef] [PubMed]

8.

H. Y. Zhang, Y. P. Han, and G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]

9.

L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]

10.

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]

11.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112(1), 1–27 (2011). [CrossRef]

12.

H. Y. Zhang, Z. X. Huang, and Y. Shi, “Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam,” Opt. Express 21(13), 15645–15653 (2013). [CrossRef] [PubMed]

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(290.0290) Scattering : Scattering
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: November 25, 2013
Revised Manuscript: January 23, 2014
Manuscript Accepted: January 23, 2014
Published: February 7, 2014

Citation
Mingjun Wang, Huayong Zhang, Guosheng Liu, Yingle Li, and Qunfeng Dong, "Reflection and transmission of Gaussian beam by a uniaxial anisotropic slab," Opt. Express 22, 3705-3711 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3705


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References

  1. J. W. Graham, J. K. Lee, “Reflection and transmission from biaxially anisotropic-isotropic interfaces,” Prog. Electromagn. Res. 136, 681–702 (2013). [CrossRef]
  2. J. F. Dong, J. Li, “The reflection and transmission of electromagnetic waves by a uniaxial chiral slab,” Prog. Electromagn. Res. 127, 389–404 (2012). [CrossRef]
  3. J. J. Stamnes, V. Dhayalan, “Double refraction of a Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 29(4), 486–497 (2012). [CrossRef] [PubMed]
  4. M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, D. Velauthapillai, D. Jiang, X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26(3), 691–698 (2009). [CrossRef] [PubMed]
  5. J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250(4–6), 231–240 (2005). [CrossRef]
  6. G. S. Sithambaranathan, J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209(1–3), 55–67 (2002). [CrossRef]
  7. G. S. Sithambaranathan, J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18(7), 1670–1677 (2001). [CrossRef] [PubMed]
  8. H. Y. Zhang, Y. P. Han, G. X. Han, “Expansion of the electromagnetic fields of a shaped beam in terms of cylindrical vector wave functions,” J. Opt. Soc. Am. B 24(6), 1383–1391 (2007). [CrossRef]
  9. L. W. Davis, “Theory of electromagnetic beam,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]
  10. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beam in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am. A 16(7), 1641–1650 (1999). [CrossRef]
  11. G. Gouesbet, J. A. Lock, G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112(1), 1–27 (2011). [CrossRef]
  12. H. Y. Zhang, Z. X. Huang, Y. Shi, “Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam,” Opt. Express 21(13), 15645–15653 (2013). [CrossRef] [PubMed]

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