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

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References

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  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]

2013 (2)

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

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]

2012 (2)

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]

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]

2011 (1)

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]

2009 (1)

2007 (1)

2005 (1)

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]

2002 (1)

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]

2001 (1)

1999 (1)

1979 (1)

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

Davis, L. W.

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

Dhayalan, V.

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]

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]

Dong, J. F.

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]

Frette, Ø.

Gouesbet, G.

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]

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]

Graham, J. W.

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

Gréhan, G.

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]

Han, G. X.

Han, Y. P.

Huang, Z. X.

Jain, M.

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]

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]

Jiang, D.

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]

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]

Lee, J. K.

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

Li, J.

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]

Lock, J. A.

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]

Lotsberg, J. K.

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]

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]

Shi, Y.

Sithambaranathan, G. S.

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]

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]

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]

Stamnes, J. J.

Velauthapillai, D.

Zhang, H. Y.

Zhao, X.

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]

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]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

Opt. Commun. (2)

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]

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]

Opt. Express (1)

Phys. Rev. A (1)

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

Prog. Electromagn. Res. (2)

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

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]

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Figures (3)

Fig. 1
Fig. 1

Geometry of an incident Gaussian beam from free space on a uniaxial anisotropic slab.

Fig. 2
Fig. 2

| ( E i + E r ) / E 0 | 2 , | ( E 1 w + E 2 w ) / E 0 | 2 and | E t / E 0 | 2 for a uniaxial anisotropic slab illuminated by a TM polarized Gaussian beam.

Fig. 3
Fig. 3

Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.

Equations (35)

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E i = E 1 i + E 2 i
E 1 i = E 0 m= 0 π 2 [ I m,TE (ζ) m mλ (1) (h)+ I m,TM (ζ) n mλ (1) (h) ]exp(ihz)dζ
I m,TE = (i) m+1 k 0 n=| m | (nm)! (n+m)! 2n+1 2n(n+1) g n [ m 2 P n m (cosβ) sinβ P n m (cosζ) sinζ + d P n m (cosβ) dβ d P n m (cosζ) dζ ]
I m,TM = (i) m+1 k 0 m n=| m | (nm)! (n+m)! 2n+1 2n(n+1) g n [ P n m (cosβ) sinβ d P n m (cosζ) dζ + d P n m (cosβ) dβ P n m (cosζ) sinζ ]
g n = 1 1+2is z 0 / w 0 exp(i k 0 z 0 )exp[ s 2 (n+1/2) 2 1+2is z 0 / w 0 ]
E r = E 0 m= 0 π 2 [ a m ζ) m mλ (1) (h)+ b m (ζ) n mλ (1) (h) ]exp(ihz)dζ
E w = E 0 q=1 2 m= G mq ( θ k )[ A q e ( θ k ) m m λ q (1) + B q e ( θ k ) n m λ q (1) + C q e ( θ k ) l m λ q (1) ] e i h q z d θ k
a 1 2 = ω 2 ε t μ 0 , a 2 2 = ω 2 ε z μ 0
λ q = k q sin θ k , h q = k q cos θ k
k 1 = a 1 , k 2 = a 1 a 2 1 a 1 2 sin 2 θ k + a 2 2 cos 2 θ k
A 1 e ( θ k )=1, B 1 e ( θ k )= C 1 e ( θ k )= A 2 e ( θ k )=0
B 2 e ( θ k )=i a 1 2 sin 2 θ k + a 2 2 cos 2 θ k a 1 2 sin θ k
C 2 e ( θ k )= a 1 2 a 2 2 a 1 2 sin θ k cos θ k
E 1 w = E 0 q=1 2 m= 0 π 2 E mq (ζ)[ α q e (ζ) m mλ (1) ( h q )+ β q e (ζ) n mλ (1) ( h q )+ γ q e (ζ) l mλ (1) ( h q )] e i h q z dζ
h 1 = a 1 2 λ 2 , h 2 = a 1 a 2 a 2 2 λ 2
k 1 = a 1 , k 2 = 1 a 2 a 1 2 a 2 2 ( a 1 2 a 2 2 ) λ 2
α 1 e (ζ)=1, β 1 e (ζ)= γ 1 e (ζ)= α 2 e (ζ)=0
β 2 e (ζ)=i a 2 3 λ 1 a 1 2 a 2 2 ( a 1 2 a 2 2 ) λ 2
γ 2 e (ζ)= ( a 1 2 a 2 2 ) a 2 a 1 λ a 2 2 λ 2 a 1 2 a 2 2 ( a 1 2 a 2 2 ) λ 2
E 2 w = E 0 q=1 2 m= 0 π 2 F mq (ζ)[ α q e (ζ) m mλ (1) ( h q )+ β q e (ζ) n mλ (1) ( h q ) γ q e (ζ) l mλ (1) ( h q )] e i h q z dζ
E t = E 0 m= 0 π 2 [ c m (ζ) m mλ (1) (h)+ d m (ζ) n mλ (1) (h) ]exp(ihz)dζ
H= 1 iω μ 0 ×E, [ m mλ e ihz m mλ e ihz ]= 1 k ×[ n mλ e ihz m mλ e ihz ]
E 1r i + E r r = E 1r w + E 2r w E 1ϕ i + E ϕ r = E 1ϕ w + E 2ϕ w H 1r i + H r r = H 1r w + H 2r w H 1ϕ i + H ϕ r = H 1ϕ w + H 2ϕ w } at z=0
E 1r w + E 2r w = E r t E 1ϕ w + E 2ϕ w = E ϕ t H 1r w + H 2r w = H r t H 1ϕ w + H 2ϕ w = H ϕ t } at z=d
I m,TE (ζ)+ a m (ζ)= E m1 (ζ)+ F m1 (ζ)
I m,TM (ζ) h k 0 b m (ζ) h k 0 = E m2 (ζ)[ β 2 e (ζ) h 2 k 2 i γ 2 e (ζ)] F m2 (ζ)[ β 2 e (ζ) h 2 k 2 i γ 2 e (ζ)]
I m,TE (ζ) h k 0 a m (ζ) h k 0 = h 1 k 0 E m1 (ζ) h 1 k 0 F m1 (ζ)
I m,TM (ζ)+ b m (ζ)= E m2 (ζ) k 2 k 0 β 2 e (ζ)+ F m2 (ζ) k 2 k 0 β 2 e (ζ)
E m1 (ζ) e i h 1 d + F m1 (ζ) e i h 1 d = c m (ζ) e ihd
E m2 (ζ)[ β 2 e (ζ) h 2 k 2 i γ 2 e (ζ)] e i h 2 d F m2 (ζ)[ β 2 e (ζ) h 2 k 2 i γ 2 e (ζ)] e i h 2 d = d m (ζ) h k 0 e ihd
E m1 (ζ) e i h 1 d F m1 (ζ) e i h 1 d = c m (ζ) h h 1 e ihd
E m2 (ζ) β 2 e (ζ) e i h 2 d + F m2 (ζ) β 2 e (ζ) e i h 2 d = d m (ζ) k 0 k 2 e ihd
| ( E i + E r ) / E 0 | 2 = ( | E r i + E r r | 2 + | E ϕ i + E ϕ r | 2 + | E z i + E z r | 2 ) / | E 0 | 2
| ( E 1 w + E 2 w ) / E 0 | 2 = ( | E 1r w + E 2r w | 2 + | E 1ϕ w + E 2ϕ w | 2 + | E 1z w + E 2z w | 2 ) / | E 0 | 2
| E t / E 0 | 2 = ( | E r t | 2 + | E φ t | 2 + | E z t | 2 ) / | E 0 | 2

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