Abstract

The induced-dipole-electric-field contribution to the refractive index at any location within a nanometer-scale dielectric is quantified by summing the electronic dipole contributions due to all the surrounding atoms in the dielectric. Using a tetragonal lattice and varying the ratio of lattice constants illustrates the important limiting chainlike and planelike behaviors. Strong polarizing effects and thus high refractive indices occur for an electric field applied along the length of a chain of atoms or applied in a planar direction to a plane of atoms. In contrast, a strong depolarizing effect and thus low refractive indices occur for an electric field applied normal to a chain of atoms or applied normal to a plane of atoms. Birefringence is increased or decreased by the simultaneous presence or absence of polarizing and depolarizing effects.

© 2007 Optical Society of America

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References

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  1. E. Mollick, "Establishing Moore's law," Ann. Hist. Comput. 28, 62-75 (2006).
    [CrossRef]
  2. J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1957).
  3. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).
  4. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, and Winston, 1968).
  5. H. A. Lorentz, The Theory of Electrons (Teubner, 1909).
  6. H. Froehlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, 2nd ed. (Oxford U. Press, 1958).
  7. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

2006 (1)

E. Mollick, "Establishing Moore's law," Ann. Hist. Comput. 28, 62-75 (2006).
[CrossRef]

1984 (1)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).

1976 (1)

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

1968 (1)

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, and Winston, 1968).

1958 (1)

H. Froehlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, 2nd ed. (Oxford U. Press, 1958).

1957 (1)

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1957).

1909 (1)

H. A. Lorentz, The Theory of Electrons (Teubner, 1909).

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, and Winston, 1968).

Froehlich, H.

H. Froehlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, 2nd ed. (Oxford U. Press, 1958).

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (Teubner, 1909).

Mollick, E.

E. Mollick, "Establishing Moore's law," Ann. Hist. Comput. 28, 62-75 (2006).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1957).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).

Ann. Hist. Comput. (1)

E. Mollick, "Establishing Moore's law," Ann. Hist. Comput. 28, 62-75 (2006).
[CrossRef]

Other (6)

J. F. Nye, Physical Properties of Crystals (Oxford U. Press, 1957).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, and Winston, 1968).

H. A. Lorentz, The Theory of Electrons (Teubner, 1909).

H. Froehlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, 2nd ed. (Oxford U. Press, 1958).

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976).

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Figures (3)

Fig. 1
Fig. 1

Single plane of dipoles oriented in the x direction and the components of the electric field due to one of the dipoles.

Fig. 2
Fig. 2

x component of the normalized-induced-dipole-electric field for an applied field in the x direction, E i n d , x x , and the z component of the normalized-induced-dipole electric field for an applied field in the z direction, E i n d , z z , versus a∕c at the reference atom at the center of the chain for single-atom-cross-section chains, nine-atom-cross-section chains, and for 25-atom-cross-section chains. All chains are 17 atoms in length. A field along the chain is highly polarizing and a field normal to the chain is highly depolarizing. Both effects are enhanced for larger values of a / c (greater separation between chains of atoms).

Fig. 3
Fig. 3

x component of the normalized-induced-dipole electric field for an applied field in the x direction, E i n d , x x , and the z component of the normalized-induced-dipole electric field for an applied field in the z direction, E i n d , z z , versus c∕a at the reference atom at the center of a single 81-atom plane, for a triple 81-atom plane, and for five 81-atom planes. A field in the plane of atoms is highly polarizing and a field normal to the plane is highly depolarizing. Both effects are enhanced for larger values of c / a (greater separation between planes of atoms).

Tables (1)

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Table 1 Normalized-Induced-Dipole-Electric Field in the Limit of an Infinitely Long Chain and an Infinitely Wide Plane of Dipoles

Equations (21)

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P = j N j p j = j N j α j E l o c a l , j ,
E l o c a l , j = E a p p l + E i n d , j ,
E i n d , j = E d e p o l + E 2 + E 3 ,
χ = P ϵ 0 E m a c r o ,
χ = 1 ϵ 0 E a p p l / P N .
n 2 = ε = 1 + 1 ϵ 0 / [ j N j α j ( 1 + E i n d , j / E a p p l ) ] N .
n 2 = ε = 1 1 [ j N j α j ( 1 + E i n d , j / E a p p l ) ] / ϵ 0 .
E = 3 ( p · r ) r r 2 p 4 π ϵ 0 r 5 ,
E z = p 4 π ϵ 0 r 3 ( 2   cos   θ r ^ + sin   θ θ ^ ) ,
n x = n x ( E i n d , j x ) , n y = n y ( E i n d , j y ) , n z = n z ( E i n d , j z ) ,
( Δ n ) b i r e f = n z n x .
E i n d , x x = x ^ p 4 π ϵ 0 { ( k = M z M z + i + = 1 M x + j + = 1 M y + ψ x x ( θ , ϕ ) [ ( i + a ) 2 + ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = π + tan 1 ( j + / i + ) , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + i = 1 M x j + = 1 M y + ψ x x ( θ , ϕ ) [ ( i a ) 2 + ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = tan 1 ( j + / i ) , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + i = 1 M x j = 1 M y ψ x x ( θ , ϕ ) [ ( i a ) 2 + ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = tan 1 ( j / i ) , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + i + = 1 M x + j = 1 M y ψ x x ( θ , ϕ ) [ ( i + a ) 2 + ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = π tan 1 ( j / i + ) , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + i + = 1 M x + ψ x x ( θ , ϕ ) [ ( i + a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = π , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + i = 1 M x ψ x x ( θ , ϕ ) [ ( i a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = 0 , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + j + = 1 M y + ψ x x ( θ , ϕ ) [ ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = π / 2 , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = M z M z + j = 1 M y ψ x x ( θ , ϕ ) [ ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | ϕ = π / 2 , θ = θ for k < 0 , θ = θ + for k 0 + | ( k = 1 M z + ψ x x ( θ , ϕ ) ( k c ) 3 ) | ϕ = 0 , θ = π { + | ( k = M z 1 ψ x x ( θ , ϕ ) ( | k | c ) 3 ) | ϕ = 0 , θ = 0 } ,
ψ x x ( θ , ϕ ) = ( 3 sin 2 θ 1 ) cos 2 ϕ sin 2 ϕ ,
θ = tan 1 [ ( i ± ) 2 + ( j ± ) 2 ] 1 / 2 a / | k | c ,
θ + = π / 2 + tan 1 | k | c / [ ( i ± ) 2 + ( j ± ) 2 ] 1 / 2 a ,
ψ y x ( θ , ϕ ) = 3     sin 2 θ   cos   ϕ   sin   ϕ .
ψ z x ( θ , ϕ ) = 3   cos   θ   sin   θ   cos   ϕ .
E i n d , z z = z ^ p 4 π ϵ 0 { ( k = M z M z + i + = 1 M x + j + = 1 M y + ψ z z ( θ ) [ ( i + a ) 2 + ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + i = 1 M x j + = 1 M y + ψ z z ( θ ) [ ( i a ) 2 + ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + i = 1 M x j = 1 M y ψ z z ( θ ) [ ( i a ) 2 + ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + i + = 1 M x + j = 1 M y ψ z z ( θ ) [ ( i + a ) 2 + ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + i + = 1 M x + ψ z z ( θ ) [ ( i + a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + i = 1 M x ψ z z ( θ ) [ ( i a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + j + = 1 M y + ψ z z ( θ ) [ ( j + a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = M z M z + j = 1 M y ψ z z ( θ ) [ ( j a ) 2 + ( k c ) 2 ] 3 / 2 ) | θ = θ for k < 0 , θ = θ +  for  k 0 + | ( k = 1 M z + ψ z z ( θ , ϕ ) ( k c ) 3 ) | ϕ = 0 , θ = π + { | ( k = M z 1 ψ z z ( θ , ϕ ) ( | k | c ) 3 ) | ϕ = 0 , θ = 0 } ,
ψ z z ( θ ) = 3 cos 2 θ 1 ,
ψ x z ( θ , ϕ ) = ψ z x ( θ , ϕ ) = 3 cos θ sin θ cos ϕ .
ψ y z ( θ , ϕ ) = 3 cos θ sin θ sin ϕ .

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