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Tunable Goos-H{a}nchen shift and polarization beam splitter in electro-optic crystals



Tunable Goos-H¨ anchen shift and polarization beam splitter in electro-optic crystals
Xi Chen1 ? , Ming Shen1 , Zhen-Fu Zhang1 , and Chun-Fang Li1,2 ?
Department of Physics, Shangha

i University, Shanghai 200444, People’s Republic of China and 2 State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics of CAS, Xi’an 710119, People’s Republic of China (Dated: September 4, 2008) We investigate the tunable Goos-H¨ anchen shifts and polarization beam splitter in electro-optic crystals, based on the Pockels e?ect. It is found that large positive and negative lateral shifts can be easily controlled by adjusting the permittivity tensor, which is modulated by the external applied electric ?eld. These phenomena will lead to an alternative way to realize polarization beam splitter.
1

arXiv:0809.0774v1 [physics.optics] 4 Sep 2008

PACS numbers: 42.25.Bs; 78.20.Jq; 42.25.Gy; 42.79.Fm

It is well known that a light beam totally re?ected from an interface between two dielectric media undergoes lateral displacement from the position predicted by geometrical optics [1]. This phenomenon was referred to as the Goos-H¨ anchen (GH) shift [2] and was theoretically explained ?rst by Artmann [3]. Since the investigations of the GH shifts have been extended to partial re?ection [4, 5, 6, 7, 8, 9, 10, 11] including di?erent slab con?gurations containing various materials (such as weakly absorbing media [12], gain media [13], left-handed media [14, 15] and anisotropic metamaterial media [16, 17]), much attention has been paid to achieve a large (positive and negative) lateral shift, which leads to the potential applications in integrated optics [2], electromagnetic communication system [18], and optical sensors [19, 20]. For applications in optical devices, on the other hand, it is important to realize the tunability of lateral shift, that is, to control the lateral shift in a ?xed con?guration or device by external ?eld. More recently, Wang et. al. [21] reported that the lateral shift can be modulated by a coherent control ?eld, which is applied onto the two-level atoms inside a cavity. Actually, the controllable lateral shifts is important for the further applications in ?exible optical-beam steering and optical devices in information processing. Therefore, the main purpose of this paper is to investigate the control of the GH shift in electro-optic crystals, based on the Pockels e?ect [22], which o?ers opportunity for tuning the optical response characteristics of materials in photonic band-gap engineering [23, 24], composite materials [25, 26] and surface-wave propagation [27], due to the linear change of the refractive indices caused by application of an external electric ?eld. Consider a light beam of angular frequency ω incident on a slab of electro-optic crystal in the air with an incidence angle θ0 speci?ed by the inclination of the beam with respect to the z axis, as shown in Fig. 1, where the thickness, relative permittivity and relative permeFIG. 1: Schematic diagram of positive and negative lateral shift of TE and TM polarized light beams transmitted through a slab of electro-optic crystal with external electric ?eld in the y direction.

ability of the nonmagnetic electro-optic crystal slab, are denoted by d, ε ?1 , and ?0 , respectively. In the case of TE (TM) polarization, the electric (magnetic) ?eld of the plane wave component of the incident beam is assumed to Ψin (x) = A exp(ik · x), where k ≡ (kx , kz ) = (k sin θ, k cos θ), k = (ε0 ?0 )1/2 ω/c is the wave number in the air, ε0 , and ?0 are the relative permittivity and permeability of the air, c is the speed of light in vacuum, and θ stands for the incidence angle of the plane wave under consideration. For the sake of simplicity, suppose we have a tetragonal (point group ? 42m) uniaxial crystal. The electro-optic crystal with optic axis in the z direction has the following index ellipsoid equation: x2 y2 z2 + 2 + 2 = 1, 2 no no ne and the relative permittivity tensor is, ? ? εx 0 0 ε ?1 = ? 0 εy 0 ? , 0 0 εz (1)

? Email ? Email

address: xchen@shu.edu.cn address: c?i@shu.edu.cn

(2)

2
2 where εx = εy = n2 o , εz = ne , no and ne are the refractive indices for ordinary and extraordinary waves inside the anisotropic slab, respectively. In a electric ?eld Ey along the direction of y , the index ellipsoid equation becomes

y2 z2 x2 + + + 2γ41 Ey xz = 1. n2 n2 n2 o o e

(3)

By making coordinate transformations, as shown in Fig. 1, x′ z′ = cos ? ? sin ? sin ? cos ? x z (4)

the index ellipsoid equation (3) becomes x′2 y ′2 z ′2 + 2 + 2 = 1, 2 nx ′ ny ′ nz ′ where n2 x′ ≡ ε′ x = 1 1 cos2 ? + 2 sin2 ? + γ41 Ey sin 2? n2 n o e n2 y′ and
′ n2 z ′ ≡ εz = ?1

γ63 are quite di?erent for the tetragonal uniaxial crystals such as ADP and KDP with the others γij = 0, it clearly makes sense to utilize the two coe?cients mentioned above to control the lateral shift, taking account into di?erent crystal cuts. It is also noted that the expressions obtained above are valid for the cubic crystal of point group ? 43m, the symmetry group of such common materials as GaAs, where no = ne , γ41 = γ63 . In order to calculate the lateral shifts, the new relative permittivity tensors in the original xyz frame is expressed as the following form: ? ? a 0 f ?′ = ? 0 ε′ ?, ε (10) y 0 f 0 b
2 2 2 ′ 2 ′ ′ where a = ε′ x cos ? + εz sin ?, b = εx sin ? + εz cos ?, ′ ′ f = (εz ? εx ) sin ? cos ?. Then, the dispersion equations are given by 2 kx + k ′ z = ε′ y ?0 2

(5)

, and

ω2 , c2

(for TE wave),

(11)



ε′ y

=

n2 o,

2 f a 2 2 ′ ?0 ω k ′ z +2 kx k ′ z + kx , (for TM wave). (12) = ε′ x εz b b b c2 ?1

1 1 sin2 ? + 2 cos2 ? ? γ41 Ey sin 2? n2 n o e

where k ′ z for TM polarized wave is also expressed by , k ′ z± = ?α1 ± α2 ,

when the angle ? is determined by tan 2? = ? 2γ41 Ey 1 . 1 n2 ? n2
e o

(6)

Moreover, when the optic axis is changed to be in the y direction, the index ellipsoid equation is given by x2 y2 z2 + + = 1. n2 n2 n2 o e o (7)

√ 2 ′ where α1 = f kx /b, α2 = γk/b, γ = ε′ x εz (b?0 ? sin θ ). Thus, the ?eld of the corresponding transmitted plane wave is found, according to Maxwell’s equations and the boundary conditions, to be Ψt (x) = tA exp{i[kx x+kz (z ? d)]}, where the transmission coe?cient of TM polarized light beam t = ei(φ?α1 d) /f is determined by the following complex number, f eiφ = cos α2 d + and tan φ = 1 2 χ kz 1 α2 + α2 χ kz tan α2 d, i 2 χ kz 1 α2 + α2 χ kz sin α2 d,

thus the index ellipsoid equation in a ?eld along the direction of y is given by y2 z2 x2 + 2 + 2 + 2γ63 Ey xz = 1. 2 no ne no (8)

Let ? = ?π/4 in the coordinate transformations (4), the above index ellipsoid equation becomes x′2 y ′2 z ′2 + 2 + 2 = 1, 2 nx ′ ny ′ nz ′ (9)

′ where χ = (ε′ x εz )/(bε0 ). Clearly, the phase shift of the transmitted beam at x = d with respect to the incident beam at x = 0 is equal to φ ? α1 d, thus the lateral shift of the TM polarized light beam is de?ned as ?d(φ ? α1 d)/dkx |θ=θ0 [3, 10], and is given by

where nx′ = 1/(1/n2 o ? γ63 Ey ), ny ′ = ne , and nz ′ = 1/(1/n2 + γ E ). As electro-optic coe?cients γ41 and 63 y o

sT M = s + s, where

(13)

3

s =

′ d tan θ0 2 2 f0

′ ε′ x εz 2 b

χ

kz0 1 α20 + α20 χ kz0

?

1?

′ ε′ x εz

b2 α2 20 2 kz 0

χ

kz0 1 α20 ? α20 χ kz0

sin 2α20 d , 2α20 d

(14)

′ s = (f /b)d, and tan θ0 = kx0 /α20 . It is noted that the subscript 0 in this paper denotes values taken at kx = kx0 , namely, θ = θ0 . Similarly, the amplitude transmission coe?cient t = eiφ /f of the TE polarized beam is determined by the following complex number, ′ f eiφ = cos kz d+

where the phase shift is given by tan φ = 1 2 χ
′ kz 1 kz + ′ kz χ kz ′ tan kz d,

(15)

i 2

χ

′ kz 1 kz + ′ kz χ kz

′ sin kz d,

and χ = 1. The lateral shift of TE polarized beam is obtained by

sT E =

′ d tan θ0 2 2 f0

χ

′ kz0 1 kz 0 + ′ kz0 χ kz0

?

1?

′2 kz 0 2 kz0

χ

′ kz0 1 kz 0 ? ′ kz0 χ kz0

′ sin 2kz 0d , ′ 2kz0 d

(16)

′ where tan θ0 = kx0 /kz0 . The lateral shifts of TM and TE polarized light beams presented here depends not only ′ on θ0 and d, but on ε′ x , εz , and ?. From Eqs. (13) and (16), it is shown that the lateral shifts can be negative when incidence angle is larger than the threshold of angle [10]. In what as follows, the properties of the lateral shifts are discussed. For the tetragonal uniaxial crystals, we have n0 = 1.5079, ne = 1.4683, and γ41 = 23.76 × 10?12 m/V, γ63 = 8.56 × 10?12 m/V for ADP and n0 = 1.5115, ne = 1.4698, and γ41 = 8.77 × 10?12 m/V, γ63 = 10.3 × 10?12 m/V for KDP at λ = 546nm [22].
20 15 10 50 0



symmetry of point group considered here. Moreover, the negative and positive lateral shifts of TM polarized beam depend on the applied electric ?elds at di?erent angles is shown in Fig. 3, where d = 1.5λ and other parameters are the same as in Fig. 2. It is due to the fact that the lateral shift for TM polarized light beam presented here ′ is closely related to ε′ x , εz , and ?, which can be modulated by applied electric ?eld, based on the Pockles e?ect. Thus, we can easily control the lateral shift via applied electric ?elds, once one chooses the structure.
5 4 3

sTE /λ and sTM /λ

?50

5 ?100 0 ?150

2 1

?10 ?15 ?20 55

?200

sTM /λ
65 70

?5

0 ?1

(a)
60 65 70 75 80 85 90

?250

(b)
60 75 80 85 90

θ (degree)
0

?300 55

θ0 (degree)

?2 ?3

FIG. 2: Dependence of the lateral shifts on the incidence angle θ0 in the slab of ADP crystal with the optic axis in the y direction, where (a) d = 5λ and (b) d = 10λ. Solid curve represents sT E . Dotted and dashed curves represent sT M corresponding to Ey = 0 and 20GV, respectively.

?4 ?5 0 5 10 15 20

Ey (GV)

Fig. 2 indicates that the lateral shifts of TM and TE polarized light beams are quite di?erent in the ADP crystal with the optical axis in the direction y axis. It is also shown that the enhanced lateral shift of TM polarized beam can be controlled by the applied electric ?eld, while the lateral shift of TE polarized beam in this case is independent of the applied electric ?eld, due to the

FIG. 3: Dependence of the lateral shifts on the electric ?eld at incidence angle θ0 = 70? (dashed curves) and θ0 = 80? (solid curve), where d = 1.5λ and other parameters are the same as in Fig. 2.

Fig. 4 further shows the electric control of the lateral shifts in the case of di?erent crystal cuts. Since the modi?cation of the permittivity tensor by the applied electric

4 ?eld is quanti?ed through 18 electro-optic coe?cients, not all of which may be independent of each other, depending on the point group symmetry [22], the lateral shifts depend on the di?erent electro-optic coe?cients relating to crystal cut, as shown in Fig. 4, where solid and dashed curves correspond to the optical axis in different z and y directions, respectively. It is shown that the larger electro-optic coe?cient make the tunability of lateral shift easier at a ?nite electric ?eld. By comparison between Fig. 4 (a) and (b), it is implied that we can choose the crystal geometry in terms of the relative electro-optic coe?cient used to control the lateral shift e?ciently when the electrode is ?xed.
5 4 3 2 1 4.5

3

2.5

2

20 GV 15 GV 10 GV 5 GV 0 GV

D/λ

1.5

1

0.5

0 0

10

20

30

40 0

50

60

70

80

90

θ (degree)

(a)

ADP

4 3.5 3 2.5 2 1.5 1 0.5 0

(b) KDP

sTM /λ

FIG. 5: Dependence of D on the incident angle θ0 under di?erent applied electric ?eld, where all the parameters are the same as in Fig. 3.

0 ?1 ?2 ?3 ?4 ?5 0 5 10 15 20

?0.5 0

5

10

15

20

Ey (GV)

Ey (GV)

FIG. 4: Dependence of the lateral shifts on the electric ?eld in case of di?erent crystal cuts for (a) ADP and (b) KDP, where d = 1.5λ, solid and dashed curves correspond to the optical axis in the z and y directions, respectively.

of electro-optical crystals. In a word, we present a useful way to control the GH shifts in electro-optic crystal via applied electric ?eld. With the development of crystal materials, we believe that these novel phenomena will lead to an alternative way to realize polarization beam splitter, which is essential optical components in optical systems and plays an important role in optical communication, optical recording and integrated optical circuits.

Next, we have a brief look at the polarization beam splitter via tunable GH shift. In the above discussions, it is clear that the lateral shifts depends on the polarization state of the beam, which leads to the separation of the two orthogonal polarizations of light beam. To illustrate the in?uence of the applied electric ?eld on polarization beam splitter in electro-optic crystal, we de?ne the distance, as shown in Fig. 1, D = |sT M ? sT E | × sin θ0 , (17)

Acknowledgements

X. Chen is grateful to A. Lakhtakia for providing useful information and helpful suggestions. This work was supported in part by the Shanghai Rising-Star Program (08QA14030), Shanghai Educational Development Foundation (2007CG52), and the Shanghai Leading Academic Discipline Program (T0104).

which depends on the applied electric ?eld, as shown in Fig. 5. It is shown that the distance D will be enhanced with the applied electric ?eld increasing. Using this phenomena, the high e?cient ultra-compact polarization beam splitter can be achieved by lateral shifts modulated by the applied electric ?eld. In conclusion, we investigate the tunable lateral shifts and polarization beam splitter based on the Pockels e?ect in cubic and tetragonal electro-optic crystals. It is found that the large positive and negative lateral shifts of TM polarized beam can be controlled by adjusting the permittivity tensor, which is modulated by external electric ?eld. The controllable lateral shift is also related to the crystal cuts, since the electro-optic coe?cient depends on its symmetry of point group. As a matter fact, the lateral shifts of TE and TM polarized beams can also be simultaneously controlled by electric ?eld in other types

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