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Imbert–Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media
Annals of Physics 335 (2013) 33–46
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Imbert–Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media Guoding Xu ∗ , Yuting Xiao, Jun Li, Hongmin Mao, Jian Sun, Tao Pan Department of Physics, Suzhou University of Science and Technology, Suzhou 215009, PR China
highlights • • • • •
We discuss the IF shifts of a beam reflected from a uniaxially anisotropic chiral medium interface. The shifts can abruptly change near the critical angle of incidence. The cross-polarized reflection coefficient has a greater effect on the spatial shift than on the angular shift. The sign of the angular shift depends only on that of the difference between the co-polarized reflectivity. The crossovers of the shifts can be realized between positive and negative.
article
info
Article history: Received 2 March 2013 Accepted 23 April 2013 Available online 4 May 2013 Keywords: Imbert–Fedorov shifts Gaussian beam Uniaxially anisotropic chiral medium Reflection matrix
∗
abstract We study the Imbert–Fedorov (IF) shifts of a reflected Gaussian beam from uniaxially anisotropic chiral media (UACM), where the chirality appears only in one direction and the host medium is a uniaxial crystal or an electric plasma. The numerical results are presented for three kinds of UACM, respectively. It is found that the IF shifts are closely related to the propagation properties of the two eigenwaves in the UACM. In general, when either of the eigenwaves is totally reflected, the IF shifts can change abruptly near the critical angle. The cross-polarized reflection coefficient has a greater effect on the spatial IF (SIF) shift than on the angular IF (AIF) shift, and the sign of the AIF shift depends mainly on that of the difference between the co-polarized reflectivity. By designing artificially the electromagnetic parameters of the UACM, we can control the IF shifts and acquire their more abundant properties. © 2013 Elsevier Inc. All rights reserved.
Corresponding author. Tel.: +86 512 68181239; fax: +86 512 68418473. E-mail address: [email protected] (G. Xu).
0003-4916/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aop.2013.04.015
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
1. Introduction It has been long known that the behavior of a bound light beam on reflection and transmission at a dielectric interface differs from the geometric-optics prediction. The beam can experience a shift parallel or perpendicular to the plane of incidence, which stems from diffractive corrections [1]. The parallel shift is known as the Goos–Hänchen (GH) effect [2] and the transverse one as the Imbert–Fedorov (IF) effect [3]. The two effects were originally investigated in total internal reflection, but recent research indicates that they can also appear in partial dielectric reflection or transmission [4–8]. For a linearly polarized incident beam, the IF effect is also regarded as the spin Hall effect of light (SHEL) because of its resemblance to the electron spin Hall effect, where the spin of electrons is replaced by the spin of photons (i.e., polarization) and the electric potential gradient by the refractive index gradient [4,5,8]. Recently, the SHEL has been successfully explained in theory [9,10] and verified in experiment via quantum weak measurements [4]. The effect is very tiny but detectable polarization-dependent shift, and can find its potential applications in metrology. Therefore, a number of studies have been performed for different beam geometries and various interface circumstances, including air–glass interface [4,11], metamaterial interfaces [8,12], isotropic–uniaxial interfaces [13,14], and anisotropic interfaces [15–17], etc. As is known, the IF shifts are closely related to the optical properties of an interface, which depend on the electromagnetic parameters of materials constituting the interface, such as the permittivity and the permeability. Consequently, a suitable material with distinctive electromagnetic properties might offer more abundant behaviors of the IF shifts. In this respect, a chiral medium might be a desirable candidate. Chiral media have been known and studied from the early nineteenth century in many fields, such as optics, chemistry, particle physics, life sciences and material sciences [18]. Because of their unique properties, availability, and potential diverse applications in wave propagation, radiation, guidance and scattering, such media have received a lot of attention in the past years. At microwave frequencies, the chiral media can be realized artificially by using miniature wire spirals or conducting springs, which provide additional interaction of electric and magnetic fields inside the medium. Thus, the media can possess artificially tunable electromagnetic properties which are not displayed by a conventional isotropic medium. Very recently, Wang et al. [19] have studied the SHEL in the isotropic chiral metamaterials; their findings show that not only the spin-dependent displacements of the reflected beam can reach several tens of wavelengths at certain angle of incidence, but also the reversed effect for the transmitted beams can be realized by tuning the chirality parameter. In fact, the previous many studies have only focused on isotropic chiral media [17,20–22], which are, however, very difficult to obtain in practice because the chiral particles are usually anisotropic. Recently, a uniaxially anisotropic chiral medium (UACM) has been investigated [23], where the chirality appears only in one direction and the host medium is a uniaxial crystal or an electric plasma. Such a chiral medium is obviously easy to fabricate artificially; in particular, its electromagnetic parameters, such as permittivity, permeability and chirality parameter, can be designed in a desired way. It was shown that negative phase or/and group refractions may be acquired in both eigenwaves for different electromagnetic parameters [23]. Thanks to the flexibility provided by the UACM in design, we thus believe that it is possible to control the IF shifts by use of the UACM. This paper is organized as follows. In Section 2, using an eigenvalue method, we derive the reflection matrix of the electromagnetic wave on the dielectric–UACM interface within the framework of the 4 × 4 matrix [24–28]. Based on the reflection matrices, the expressions are given for the spatial IF (SIF) shift and the angular IF (AIF) shift. In Section 3, we examine the effects of the propagation behaviors of the two eigenwaves in the UACM on the IF shifts, and discuss numerically the dependence of the IF shifts on the angle of incidence for three different sets of electromagnetic parameters, which provide some interesting electromagnetic properties. Finally, we summarize the main conclusions in Section 4. 2. Basic theory 2.1. The reflection matrix The IF shifts of a spatially bound light beam have been discussed by virtue of different methods. Physically, the most satisfying approach perhaps uses the fundamental law of conservation of angular
G. Xu et al. / Annals of Physics 335 (2013) 33–46
35
momentum [9,10,29,30]. Based on Noether’s theorem, the total (spin and orbital) angular momentum must be conserved along the axis of symmetry of the system which is, for an obliquely incident beam on a planar optical interface, the normal to the interface. To conserve total angular momentum, the centroids of the reflected and transmitted components of an incident circularly polarized beam undergo the shifts along the direction perpendicular to the plane of incidence. A more typical method to determine the IF shifts consists in decomposing the incident, reflected, and transmitted beams of finite cross section into their plane wave constituents and applying the Fresnel relations to the s and p components of the individual plane waves. The spatial dependence of the reflected and transmitted beams is retrieved by summing over all partial plane waves which can be done analytically by using the paraxial approximation [4,6,31–33]. As is known, in a general anisotropic or chiral medium s and p waves are not the eigenmodes of Maxwell’s equations, so that the reflected wave must be a combination of the perpendicular and parallel components so as to meet the boundary conditions. Hence, we must use the so-called reflection matrix to describe the reflection of a wave. Below, we first derive the reflection matrix. We should emphasize here that based on the Fourier analysis, the incoming Gaussian beam can be expressed as the superposition of a continuous spectrum of uniform plane wave components propagating in a wide range of directions, and therefore the reflection matrix can be derived by the usual plane-wave method. Let an isotropic medium of refractive index n and the UACM occupy the z < 0 and the z > 0 regions, respectively, as shown in Fig. 1(a). Consider the central a plane-wave component of the Gaussian beam of angular frequency ω impinging on the z = 0 interface at the angle of incidence θ from the isotropic medium side. Clearly, the z-axis is normal to the interface. Assuming O − xz is the plane of incidence, then the fields can be expressed as F = F(z )ei(kx x−ωt ) , where F represents the electric field E or the magnetic field H; kx = k0 n sin θ and k0 = ω/c. It is known that the electromagnetic response of the UACM is still described by the ordinary Maxwell’s equations, but the constitutive relations which define the electric displacement D and the magnetic induction B are written as [23] ↔
D = ε0 ( ε ·E + iγ zˆ zˆ · η0 H),
(1a)
↔ √ (1b) B = ε0 µ0 (µ ·η0 H − iγ zˆ zˆ · E), √ where η0 = µ0 /ε0 is the wave impedance in vacuum and γ is the chirality parameter. Eqs. (1) imply
that the chiral particles (wire spirals, copper springs or Swiss-roll array) are aligned with the z-axis. ↔
For a nonmagnetic uniaxial host medium, the permeability tensor µ is the 3 × 3 identity matrix, and ↔
the permittivity tensor ε can be cast in the form ↔
ε=
εt 0 0
0
εt 0
0 0
(2)
εz ,
where εt and εz are the permittivity in the transverse and the longitudinal directions, respectively, and they can be either positive or negative, depending on the host medium. According to the continuity of the tangential field components across the boundary, we can introduce a four-component column vector ψ(z ) = (Ex , Ey , η0 Hx , η0 Hy )T , where the superscript ‘T ’ stands for the transpose operator. From Maxwell’s equations and Eqs. (1) and (2), it follows that ψ obeys the matrix ordinary differential equation dψ(z )
= ik0 M ψ(z ),
dz
(3)
where M is a 4 × 4 matrix in the form
0 0 M = 0
εt
−i γ a 0 aε z − ε t 0
0 −1 0 0
1−a 0 , −iγ a 0
(4)
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
b
Fig. 1. (a) Schematic of a light beam reflection on the dielectric–UACM interface. (b) An illustration of the IF shifts. Here, the dashed line with arrow denotes the reflected beam predicted by the geometric optics, and the solid line with arrow stands for the practical reflected beam. yr , Θr and Y = yr + ℓΘr represent the SIF, the AIF and the total IF shifts of a reflected beam, respectively. In this figure, both yr and Θr are positive.
with a = n2 sin2 θ /(εz −γ 2 ). After some lengthy algebra, the eigenvalue equation of M can be derived as
(q2 − εt + aεz )(q2 − εt + aεt ) = εt a2 γ 2 ,
(5)
which actually defines the dispersion relations of two eigenwaves in the UACM q2
εt
+
with δ± =
k2x δ± k20
(εz − γ 2 )
1+
εz εt
±
= 1,
(6)
εt sign( ε −γ 2) z
(1 −
εz 2 ) εt
+
4γ 2
εt
/2. The dispersion relations described by Eq.
(6) may be elliptical curve, one-sheet hyperbola or two-sheet hyperbola, depending on the sign and magnitude of the parameters εt , εz and γ . If the four solutions of Eq. (5) or (6), i.e., the eigenvalues of M, are denoted by qℓ (ℓ = 1 ∼ 4), then k0 qℓ are the propagation constants in the z-direction in the UACM. Let q1 and q2 correspond to the two forward waves (eigenwaves), respectively, and then q3 = −q1 and q4 = −q2 to the two backward (or reflected) waves. In this way, we can write q1 = ± εt (1 − aδ+ ),
q2 = ± εt (1 − aδ− ),
(7)
where the positive or the negative square root tied to q1 and q2 is chosen by examining the direction of the energy flow carried by the two waves in the UACM. For convenience, we hereafter label the two eigenwaves as q1 wave and q2 wave. They can either propagate in the UACM or be totally reflected, depending on the parameters εt , εz , γ and the angle of incidence. The critical angles of incidence for the two eigenwaves, if they exist, can be determined by putting q1 = 0 or q2 = 0. This yields sin θc ± = (εz − γ 2 )/δ± /n, respectively. The general solution of Eq. (3) is straightforward and expressible as [24–28]
ψ(z ) = Veik0 Qz ψ0 ,
(8)
where ψ0 is a 4 × 1 constant column vector determined by the boundary conditions; Q and V are the 4 × 4 matrices that consist of the four eigenvalues and eigenvectors of M, respectively. For the
G. Xu et al. / Annals of Physics 335 (2013) 33–46
37
geometry shown in Fig. 1(a), we arrange the eigenvalues and eigenvectors such that the first two columns correspond to the reflected waves and the last two columns to the transmitted waves. Then, we have
−Q4
Q =
O Q4
O
,
V =
V1 V3
V2 V4
,
(9)
where the 2 × 2 sub-matrices Q4 = diag(q1 , q2 ), O consists of four zeros and iγ q2
1
V1 =
V2 =
1
q1 (1 − δ− ) iγ
q2
1 − δ− εt −
V3 =
−
q1
iγ
−
V4 =
1 − δ−
iγ
q1
iγ q2
εt (1 − δ− ) ,
(10a)
1
q1 (1 − δ− )
iγ
1 − δ− εt
,
−
1
εt (1 − δ− ) ,
iγ
−q2 −
iγ
.
(10b)
1 − δ−
It should be noted that Eqs. (10) still hold for the isotropic medium lying in z < 0 region. Therefore, we can acquire, only setting γ = 0 and εt = εz = n2 , the V -matrix of the isotropic medium
I2 V0
Vn =
I2 −V0
,
(11)
where I2 is the 2 × 2 identity matrix and n cos θ 0
0 −n cos−1 θ
V0 =
.
(12)
At the z = 0 interface, the boundary conditions imply ψ(0− ) = ψ(0+ ), which yields [27]
I2 V0
I2 −V0
Rxy I2
=
V1 V3
V2 V4
O Txy
,
(13)
where the 2 × 2 matrices O consists of four zeros and, Rxy and Txy are defined, respectively, by Ex(r )
= Rxy
Ey(r ) (j)
Ex(i)
,
Ey(i)
Ex(t )
Ey(t )
= Txy
Ex(i) Ey(i)
,
(14)
(j)
where Ex , Ey (j = i, r , t ) are the tangential electric field components, and i, r , t stand for the incident, reflected, transmitted waves, respectively. From Eq. (13), we derive Rxy = (V0 − V4 V2−1 )−1 (V0 + V4 V2−1 ).
(15)
Usually, the reflection matrix R is defined as
R=
rpp rsp
rps rss
,
(16)
satisfying
Ep(r ) Es(r )
=R
Ep(i) Es(i)
,
(17)
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
where rℓm is the ratio of the amplitudes of a reflected field with ℓ polarization to an incident field with m polarization, and p and s denote the transverse magnetic (TM) and the transverse electric (TE) polarizations, respectively. Referring to Fig. 1(a), we readily obtain
Ex(i)
Ey(i)
Ep(i)
Ex(r )
= Ui
Es(i)
cos θ 0
0 , 1
;
Ey(r )
= Ur
− cos θ
Ep(r )
Es(r )
,
(18)
with
Ui =
Ur =
0 . 1
0
(19)
From Eqs. (14), (17) and (18), we have the reflection matrix R = Ur−1 Rxy Ui .
(20)
After some derivations, the elements of R are determined by
rpp = −
(n cos θ + gq2 ) n cos−1 θ − g qε1t − m2 εt2 (1 − δ− )2 ∆
rsp = −rps =
rss =
2inmεt (1 − δ− )
∆
,
(21a)
,
(21b)
(n cos θ − gq2 ) n cos−1 θ + g qε1t − m2 εt2 (1 − δ− )2 ∆
,
(21c) γ (1−q /q )
2 1 with ∆ = (n cos θ + gq2 )(n cos−1 θ + g qεt ) + m2 εt2 (1 − δ− )2 and m = ε (1−δ )2 +γ 2 q /q , g = 1 + γ m. 1 t − 2 1 It is easy to show that if the UACM is replaced by another isotropic medium, Eqs. (21) can, by setting γ = 0, εt = εz = n′2 and then m = 0, g = 1, reduce to the familiar Fresnel reflection coefficients on the interface delimiting two isotropic media.
2.2. The IF shifts With the elements of reflection matrix given above, we can present the expressions for the SIF shift yr and the AIF shift Θr . Usually, Θr ≪ 1; thus the total IF shift observed at distance ℓ from the actual reflecting point can be expressed as a linear combination of yr and Θr , viz., Y = yr + ℓΘr (see Fig. 1(b)), where yr and Θr are given in the paraxial approximation, respectively, by [17] yr =
Γ k0 S
cot θ ,
(22)
and
Θr =
1N 2 S
θ02 ,
(23)
with
Γ = Im[fp fs∗ (|rpp |2 + |rss |2 + 2rpp rss∗ + 4|rsp |2 ) ∗ ∗ + (rpp + 3rss )rsp + 2(rpp rsp + rss∗ rsp )|fp |2 ], 2 2 ∗ S = rpp fp + |rss fs |2 + rsp − 2 Re[fp fs∗ (rpp rsp − rss∗ rsp )],
(24b)
∗ N = Re[fp fs∗ (|rpp |2 − |rss |2 ) + (|fp |2 − |fs |2 )(rpp rsp − rss∗ rsp )],
(24c)
(24a)
and where Im(. . .) and Re(. . .) mean the imaginary and the real parts of a complex number, respectively; θ0 = 2/(k0 w0 ) is the angular spread of the incident Gaussian beam of the waist w0 ; fp and fs are
G. Xu et al. / Annals of Physics 335 (2013) 33–46
39
2 associated with the complex-valued unit vector fˆ = fp xˆ i + fs yˆ i with fp + |fs |2 = 1 denoting the orientation of the polarizer, which is put perpendicularly to the central wave vector k0 of the incident beam, and the asterisk (∗) marks the complex conjugate. A positive (negative) value of yr implies that the SIF shift occurs in the positive (negative) y-axis direction, while a positive (negative) value of Θr implies that the AIF shift slants towards the positive (negative) y-axis direction, as illustrated in Fig. 1(b). In what follows, by ‘the IF shifts’, we mean both the SIF shift and the AIF shift. We can see from Eqs. (22) to (24) that in an anisotropic medium, besides the co-polarized
2
reflectivity and reflection coefficients, the cross-polarized reflectivity rsp and reflection coefficients rsp play a role in the IF shifts, especially in the SIF shift, as compared with the isotropic case where rsp = rps = 0. 3. Numerical results and discussion Here, bearing on three kinds of UACM, which have different but easy fabricated electromagnetic parameters [23], we will discuss the effects of the propagation behaviors of the two eigenwaves in the UACM on the IF shifts for different electromagnetic parameters. Throughout the paper, we use right√ elliptically polarized incident beams, denoted by (fp , fs ) = (1, 1 + i)/ 3, and other incident beams, such as linearly and circularly polarized beams, can be tackled similarly. Assuming the z < 0 region is free space, then n = 1. As is known, the paraxial approximation requires that the transverse size of a beam is much greater than its wavelength λ [34]; as a consequence, in the following numerical calculations we choose the waist of incident beam w0 = 50λ. 3.1. Case I: εt > 0, εz > 0 In this case, the host medium is a conventional right-handed material. First, we assume γ 2 < εz , so δ± are always positive. Then, the dispersion relations for the two eigenwaves are both elliptical curves, which can be determined from Eq. (6). When the angle of incidence θ < min(θc + , θc − ), both the eigenwaves can propagate in the UACM; otherwise, both or one of the eigenwaves becomes evanescent. Such situations are illustrated in Fig. 2, where εt = 3, εz = 2 and γ = 0.5 for 2(a)–(d) and γ = 1.2 for 2(e)–(h). For γ = 0.5, since neither θc + nor θc − exists, the two eigenwaves can propagate
2 in the UACM; in the situation, we get the cross-polarized reflectivity rsp ≈ 0 due to the weak
2
chirality and a pseudo-Brewster angle near θ = 65.8° for p-polarized wave, where rpp ≈ 0 (see Fig. 2(b)). These results indicate that the two eigenwaves propagate just as in a conventional uniaxial medium. Hence, we reckon that the variation of IF shifts with the angle of incidence should also be similar to that in a uniaxial medium [11,35], as is numerically verified in Fig. 2(c) and (d). In addition, we find from Fig. 2(c) and (d) that the SIF shift is negative and attains its maximum near the pseudo∗ − rss∗ rsp is Brewster angle; the AIF shift is also negative, which is due to the fact that the quantity rpp rsp
2
usually tiny, so the sign of the AIF shift relies mainly on that of the difference rpp − |rss |2 (see (24c)). For γ = 1.2, the q2 wave is propagative, but the q1 wave is totally reflected when θ > θc + (≈37.0°). We can remark from Fig. 2(g) and (h) that both the SIF and the AIF shifts attain their positive maxima at the critical angle θc + . In particular, the SIF and AIF shifts can switch between positive and negative values, which differs from the case γ = 0.5. The results arise evidently from the change in the propagation properties of the q1 wave in the UACM, which leads to the abrupt change in the properties 2 of reflected wave, for example, the cross-polarized reflectivity rsp is significantly enhanced, and
2 |rss |2 exceeds rpp in magnitude. When γ 2 > εz , i.e., the chirality becomes stronger, we always have δ+ < 0 and δ− > 0. Therefore, the dispersion curve for the q1 wave is an ellipse, while that for the q2 wave is a two-sheet hyperbola, implying that the q2 wave always propagates for an arbitrary angle of incidence, and the q1 wave is totally reflected for θ ≥ θc + . This situation is plotted in Fig. 2(i)–(l), where γ = 1.5 and εt and εz remain unchanged. For such parameters, since θc + does not exist, both the eigenwaves are propagative. In consequence, the behaviors of the reflected wave are quite analogous to that in the case γ = 0.5. However, the IF shifts, particularly the SIF shift, are very dissimilar to the case γ = 0.5,
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
i
b
f
j
c
g
k
d
h
l
Fig. 2. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 3, εz = 2 and γ = 0.5 for (a)–(d), γ = 1.2 for (e)–(h) and γ = 1.5 for (i)–(l). (a), (e), (i) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b), (f), (j)
2
2
The reflectivity. The solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g), (k) The SIF shift and (d), (h), (k) the AIF shift (in s).
as can seen by comparing Fig. 2(k) with Fig. 2(c). This difference might be attributed to the role that
2 rsp plays in the situation. Therefore, it is believed that the cross-polarized reflection coefficient rsp has a greater effect on the SIF shift than on the AIF shift. 3.2. Case II: εt > 0, εz < 0 When the host medium contains resonant dipoles directed only to the z direction, the UACM has
εt > 0, εz < 0 [23]. We easily get δ+ < 0 and δ− > 0, and then the dispersion curves for the q1
wave and the q2 wave are an ellipse and a two-sheet hyperbola, respectively. This indicates that the q2 wave always propagates for an arbitrary angle of incidence, and the q1 wave is totally reflected for θ ≥ θc + . In this sense, the case seems to be similar to that for εt > 0, εz > 0 and γ 2 > εz . However, owing to very distinct physical parameters, the propagation properties and the IF shifts of the reflected
G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
b
f
c
g
d
h
41
Fig. 3. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 0.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
wave should be dissimilar. To examine this, we depict the case in Fig. 3, where εt = 0.3, εz = −2 and, γ = 0.8 for Fig. 3(a)–(d) and γ = 2.8 for 3(e)–(h). For γ = 0.8, we can see from Fig. 3(b)–(d) that the properties and the IF shifts of the reflected wave all change abruptly at the critical angle of
2
incidence for the q1 wave θc + = 38.1°. Since |rss |2 is always larger than rpp for an arbitrary angle of incidence, the AIF shift is negative with its maximum at θc + = 38.1°. For γ = 2.8, θc + does not
2
exist, and thus both the eigenwaves are propagative in the situation. Because of rpp > |rss |2 for any angle of incidence, the AIF shift is always positive, as shown in Fig. 3(f) and (h). Furthermore,
2
when θ > 60°, rsp is observable; consequently, in the range of the angles of incidence, the SIF shift exhibits remarkable changes which are absent in a conventional isotropic medium.
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
As is known, the electromagnetic parameters affect the propagation properties of the two eigenwaves in the UACM and thus affect the IF shifts. To this end, we alter εt and keep εz and γ unchanged to explore the IF shifts. The situation is plotted in Fig. 4, where we use εt = 1.3 instead of 0.3 in Fig. 3, γ = 0.8 (for Fig. 4(a)–(d)) and γ = 2.8 (for Fig. 4(e)–(h)). In the calculation, we find that for either of the two γ values, θc + does not exist, and thus both the eigenwaves are propagative; the SIF shifts change similarly. Specifically, the SIF shifts are positive for a smaller θ and negative for a larger θ . Additionally, the sign of the AIF shift can be still interpreted in the same way as above. Obviously, the result is distinct from the situation in Fig. 3, where neither the SIF shift nor the AIF shift can switch between positive and negative values. 3.3. Case III: εt < 0, εz > 0 When the host medium contains resonant dipoles directed towards the transverse directions, the UACM possesses εt < 0 and εz > 0 [23]. The case can provide very different propagation properties for varied electromagnetic parameters. First, for the weak chirality, i.e., γ 2 < εz , we can obtain δ+ < 0 and δ− > 0, which means that q1 is purely imaginary, viz., the q1 wave is evanescent, and the dispersion curve for the q2 wave is a one-sheet hyperbola. We should stress here that only when θ ≤ θc − in the situation, can the q2 wave be totally reflected, as is quite different from the usual situation. Moreover, to meet the conversation law of energy both q1 and q2 are here taken the negative square roots in Eq. (7). Aiming at γ = 0.3 and 1.3, respectively, we depict the situation in Fig. 5 for εt = −3, εz = 2. For γ = 0.3, as shown in Fig. 5(a)–(d), it is clear that θc − does not exist, and hence both the eigenwaves are evanescent, implying that the q1 wave and the q2 wave are totally reflected for an arbitrary angle of incidence. The
2
2
2
2
result is confirmed in Fig. 5(b), where |rss |2 + rps = 1 and rpp + rsp = 1. Owing to rsp ≈ 0,
2
the co-polarized reflectivity |rss |2 and rpp are nearly identical, so that the AIF shift is roughly null, which can be seen in Fig. 5(d). We find from Fig. 5(c) that for θ < 70°, the SIF shift is positive with its maximum near θ = 40°. Next, for γ = 1.3, Fig. 5(e) and (f) show that when θ ≤ θc − (=49.97°), the q2 wave is totally reflected, thus leading to the abrupt changes in the reflectivity near θc − . In particular,
2
as compared with other above cases, rsp here becomes quite appreciable. These might be the factors that induce the complicated changes in the SIF shift which, e.g., crosses twice between positive and negative values. Similarly, the change in the AIF shift can be reasonably accounted for. When the chirality becomes stronger, i.e., γ 2 > εz , there exist three situations, which are discussed, respectively, as follows.
(i) When 1 + εz /εt ≥ 2 (εz − γ 2 )/εt , δ± are always positive. Correspondingly, both q1 and q2 are purely imaginary and taken the negative square roots (see Eq. (7)), namely, the two eigenwaves are totally reflected. The situation is plotted in Fig. 6(a)–(d), where εt = −3, εz = 2 and γ = 1.43. As 2 2 2 2 expected, Fig. 6(b) shows up the total reflection: rpp + rsp = 1 (note that |rss |2 = rpp , rsp =
2 rps here). Therefore, the AIF is nearly null. In addition, we get the negative SIF shift in the situation. (ii) When −2 (εz − γ 2 )/εt < 1 + εz /εt < 2 (εz − γ 2 )/εt , δ± are complex, so that both q1
and q2 are complex and taken the negative square roots. We plot the situation in Fig. 6(e)–(h) for εt = −3, εz = 2 and γ = 1.6. Clearly, both the SIF and the AIF shifts are enormous near θ = 0° (i.e., normal incidence), but vanish rapidly with increasing θ .
(iii) When 1 + εz /εt ≤ −2 (εz − γ 2 )/εt , δ± are always negative. The dispersion curves for the two eigenwaves are both one-sheet hyperbolae. When θ ≤ θc + (θc − ), the q1 (q2 ) is totally reflected. Then, the angle of incidence can be divided into three intervals: θ ≤ min(θc + , θc − ), in which both the eigenwaves are totally reflected; min(θc + , θc − ) < θ < max(θc + , θc − ), where one eigenwave is propagative, and the other is evanescent, and θ ≥ max(θc + , θc − ) where both the eigenwaves are propagative. Notably, q1 is here taken the negative square root, while q2 taken the positive square root. Such a situation is illustrated in Fig. 6(i)–(l) for εt = −1.2, εz = 2 and γ = 1.45. We can see that the reflectivity curves and the AIF shift are also divided into three parts corresponding, respectively, to the above three intervals. Only in the interval min(θc + , θc − ) < θ < max(θc + , θc − ), is the AIF shift
G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
b
f
c
g
d
h
43
Fig. 4. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 1.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
observable, as seen in Fig. 6(l). The SIF shift is similar to that in case (ii), that is, it is enormous near θ = 0°, but vanishes quickly with increasing θ . 4. Conclusion In this work, we first derive the reflection matrix of a plane electromagnetic wave (or saying the central plane-wave component of a Gaussian beam) on the dielectric–UACM interface by virtue of the eigenvalue method of a 4 × 4 matrix. Based on the reflection matrix, the IF shifts of a Gaussian beam
44
G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
b
f
c
g
d
h
Fig. 5. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 0.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
reflected from the UACM have been numerically discussed. The numerical results have been presented for three kinds of UACM: I εt > 0, εz > 0, II εt > 0, εz < 0 and III εt < 0, εz > 0. In every case, for different electromagnetic parameters which may induce the interesting propagation properties of the two eigenwaves in the UACM, we have discussed some possible behaviors of the IF shifts. After examining these results, we have found that in general, the IF shifts can abruptly change near the critical angle at which either of the eigenwaves is totally reflected. The cross-polarized reflection coefficient has a greater effect on the SIF shift than on the AIF shift, and the sign of the AIF shift depends
2
only on that of the difference rpp − |rss |2 . By adjusting the angle of incidence or the electromagnetic parameters of the UACM, such as permittivity, permeability and chirality, the crossovers of the IF shifts
G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
i
b
f
j
c
g
k
d
h
l
45
Fig. 6. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = −3, εz = 2 and γ = 1.43 for (a)–(d), γ = 1.6 for (e)–(h) and εt = −1.2, εz = 2, γ = 1.45 for (i)–(l). (a), (e), (i) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 ,
2
2
respectively. (b), (f), (j) The reflectivity. The solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g), (k) The SIF shift and (d), (h), (k) the AIF shift (in s).
between positive and negative values can be realized. In short, the IF shifts are closely related to the propagation properties of the two eigenwaves in the UACM, while the propagation properties depend greatly on the electromagnetic parameters. Thus, due to its flexibility in design the UACM opens up possibilities to control the IF shifts. Acknowledgments We would like to express our gratitude to the referee, whose searching questions and suggestions helped us elaborate and improve the manuscript. This work is supported by the ‘Qinglan’ Project of Jiangsu Province. References [1] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley & Sons, New York, 1998. [2] F. Goos, H. Hänchen, Ann. Phys. (Leipzig) 1 (1947) 333.
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Annals of Physics journal homepage: www.elsevier.com/locate/aop
Imbert–Fedorov shifts of a Gaussian beam reflected from uniaxially anisotropic chiral media Guoding Xu ∗ , Yuting Xiao, Jun Li, Hongmin Mao, Jian Sun, Tao Pan Department of Physics, Suzhou University of Science and Technology, Suzhou 215009, PR China
highlights • • • • •
We discuss the IF shifts of a beam reflected from a uniaxially anisotropic chiral medium interface. The shifts can abruptly change near the critical angle of incidence. The cross-polarized reflection coefficient has a greater effect on the spatial shift than on the angular shift. The sign of the angular shift depends only on that of the difference between the co-polarized reflectivity. The crossovers of the shifts can be realized between positive and negative.
article
info
Article history: Received 2 March 2013 Accepted 23 April 2013 Available online 4 May 2013 Keywords: Imbert–Fedorov shifts Gaussian beam Uniaxially anisotropic chiral medium Reflection matrix
∗
abstract We study the Imbert–Fedorov (IF) shifts of a reflected Gaussian beam from uniaxially anisotropic chiral media (UACM), where the chirality appears only in one direction and the host medium is a uniaxial crystal or an electric plasma. The numerical results are presented for three kinds of UACM, respectively. It is found that the IF shifts are closely related to the propagation properties of the two eigenwaves in the UACM. In general, when either of the eigenwaves is totally reflected, the IF shifts can change abruptly near the critical angle. The cross-polarized reflection coefficient has a greater effect on the spatial IF (SIF) shift than on the angular IF (AIF) shift, and the sign of the AIF shift depends mainly on that of the difference between the co-polarized reflectivity. By designing artificially the electromagnetic parameters of the UACM, we can control the IF shifts and acquire their more abundant properties. © 2013 Elsevier Inc. All rights reserved.
Corresponding author. Tel.: +86 512 68181239; fax: +86 512 68418473. E-mail address: [email protected] (G. Xu).
0003-4916/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.aop.2013.04.015
34
G. Xu et al. / Annals of Physics 335 (2013) 33–46
1. Introduction It has been long known that the behavior of a bound light beam on reflection and transmission at a dielectric interface differs from the geometric-optics prediction. The beam can experience a shift parallel or perpendicular to the plane of incidence, which stems from diffractive corrections [1]. The parallel shift is known as the Goos–Hänchen (GH) effect [2] and the transverse one as the Imbert–Fedorov (IF) effect [3]. The two effects were originally investigated in total internal reflection, but recent research indicates that they can also appear in partial dielectric reflection or transmission [4–8]. For a linearly polarized incident beam, the IF effect is also regarded as the spin Hall effect of light (SHEL) because of its resemblance to the electron spin Hall effect, where the spin of electrons is replaced by the spin of photons (i.e., polarization) and the electric potential gradient by the refractive index gradient [4,5,8]. Recently, the SHEL has been successfully explained in theory [9,10] and verified in experiment via quantum weak measurements [4]. The effect is very tiny but detectable polarization-dependent shift, and can find its potential applications in metrology. Therefore, a number of studies have been performed for different beam geometries and various interface circumstances, including air–glass interface [4,11], metamaterial interfaces [8,12], isotropic–uniaxial interfaces [13,14], and anisotropic interfaces [15–17], etc. As is known, the IF shifts are closely related to the optical properties of an interface, which depend on the electromagnetic parameters of materials constituting the interface, such as the permittivity and the permeability. Consequently, a suitable material with distinctive electromagnetic properties might offer more abundant behaviors of the IF shifts. In this respect, a chiral medium might be a desirable candidate. Chiral media have been known and studied from the early nineteenth century in many fields, such as optics, chemistry, particle physics, life sciences and material sciences [18]. Because of their unique properties, availability, and potential diverse applications in wave propagation, radiation, guidance and scattering, such media have received a lot of attention in the past years. At microwave frequencies, the chiral media can be realized artificially by using miniature wire spirals or conducting springs, which provide additional interaction of electric and magnetic fields inside the medium. Thus, the media can possess artificially tunable electromagnetic properties which are not displayed by a conventional isotropic medium. Very recently, Wang et al. [19] have studied the SHEL in the isotropic chiral metamaterials; their findings show that not only the spin-dependent displacements of the reflected beam can reach several tens of wavelengths at certain angle of incidence, but also the reversed effect for the transmitted beams can be realized by tuning the chirality parameter. In fact, the previous many studies have only focused on isotropic chiral media [17,20–22], which are, however, very difficult to obtain in practice because the chiral particles are usually anisotropic. Recently, a uniaxially anisotropic chiral medium (UACM) has been investigated [23], where the chirality appears only in one direction and the host medium is a uniaxial crystal or an electric plasma. Such a chiral medium is obviously easy to fabricate artificially; in particular, its electromagnetic parameters, such as permittivity, permeability and chirality parameter, can be designed in a desired way. It was shown that negative phase or/and group refractions may be acquired in both eigenwaves for different electromagnetic parameters [23]. Thanks to the flexibility provided by the UACM in design, we thus believe that it is possible to control the IF shifts by use of the UACM. This paper is organized as follows. In Section 2, using an eigenvalue method, we derive the reflection matrix of the electromagnetic wave on the dielectric–UACM interface within the framework of the 4 × 4 matrix [24–28]. Based on the reflection matrices, the expressions are given for the spatial IF (SIF) shift and the angular IF (AIF) shift. In Section 3, we examine the effects of the propagation behaviors of the two eigenwaves in the UACM on the IF shifts, and discuss numerically the dependence of the IF shifts on the angle of incidence for three different sets of electromagnetic parameters, which provide some interesting electromagnetic properties. Finally, we summarize the main conclusions in Section 4. 2. Basic theory 2.1. The reflection matrix The IF shifts of a spatially bound light beam have been discussed by virtue of different methods. Physically, the most satisfying approach perhaps uses the fundamental law of conservation of angular
G. Xu et al. / Annals of Physics 335 (2013) 33–46
35
momentum [9,10,29,30]. Based on Noether’s theorem, the total (spin and orbital) angular momentum must be conserved along the axis of symmetry of the system which is, for an obliquely incident beam on a planar optical interface, the normal to the interface. To conserve total angular momentum, the centroids of the reflected and transmitted components of an incident circularly polarized beam undergo the shifts along the direction perpendicular to the plane of incidence. A more typical method to determine the IF shifts consists in decomposing the incident, reflected, and transmitted beams of finite cross section into their plane wave constituents and applying the Fresnel relations to the s and p components of the individual plane waves. The spatial dependence of the reflected and transmitted beams is retrieved by summing over all partial plane waves which can be done analytically by using the paraxial approximation [4,6,31–33]. As is known, in a general anisotropic or chiral medium s and p waves are not the eigenmodes of Maxwell’s equations, so that the reflected wave must be a combination of the perpendicular and parallel components so as to meet the boundary conditions. Hence, we must use the so-called reflection matrix to describe the reflection of a wave. Below, we first derive the reflection matrix. We should emphasize here that based on the Fourier analysis, the incoming Gaussian beam can be expressed as the superposition of a continuous spectrum of uniform plane wave components propagating in a wide range of directions, and therefore the reflection matrix can be derived by the usual plane-wave method. Let an isotropic medium of refractive index n and the UACM occupy the z < 0 and the z > 0 regions, respectively, as shown in Fig. 1(a). Consider the central a plane-wave component of the Gaussian beam of angular frequency ω impinging on the z = 0 interface at the angle of incidence θ from the isotropic medium side. Clearly, the z-axis is normal to the interface. Assuming O − xz is the plane of incidence, then the fields can be expressed as F = F(z )ei(kx x−ωt ) , where F represents the electric field E or the magnetic field H; kx = k0 n sin θ and k0 = ω/c. It is known that the electromagnetic response of the UACM is still described by the ordinary Maxwell’s equations, but the constitutive relations which define the electric displacement D and the magnetic induction B are written as [23] ↔
D = ε0 ( ε ·E + iγ zˆ zˆ · η0 H),
(1a)
↔ √ (1b) B = ε0 µ0 (µ ·η0 H − iγ zˆ zˆ · E), √ where η0 = µ0 /ε0 is the wave impedance in vacuum and γ is the chirality parameter. Eqs. (1) imply
that the chiral particles (wire spirals, copper springs or Swiss-roll array) are aligned with the z-axis. ↔
For a nonmagnetic uniaxial host medium, the permeability tensor µ is the 3 × 3 identity matrix, and ↔
the permittivity tensor ε can be cast in the form ↔
ε=
εt 0 0
0
εt 0
0 0
(2)
εz ,
where εt and εz are the permittivity in the transverse and the longitudinal directions, respectively, and they can be either positive or negative, depending on the host medium. According to the continuity of the tangential field components across the boundary, we can introduce a four-component column vector ψ(z ) = (Ex , Ey , η0 Hx , η0 Hy )T , where the superscript ‘T ’ stands for the transpose operator. From Maxwell’s equations and Eqs. (1) and (2), it follows that ψ obeys the matrix ordinary differential equation dψ(z )
= ik0 M ψ(z ),
dz
(3)
where M is a 4 × 4 matrix in the form
0 0 M = 0
εt
−i γ a 0 aε z − ε t 0
0 −1 0 0
1−a 0 , −iγ a 0
(4)
36
G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
b
Fig. 1. (a) Schematic of a light beam reflection on the dielectric–UACM interface. (b) An illustration of the IF shifts. Here, the dashed line with arrow denotes the reflected beam predicted by the geometric optics, and the solid line with arrow stands for the practical reflected beam. yr , Θr and Y = yr + ℓΘr represent the SIF, the AIF and the total IF shifts of a reflected beam, respectively. In this figure, both yr and Θr are positive.
with a = n2 sin2 θ /(εz −γ 2 ). After some lengthy algebra, the eigenvalue equation of M can be derived as
(q2 − εt + aεz )(q2 − εt + aεt ) = εt a2 γ 2 ,
(5)
which actually defines the dispersion relations of two eigenwaves in the UACM q2
εt
+
with δ± =
k2x δ± k20
(εz − γ 2 )
1+
εz εt
±
= 1,
(6)
εt sign( ε −γ 2) z
(1 −
εz 2 ) εt
+
4γ 2
εt
/2. The dispersion relations described by Eq.
(6) may be elliptical curve, one-sheet hyperbola or two-sheet hyperbola, depending on the sign and magnitude of the parameters εt , εz and γ . If the four solutions of Eq. (5) or (6), i.e., the eigenvalues of M, are denoted by qℓ (ℓ = 1 ∼ 4), then k0 qℓ are the propagation constants in the z-direction in the UACM. Let q1 and q2 correspond to the two forward waves (eigenwaves), respectively, and then q3 = −q1 and q4 = −q2 to the two backward (or reflected) waves. In this way, we can write q1 = ± εt (1 − aδ+ ),
q2 = ± εt (1 − aδ− ),
(7)
where the positive or the negative square root tied to q1 and q2 is chosen by examining the direction of the energy flow carried by the two waves in the UACM. For convenience, we hereafter label the two eigenwaves as q1 wave and q2 wave. They can either propagate in the UACM or be totally reflected, depending on the parameters εt , εz , γ and the angle of incidence. The critical angles of incidence for the two eigenwaves, if they exist, can be determined by putting q1 = 0 or q2 = 0. This yields sin θc ± = (εz − γ 2 )/δ± /n, respectively. The general solution of Eq. (3) is straightforward and expressible as [24–28]
ψ(z ) = Veik0 Qz ψ0 ,
(8)
where ψ0 is a 4 × 1 constant column vector determined by the boundary conditions; Q and V are the 4 × 4 matrices that consist of the four eigenvalues and eigenvectors of M, respectively. For the
G. Xu et al. / Annals of Physics 335 (2013) 33–46
37
geometry shown in Fig. 1(a), we arrange the eigenvalues and eigenvectors such that the first two columns correspond to the reflected waves and the last two columns to the transmitted waves. Then, we have
−Q4
Q =
O Q4
O
,
V =
V1 V3
V2 V4
,
(9)
where the 2 × 2 sub-matrices Q4 = diag(q1 , q2 ), O consists of four zeros and iγ q2
1
V1 =
V2 =
1
q1 (1 − δ− ) iγ
q2
1 − δ− εt −
V3 =
−
q1
iγ
−
V4 =
1 − δ−
iγ
q1
iγ q2
εt (1 − δ− ) ,
(10a)
1
q1 (1 − δ− )
iγ
1 − δ− εt
,
−
1
εt (1 − δ− ) ,
iγ
−q2 −
iγ
.
(10b)
1 − δ−
It should be noted that Eqs. (10) still hold for the isotropic medium lying in z < 0 region. Therefore, we can acquire, only setting γ = 0 and εt = εz = n2 , the V -matrix of the isotropic medium
I2 V0
Vn =
I2 −V0
,
(11)
where I2 is the 2 × 2 identity matrix and n cos θ 0
0 −n cos−1 θ
V0 =
.
(12)
At the z = 0 interface, the boundary conditions imply ψ(0− ) = ψ(0+ ), which yields [27]
I2 V0
I2 −V0
Rxy I2
=
V1 V3
V2 V4
O Txy
,
(13)
where the 2 × 2 matrices O consists of four zeros and, Rxy and Txy are defined, respectively, by Ex(r )
= Rxy
Ey(r ) (j)
Ex(i)
,
Ey(i)
Ex(t )
Ey(t )
= Txy
Ex(i) Ey(i)
,
(14)
(j)
where Ex , Ey (j = i, r , t ) are the tangential electric field components, and i, r , t stand for the incident, reflected, transmitted waves, respectively. From Eq. (13), we derive Rxy = (V0 − V4 V2−1 )−1 (V0 + V4 V2−1 ).
(15)
Usually, the reflection matrix R is defined as
R=
rpp rsp
rps rss
,
(16)
satisfying
Ep(r ) Es(r )
=R
Ep(i) Es(i)
,
(17)
38
G. Xu et al. / Annals of Physics 335 (2013) 33–46
where rℓm is the ratio of the amplitudes of a reflected field with ℓ polarization to an incident field with m polarization, and p and s denote the transverse magnetic (TM) and the transverse electric (TE) polarizations, respectively. Referring to Fig. 1(a), we readily obtain
Ex(i)
Ey(i)
Ep(i)
Ex(r )
= Ui
Es(i)
cos θ 0
0 , 1
;
Ey(r )
= Ur
− cos θ
Ep(r )
Es(r )
,
(18)
with
Ui =
Ur =
0 . 1
0
(19)
From Eqs. (14), (17) and (18), we have the reflection matrix R = Ur−1 Rxy Ui .
(20)
After some derivations, the elements of R are determined by
rpp = −
(n cos θ + gq2 ) n cos−1 θ − g qε1t − m2 εt2 (1 − δ− )2 ∆
rsp = −rps =
rss =
2inmεt (1 − δ− )
∆
,
(21a)
,
(21b)
(n cos θ − gq2 ) n cos−1 θ + g qε1t − m2 εt2 (1 − δ− )2 ∆
,
(21c) γ (1−q /q )
2 1 with ∆ = (n cos θ + gq2 )(n cos−1 θ + g qεt ) + m2 εt2 (1 − δ− )2 and m = ε (1−δ )2 +γ 2 q /q , g = 1 + γ m. 1 t − 2 1 It is easy to show that if the UACM is replaced by another isotropic medium, Eqs. (21) can, by setting γ = 0, εt = εz = n′2 and then m = 0, g = 1, reduce to the familiar Fresnel reflection coefficients on the interface delimiting two isotropic media.
2.2. The IF shifts With the elements of reflection matrix given above, we can present the expressions for the SIF shift yr and the AIF shift Θr . Usually, Θr ≪ 1; thus the total IF shift observed at distance ℓ from the actual reflecting point can be expressed as a linear combination of yr and Θr , viz., Y = yr + ℓΘr (see Fig. 1(b)), where yr and Θr are given in the paraxial approximation, respectively, by [17] yr =
Γ k0 S
cot θ ,
(22)
and
Θr =
1N 2 S
θ02 ,
(23)
with
Γ = Im[fp fs∗ (|rpp |2 + |rss |2 + 2rpp rss∗ + 4|rsp |2 ) ∗ ∗ + (rpp + 3rss )rsp + 2(rpp rsp + rss∗ rsp )|fp |2 ], 2 2 ∗ S = rpp fp + |rss fs |2 + rsp − 2 Re[fp fs∗ (rpp rsp − rss∗ rsp )],
(24b)
∗ N = Re[fp fs∗ (|rpp |2 − |rss |2 ) + (|fp |2 − |fs |2 )(rpp rsp − rss∗ rsp )],
(24c)
(24a)
and where Im(. . .) and Re(. . .) mean the imaginary and the real parts of a complex number, respectively; θ0 = 2/(k0 w0 ) is the angular spread of the incident Gaussian beam of the waist w0 ; fp and fs are
G. Xu et al. / Annals of Physics 335 (2013) 33–46
39
2 associated with the complex-valued unit vector fˆ = fp xˆ i + fs yˆ i with fp + |fs |2 = 1 denoting the orientation of the polarizer, which is put perpendicularly to the central wave vector k0 of the incident beam, and the asterisk (∗) marks the complex conjugate. A positive (negative) value of yr implies that the SIF shift occurs in the positive (negative) y-axis direction, while a positive (negative) value of Θr implies that the AIF shift slants towards the positive (negative) y-axis direction, as illustrated in Fig. 1(b). In what follows, by ‘the IF shifts’, we mean both the SIF shift and the AIF shift. We can see from Eqs. (22) to (24) that in an anisotropic medium, besides the co-polarized
2
reflectivity and reflection coefficients, the cross-polarized reflectivity rsp and reflection coefficients rsp play a role in the IF shifts, especially in the SIF shift, as compared with the isotropic case where rsp = rps = 0. 3. Numerical results and discussion Here, bearing on three kinds of UACM, which have different but easy fabricated electromagnetic parameters [23], we will discuss the effects of the propagation behaviors of the two eigenwaves in the UACM on the IF shifts for different electromagnetic parameters. Throughout the paper, we use right√ elliptically polarized incident beams, denoted by (fp , fs ) = (1, 1 + i)/ 3, and other incident beams, such as linearly and circularly polarized beams, can be tackled similarly. Assuming the z < 0 region is free space, then n = 1. As is known, the paraxial approximation requires that the transverse size of a beam is much greater than its wavelength λ [34]; as a consequence, in the following numerical calculations we choose the waist of incident beam w0 = 50λ. 3.1. Case I: εt > 0, εz > 0 In this case, the host medium is a conventional right-handed material. First, we assume γ 2 < εz , so δ± are always positive. Then, the dispersion relations for the two eigenwaves are both elliptical curves, which can be determined from Eq. (6). When the angle of incidence θ < min(θc + , θc − ), both the eigenwaves can propagate in the UACM; otherwise, both or one of the eigenwaves becomes evanescent. Such situations are illustrated in Fig. 2, where εt = 3, εz = 2 and γ = 0.5 for 2(a)–(d) and γ = 1.2 for 2(e)–(h). For γ = 0.5, since neither θc + nor θc − exists, the two eigenwaves can propagate
2 in the UACM; in the situation, we get the cross-polarized reflectivity rsp ≈ 0 due to the weak
2
chirality and a pseudo-Brewster angle near θ = 65.8° for p-polarized wave, where rpp ≈ 0 (see Fig. 2(b)). These results indicate that the two eigenwaves propagate just as in a conventional uniaxial medium. Hence, we reckon that the variation of IF shifts with the angle of incidence should also be similar to that in a uniaxial medium [11,35], as is numerically verified in Fig. 2(c) and (d). In addition, we find from Fig. 2(c) and (d) that the SIF shift is negative and attains its maximum near the pseudo∗ − rss∗ rsp is Brewster angle; the AIF shift is also negative, which is due to the fact that the quantity rpp rsp
2
usually tiny, so the sign of the AIF shift relies mainly on that of the difference rpp − |rss |2 (see (24c)). For γ = 1.2, the q2 wave is propagative, but the q1 wave is totally reflected when θ > θc + (≈37.0°). We can remark from Fig. 2(g) and (h) that both the SIF and the AIF shifts attain their positive maxima at the critical angle θc + . In particular, the SIF and AIF shifts can switch between positive and negative values, which differs from the case γ = 0.5. The results arise evidently from the change in the propagation properties of the q1 wave in the UACM, which leads to the abrupt change in the properties 2 of reflected wave, for example, the cross-polarized reflectivity rsp is significantly enhanced, and
2 |rss |2 exceeds rpp in magnitude. When γ 2 > εz , i.e., the chirality becomes stronger, we always have δ+ < 0 and δ− > 0. Therefore, the dispersion curve for the q1 wave is an ellipse, while that for the q2 wave is a two-sheet hyperbola, implying that the q2 wave always propagates for an arbitrary angle of incidence, and the q1 wave is totally reflected for θ ≥ θc + . This situation is plotted in Fig. 2(i)–(l), where γ = 1.5 and εt and εz remain unchanged. For such parameters, since θc + does not exist, both the eigenwaves are propagative. In consequence, the behaviors of the reflected wave are quite analogous to that in the case γ = 0.5. However, the IF shifts, particularly the SIF shift, are very dissimilar to the case γ = 0.5,
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G. Xu et al. / Annals of Physics 335 (2013) 33–46
a
e
i
b
f
j
c
g
k
d
h
l
Fig. 2. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 3, εz = 2 and γ = 0.5 for (a)–(d), γ = 1.2 for (e)–(h) and γ = 1.5 for (i)–(l). (a), (e), (i) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b), (f), (j)
2
2
The reflectivity. The solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g), (k) The SIF shift and (d), (h), (k) the AIF shift (in s).
as can seen by comparing Fig. 2(k) with Fig. 2(c). This difference might be attributed to the role that
2 rsp plays in the situation. Therefore, it is believed that the cross-polarized reflection coefficient rsp has a greater effect on the SIF shift than on the AIF shift. 3.2. Case II: εt > 0, εz < 0 When the host medium contains resonant dipoles directed only to the z direction, the UACM has
εt > 0, εz < 0 [23]. We easily get δ+ < 0 and δ− > 0, and then the dispersion curves for the q1
wave and the q2 wave are an ellipse and a two-sheet hyperbola, respectively. This indicates that the q2 wave always propagates for an arbitrary angle of incidence, and the q1 wave is totally reflected for θ ≥ θc + . In this sense, the case seems to be similar to that for εt > 0, εz > 0 and γ 2 > εz . However, owing to very distinct physical parameters, the propagation properties and the IF shifts of the reflected
G. Xu et al. / Annals of Physics 335 (2013) 33–46
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Fig. 3. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 0.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
wave should be dissimilar. To examine this, we depict the case in Fig. 3, where εt = 0.3, εz = −2 and, γ = 0.8 for Fig. 3(a)–(d) and γ = 2.8 for 3(e)–(h). For γ = 0.8, we can see from Fig. 3(b)–(d) that the properties and the IF shifts of the reflected wave all change abruptly at the critical angle of
2
incidence for the q1 wave θc + = 38.1°. Since |rss |2 is always larger than rpp for an arbitrary angle of incidence, the AIF shift is negative with its maximum at θc + = 38.1°. For γ = 2.8, θc + does not
2
exist, and thus both the eigenwaves are propagative in the situation. Because of rpp > |rss |2 for any angle of incidence, the AIF shift is always positive, as shown in Fig. 3(f) and (h). Furthermore,
2
when θ > 60°, rsp is observable; consequently, in the range of the angles of incidence, the SIF shift exhibits remarkable changes which are absent in a conventional isotropic medium.
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As is known, the electromagnetic parameters affect the propagation properties of the two eigenwaves in the UACM and thus affect the IF shifts. To this end, we alter εt and keep εz and γ unchanged to explore the IF shifts. The situation is plotted in Fig. 4, where we use εt = 1.3 instead of 0.3 in Fig. 3, γ = 0.8 (for Fig. 4(a)–(d)) and γ = 2.8 (for Fig. 4(e)–(h)). In the calculation, we find that for either of the two γ values, θc + does not exist, and thus both the eigenwaves are propagative; the SIF shifts change similarly. Specifically, the SIF shifts are positive for a smaller θ and negative for a larger θ . Additionally, the sign of the AIF shift can be still interpreted in the same way as above. Obviously, the result is distinct from the situation in Fig. 3, where neither the SIF shift nor the AIF shift can switch between positive and negative values. 3.3. Case III: εt < 0, εz > 0 When the host medium contains resonant dipoles directed towards the transverse directions, the UACM possesses εt < 0 and εz > 0 [23]. The case can provide very different propagation properties for varied electromagnetic parameters. First, for the weak chirality, i.e., γ 2 < εz , we can obtain δ+ < 0 and δ− > 0, which means that q1 is purely imaginary, viz., the q1 wave is evanescent, and the dispersion curve for the q2 wave is a one-sheet hyperbola. We should stress here that only when θ ≤ θc − in the situation, can the q2 wave be totally reflected, as is quite different from the usual situation. Moreover, to meet the conversation law of energy both q1 and q2 are here taken the negative square roots in Eq. (7). Aiming at γ = 0.3 and 1.3, respectively, we depict the situation in Fig. 5 for εt = −3, εz = 2. For γ = 0.3, as shown in Fig. 5(a)–(d), it is clear that θc − does not exist, and hence both the eigenwaves are evanescent, implying that the q1 wave and the q2 wave are totally reflected for an arbitrary angle of incidence. The
2
2
2
2
result is confirmed in Fig. 5(b), where |rss |2 + rps = 1 and rpp + rsp = 1. Owing to rsp ≈ 0,
2
the co-polarized reflectivity |rss |2 and rpp are nearly identical, so that the AIF shift is roughly null, which can be seen in Fig. 5(d). We find from Fig. 5(c) that for θ < 70°, the SIF shift is positive with its maximum near θ = 40°. Next, for γ = 1.3, Fig. 5(e) and (f) show that when θ ≤ θc − (=49.97°), the q2 wave is totally reflected, thus leading to the abrupt changes in the reflectivity near θc − . In particular,
2
as compared with other above cases, rsp here becomes quite appreciable. These might be the factors that induce the complicated changes in the SIF shift which, e.g., crosses twice between positive and negative values. Similarly, the change in the AIF shift can be reasonably accounted for. When the chirality becomes stronger, i.e., γ 2 > εz , there exist three situations, which are discussed, respectively, as follows.
(i) When 1 + εz /εt ≥ 2 (εz − γ 2 )/εt , δ± are always positive. Correspondingly, both q1 and q2 are purely imaginary and taken the negative square roots (see Eq. (7)), namely, the two eigenwaves are totally reflected. The situation is plotted in Fig. 6(a)–(d), where εt = −3, εz = 2 and γ = 1.43. As 2 2 2 2 expected, Fig. 6(b) shows up the total reflection: rpp + rsp = 1 (note that |rss |2 = rpp , rsp =
2 rps here). Therefore, the AIF is nearly null. In addition, we get the negative SIF shift in the situation. (ii) When −2 (εz − γ 2 )/εt < 1 + εz /εt < 2 (εz − γ 2 )/εt , δ± are complex, so that both q1
and q2 are complex and taken the negative square roots. We plot the situation in Fig. 6(e)–(h) for εt = −3, εz = 2 and γ = 1.6. Clearly, both the SIF and the AIF shifts are enormous near θ = 0° (i.e., normal incidence), but vanish rapidly with increasing θ .
(iii) When 1 + εz /εt ≤ −2 (εz − γ 2 )/εt , δ± are always negative. The dispersion curves for the two eigenwaves are both one-sheet hyperbolae. When θ ≤ θc + (θc − ), the q1 (q2 ) is totally reflected. Then, the angle of incidence can be divided into three intervals: θ ≤ min(θc + , θc − ), in which both the eigenwaves are totally reflected; min(θc + , θc − ) < θ < max(θc + , θc − ), where one eigenwave is propagative, and the other is evanescent, and θ ≥ max(θc + , θc − ) where both the eigenwaves are propagative. Notably, q1 is here taken the negative square root, while q2 taken the positive square root. Such a situation is illustrated in Fig. 6(i)–(l) for εt = −1.2, εz = 2 and γ = 1.45. We can see that the reflectivity curves and the AIF shift are also divided into three parts corresponding, respectively, to the above three intervals. Only in the interval min(θc + , θc − ) < θ < max(θc + , θc − ), is the AIF shift
G. Xu et al. / Annals of Physics 335 (2013) 33–46
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Fig. 4. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 1.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
observable, as seen in Fig. 6(l). The SIF shift is similar to that in case (ii), that is, it is enormous near θ = 0°, but vanishes quickly with increasing θ . 4. Conclusion In this work, we first derive the reflection matrix of a plane electromagnetic wave (or saying the central plane-wave component of a Gaussian beam) on the dielectric–UACM interface by virtue of the eigenvalue method of a 4 × 4 matrix. Based on the reflection matrix, the IF shifts of a Gaussian beam
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a
e
b
f
c
g
d
h
Fig. 5. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = 0.3, εz = −2 and γ = 0.8 for (a)–(d), γ = 2.8 for (e)–(h). (a) and (e) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 , respectively. (b) and (f) The reflectivity. The
2
2
solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g) The SIF shift and (d), (h) the AIF shift (in s).
reflected from the UACM have been numerically discussed. The numerical results have been presented for three kinds of UACM: I εt > 0, εz > 0, II εt > 0, εz < 0 and III εt < 0, εz > 0. In every case, for different electromagnetic parameters which may induce the interesting propagation properties of the two eigenwaves in the UACM, we have discussed some possible behaviors of the IF shifts. After examining these results, we have found that in general, the IF shifts can abruptly change near the critical angle at which either of the eigenwaves is totally reflected. The cross-polarized reflection coefficient has a greater effect on the SIF shift than on the AIF shift, and the sign of the AIF shift depends
2
only on that of the difference rpp − |rss |2 . By adjusting the angle of incidence or the electromagnetic parameters of the UACM, such as permittivity, permeability and chirality, the crossovers of the IF shifts
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Fig. 6. Dependence of the reduced propagation constants q1 , q2 , reflectivity and the IF shifts on the angle of incidence for εt = −3, εz = 2 and γ = 1.43 for (a)–(d), γ = 1.6 for (e)–(h) and εt = −1.2, εz = 2, γ = 1.45 for (i)–(l). (a), (e), (i) The reduced propagation constants q1 and q2 . The solid and dashed lines correspond to the real and imaginary parts of q1 and q2 ,
2
2
respectively. (b), (f), (j) The reflectivity. The solid, dotted and dashed lines correspond to |rss |2 , rpp and rsp , respectively. (c), (g), (k) The SIF shift and (d), (h), (k) the AIF shift (in s).
between positive and negative values can be realized. In short, the IF shifts are closely related to the propagation properties of the two eigenwaves in the UACM, while the propagation properties depend greatly on the electromagnetic parameters. Thus, due to its flexibility in design the UACM opens up possibilities to control the IF shifts. Acknowledgments We would like to express our gratitude to the referee, whose searching questions and suggestions helped us elaborate and improve the manuscript. This work is supported by the ‘Qinglan’ Project of Jiangsu Province. References [1] J.D. Jackson, Classical Electrodynamics, third ed., John Wiley & Sons, New York, 1998. [2] F. Goos, H. Hänchen, Ann. Phys. (Leipzig) 1 (1947) 333.
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