doi: 10.1098/rspa.2011.0081 published online 23 March 2011Proc. R. Soc. A

 M. V. Berry radiationLateral and transverse shifts in reflected dipole  

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Proc. R. Soc. Adoi:10.1098/rspa.2011.0081

Published online

Lateral and transverse shifts in reflecteddipole radiation

BY M. V. BERRY*

H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK

In-plane (lateral) and out-of-plane (transverse) shifts in the direction of arbitrarilypolarized electromagnetic waves in a denser medium, reflected totally or partially atan interface with a rarer medium, are calculated exactly, in terms of the deviation of thePoynting vector from radial. The shifts are analogous to the Goos–Hänchen and Fedorov–Imbert shifts for beams. There is a transverse shift even for unreflected dipole radiationif the polarization is not linear. With reflection, there is a transverse shift for linearpolarization, provided this is not pure transverse electric or transverse magnetic. Thecontributions from the geometrical ray, the lateral ray that interferes strongly with it, andthe large peak at the Brewster angle (for transverse magnetic polarization), are calculatedasymptotically far from the geometrical image. At the critical angle, the lowest orderasymptotics is inadequate and a more sophisticated treatment is devised, reproducingthe exact shifts accurately.

Keywords: Goos–Hänchen; Fedorov–Imbert; polarization; diffraction; asymptotics

1. Introduction

Beams reflected from surfaces can be slightly shifted, laterally (Goos–Hänchenshift (Lotsch 1970)) or transversely (Fedorov–Imbert shift (Costa deBeauregard & Imbert 1972)). These shifts have been extensively studiedtheoretically and experimentally, over many decades (Lotsch 1970), in light,sound, elastic and quantum (Hirschfelder & Christoph 1974) waves. For beams,such as Gaussian beams or the recently studied Bessel beams (Schilling 1965;Bliokh et al. 2010; Aiello & Woerdman 2011), the shifts are naturally describedas displacements of the centre of gravity or system of vortices (Dennis & Göttein preparation) of the reflected beam.

My aim here is to study a different situation: shifts not in beams but in reflectedlight diverging from localized sources such as point dipoles. In this case, the shiftsare naturally described as deflections: deviations in the direction of the Poyntingvector far from the reflecting surface, measured with respect to the radial directionfrom the geometrical image of the source. Multiplying the angular shift by thedistance to the geometrical image, the shifts appear equivalently as apparentdisplacements of the image in the plane perpendicular to the radial direction.

*tracie.anderson@bristol.ac.uk

Received 28 January 2011Accepted 21 February 2011 This journal is © 2011 The Royal Society1

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interface

h

hrr

geometricalimage

geometrical ray

source

θ

lateral ray

θcindex 1

index n < 1

R

z

Figure 1. Geometry of reflection from source in denser medium, polar coordinates of the field pointR, centred on the geometrical image, geometrical ray and lateral ray (dashed).

These deviations and displacements are vectors with two components: ‘lateral’,in the plane containing the observation point and the normal to the surface; and,orthogonal to this, ‘transverse’.

Any general theory must incorporate the polarization of the incident wave,an aspect that requires some care, as was emphasized recently (Bliokh & Bliokh2006, 2007). The treatment here makes extensive use of techniques describedin a detailed account (Brekhovskikh 1960) of reflected electric and magneticdipole radiation in terms of integrals over an angular superposition of planewaves travelling in different directions. The quantity to be calculated is thedirection (unit vector) of the Poynting vector representing the energy flow ina monochromatic electromagnetic wave, represented by complex electric andmagnetic fields E and B, namely

s = S|S| , where S = ReE∗ × B. (1.1)

S, and hence s, is calculated as a function of position, for a general class of fields,in §2. The shifts are represented by the components of s perpendicular to theradial direction.

In §3, s is calculated for dipole radiation without reflection. There is no lateralshift, but even in this simple case there is a transverse shift unless the polarizationis purely linear.

The main work, beginning in §4, concerns reflection from a planar interfacebetween two transparent dielectrics (figure 1), with the source and reflected wavein the denser medium, whose refractive index is conveniently defined as 1. Thetransmitted wave is in the rarer medium, with (relative) refractive index n < 1. Innumerical illustrations, we will take n = 2/3, representing glass. Important rolesare played by the critical angle qc(n), and, for in-plane polarization, the Brewsterangle qB(n), defined by

qc(n) = arcsin n and qB(n) = arctan n. (1.2)

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The derivation in §4 leads to exact expressions, easily evaluated numerically,for the components of S, and in particular the components sq and sf representingthe lateral and transverse shifts in polar coordinates centred on the geometricalimage. There is a lateral shift for any polarization, and also a transverseshift for any polarization, in particular linear unless this is TE (electricfield purely out-of-plane) or TM (magnetic field out-of-plane, i.e. electric fieldin-plane).

Different contributions to the shifts, associated with distinct physical processes,emerge with increasing distance r from the geometrical image. Mathematically,these contributions arise from different plane waves in the superposition, and canbe disentangled by asymptotic approximations to the angular integrals, in whichthe large parameter is kr, with k = 2p/l, where l is the wavelength and k thewavenumber, both in the denser medium.

The simplest of these contributions, derived in §5, is the geometrical optics-reflected wave, associated with the saddle-point of the integrals; the correspondingangular deviations are of the order of 1/kr, so the apparent displacements ofthe source are of the order of l. The lateral shift, non-zero only for observationdirections q > qc, is given by the well-known Artmann formula (Lotsch 1970).There is also a transverse geometrical shift, which for q > qc requires only thatthe incident wave is not purely TE or TM, and for q < qc requires the polarizationto be not purely linear in any direction.

The next contribution, significant only for TM polarization, is a large peakin the lateral shift at the Brewster angle, whose precise height and shape areobtained in §6. This is also associated with the geometrical-reflected wave butrequires more sophisticated asymptotics even though the peak represents a largeangular deviation: of order (kr)0 (approx. 8.5◦ for n = 2/3). The Brewster angle isa zero of the in-plane reflection coefficient; but, on a different Riemann sheet, qBalso corresponds to a pole, representing a surface wave in certain cases where nis complex, for example, when one medium is conducting. The surface wave doesnot contribute directly to the reflected wave—unsurprising because the reflectioncoefficient is zero—but was nevertheless the subject of controversy for many years(Kahan & Eckart 1949, 1950; Wait 1998). The reason for a strong lateral shift atqB is the denominator |S| in the normalization of the unit vector in equation (1.1):away from qB, this is dominated by the geometrical wave, which vanishes atqB itself.

The remaining contribution is the ‘lateral wave’ (Tamir & Oliner 1969; Laiet al. 1986), associated with the branch-point singularity (Brekhovskikh 1960)of the angular integral from the plane wave in the critical direction qc. Forobservation directions away from qc, the leading-order lateral wave, derived in§7, exists only for q > qc. Although the contribution of the lateral wave to thefield strengths is smaller than that of the geometrical wave, its contributionto the Poynting direction s is of the same order, because Maxwell’s equationsimply that S involves derivatives of the fields. This causes powerful interferenceoscillations, as a function of observation direction, between the lateral andgeometrical waves.

For observation at the critical direction itself (§8), the geometrical and lateralwaves coincide. In the integrals, the saddle-point coincides with the branch-point,and in the leading-order approximation the shifts are of the order of 1/(kr)3/4—that is, larger than geometrical by a factor (kr)1/4. However, this asymptotic

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regime is approached extremely slowly, especially for the TM polarization thatrequires kr � 1000. Approximating the shifts accurately for more modest (butstill large) values of kr requires a more sophisticated asymptotics (§8).

A differently defined shift, implicit in the exact fields calculated here but notcaptured by the Poynting-based analysis of the fields, is associated with themodulation of the intensity of the reflected wave caused by the angle-dependence,below qc and in particular near qB, of the reflection coefficients of the constituentplane waves. The analogous effect for Gaussian beams has been observed and isfully understood (Aiello et al. 2009; Merano et al. 2009).

2. General theory

We will study reflected fields E, B at a point R, using spherical polar coordinatescentred on the geometrical image of the source (figure 1), with the z directionchosen normal to the interface, and r representing position in planes with constantazimuth f:

R = {r , q, f} = {r , f} r = {r , q}. (2.1)

The fields will be written as a general superposition of orthogonally linearlypolarized fields, labelled i, where,

i = 1, 2 1 = TE and 2 = TM, (2.2)

with ‘transverse’ in TE and TM referring to the constant f planes. Thus, withcomplex polarization coefficients a1, a2, the fields are

E(R) = a1E1(R) + a2E2(R)

and B(R) = a1B1(R) + a2B2(R) (|a1|2 + |a2|2 = 1)

}. (2.3)

It is convenient to represent the fields as scalar waves ji , generated fromazimuth-independent electric and magnetic Hertz potentials Pi (Brekhovskikh1960) associated with the sources, that is, for the TE and TM cases, and withunit direction and polarization vectors here and hereafter represented by e,

E1(R) = iV × P1(R) = j1(r)ef

and B2(R) = ic

V × P2(R) = j2(r)c

ef

⎫⎬⎭, (2.4)

wherePi(R) = fi(r)ez . (2.5)

The scalar f (r) satisfies the scalar Helmholtz equation with wavenumber k. Thus,the scalar fields are obtained from the potentials, as

ji(r) = ief

(− sin qvr fi(r) + cos q

rvqfi(r)

). (2.6)

In §4, we will calculate the reflected scalar fields j1 and j2, which are differentbecause the Fresnel reflection coefficients differ for TE and TM polarizations.

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A simple illustration of this formulation, to which we will return in the nextsection, is the unreflected field from a point dipole:

f (r) = exp(ikr)r

⇒ j(r) = k sin qexp(ikr)

r

(1 + i

kr

), (2.7)

in which r and q are (temporarily) coordinates whose origin is the dipole sourcerather than its geometrical image.

Returning to the general case, the magnetic field B1 corresponding to E1, andthe electric field E2 corresponding to B2, are given by Maxwell’s equations:

B1(R) = − iV × E1

ck= − i

ckr2 sin q(er rvq(sin qj1) − eqr sin qvr(rj1))

and E2(R) = ick

V × B2 = ikr2 sin q

(er rvq(sin qj2) − eqr sin qvr(rj2))

⎫⎪⎪⎬⎪⎪⎭.

(2.8)A direct calculation now gives the desired Poynting vector in terms of the TEand TM scalar fields:

S(R) = 1kc

[|a1|2Imj∗

1Vj1 + |a2|2Imj∗2Vj2

+ 1kr2 sin2 q

Re(a∗1a2V(r sin qj∗

1) × V(r sin qj2))].

(2.9)

Explicitly, the separate components are

Sr = S · er = 1kc

(|a1|2 Im j∗1vrj1 + |a2|2 Im j∗

2vrj2)

Sq = S · eq = 1kcr

(|a1|2 Im j∗1vqj1 + |a2|2 Im j∗

2vqj2)

and Sf = S · ef = 1kcr2 sin q

Re[a∗1a2(vr(rj∗

1)vq(sin qj2)

− vq(sin qj∗1)vr(rj2))]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (2.10)

from which the unit Poynting direction s = srer + sqeq + sfef can be obtainedby normalization. The shifts are the component sq, which is the lateral shift,making an angle sin−1 sq with the radial direction, and the component sf, whichis the transverse shift, whose corresponding angle is sin−1 sf. The dependenceon polarization is clear: in the lateral shift, the TE and TM fields contributeseparately, and the transverse shift vanishes if the polarization is pure linear TE(a2 = 0) or pure linear TM (a1 = 0).

It is tempting in equation (2.9) to associate the lateral shift (the terms in|a1|2 and |a2|2) with the orbital current, and the transverse shift (the terms ina∗

1a2) with the spin current. But this is not compatible with a recently identifiednatural separation between orbital and spin current (Bekshaev & Soskin 2007;Berry 2009; Bliokh et al. 2010). One way to see this is to note that there can be atransverse shift for oblique linear polarizations, i.e. neither pure TE nor pure TM.

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–1.0 –0.5 0 0.5 1.0

–1.0

–0.5

0

0.5

1.0

x

x

y

Figure 2. Projection of Poynting vector direction (equation (3.2)) from a circularly polarized dipole,in the plane z = r cos q = 0.1, showing swirl near the dipole axis and divergence far away.

3. Dipole wave

Before considering the reflected wave, it is worth pointing out that there is a shifteven in the simple case of the incident wave from a dipole source with generalpolarization. Then j1(r) = j2(r) = j(r), with j(r) given by equation (2.7).Equation (2.10) gives the Poynting vector as

S = k2

cr2

[sin2 qer + 4

krIm(a∗

1a2) sin q cos q ef

], (3.1)

and hence the energy flow direction

s = er + (4/kr)Im(a∗1a2) cot q ef√

1 + ((4/kr)Im(a∗1a2) cot q)2

≈ er + 4kr

Im(a∗1a2) cot q ef. (3.2)

A similar expression has been found before (Schilling 1965) for the transverseshift of an incident beam.

This shows that there is always a transverse shift for the dipole wave, unlessthe polarization is purely linear. Therefore, it makes sense to interpret the shiftas an example (possibly the simplest) of the spin Hall effect of light (Bliokh &Bliokh 2006, 2007), that is, a manifestation of spin–orbit coupling. As mentionedearlier, this interpretation fails for the general transverse shifts of §2, and also forthe transverse shifts in reflected waves to be discussed in §4, because these canexists even for linear polarization provided this is oblique.

For kr >> 1, the energy flow is mostly radial, i.e. diverging from the source, butwith a gentle swirl around the dipole axis (figure 2). The swirl gets stronger as kr

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gets smaller, and the z-axis is a singularity of the unit vector field s (though notof the Poynting vector S). The sense of swirl is opposite in the two hemispheres,so there is no torque on the dipole source. The region of swirl corresponds to theinterior of a paraboloid, a measure of whose radius is the distance from the axisfor which sr sin q = sf:

rswirl =√

zIma∗1a2

k. (3.3)

It is worth noticing that the helicity (Moffatt & Ricca 1992) of the Poyntingfield (equation (3.1)), that is,

S · V × S = − 4k3

c2r6sin4 qIm(a∗

1a2), (3.4)

is non-zero. This implies that S cannot be written in the form S = a(R)VRb(R),so there are no exact ‘wavefronts’, which are contour surfaces of a smooth functionnormal to S.

4. Reflected wave

The standard way to obtain the reflected (also transmitted) fields is torepresent the Hertz potential (equation (2.7)) as a superposition of plane wavestravelling in directions q′, that is (after integrating over azimuth directions f(Brekhovskikh 1960)),

exp(ikr)r

= ik∫C+

dq′ sin q′J0(kr sin q sin q′) exp(ikr cos q cos q′)

= ik2

∫C

dq′ sin q′H (1)0 (kr sin q sin q′) exp(ikr cos q cos q′),

(4.1)

where J and H denote standard Bessel functions (DLMF 2010). The contours areshown in figure 3, with the complex segments corresponding to the unavoidableevanescent waves in the superposition. Each constituent plane wave is reflectedwith its Fresnel reflection coefficient Ri(q′) corresponding to direction q′ andpolarization i = 1 or i = 2. In cartesian coordinates with z = 0 representing theinterface, with the source at z = h (figure 1), the plane waves reflect according to

exp{i(kxx + kyy − (z − h)

) √k2 − k2

x − k2y

}⇒ Ri(q) exp

{i(kxx + kyy + (h + z)

)√k2 − k2

x − k2y

},

(4.2)

where here and hereafter square roots are interpreted as

√F = √|F | ×

(+1 (F ≥ 0)+i (F < 0)

). (4.3)

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cut

cut

Re q¢qc

–qc

Im q¢

C+

C

1–2π

12π

Figure 3. Integration contours and branch cuts in the q′ plane.

The Fresnel coefficients are (Born & Wolf 2005)

R1(q) = cos q −√

n2 − sin2 q

cos q +√

n2 − sin2 qand R2(q) = n2 cos q −

√n2 − sin2 q

n2 cos q +√

n2 − sin2 q, (4.4)

with branch-points at the critical angles ±qc(n) (equation (1.2)) and, for R2, azero at the Brewster angle qB(n). In the regime q > qc of total reflection, |Ri| = 1and it is convenient to write

Ri(q) = exp{imi(q)},m1(q) = −2 arctan

(√tan2 q − n2 sec2 q

)

and m2(q) = −2 arctan(

1n2

√tan2 q − n2 sec2 q

)

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (4.5)

With the notation

s = sin q, c = cos q, s′ = sin q′ and c′ = cos q′, (4.6)

(for this section only, and with s not to be confused with the Poynting unitvector s), the Hertz potentials are

fi(r , q) = ik∫C+

dq′s′Ri(q′)J0(krss′) exp(ikrcc′)

= ik2

∫C

dq′s′Ri(q′)H (1)0 (krss′) exp(ikrcc′).

(4.7)

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Bessel identities (DLMF 2010) and equation (2.6) now lead to the followingexpressions for the scalar waves representing the two fields and the derivativesinvolved in S:

ji = −k2Ki , vr rj1 = −k3r(sI0,i + icI1,i) and vqsji = −k3rs(cI0,i − isI1,i),(4.8)

where the integrals are

Ki =∫C+

dq′s′2Ri(q′)J1(krss′) exp(ikrcc′)

I0,i =∫C+

dq′s′3Ri(q′)J0(krss′) exp(ikrcc′)

and I1,i =∫C+

dq′s′2c′Ri(q′)J1(krss′) exp(ikrcc′)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (4.9)

Because of the branch-points, there are cuts in the plane of the integrationvariable q′; the cuts shown in figure 3 are consistent with the specification(equation (4.3)) of the square roots. Thus, we obtain the exact expressions forthe components (equation (2.10)) of S:

Sr = k4

m0c(|a|2ImK ∗

1 (sI0,1 + icI1,1) + |b|2ImK ∗2 (sI0,2 + icI1,2))

Sq = k4

m0c(|a|2ImK ∗

1 (cI0,1 − isI1,1) + |b|2ImK ∗2 (cI0,2 − isI1,2))

and Sf = k4

m0cIma∗b(I ∗

0,2I1,2 + I ∗1,1I0,2)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (4.10)

As already noticed (Brekhovskikh 1960), these exact formulae do not involvethe distance h between the source and the interface. This implies that thereflected fields and shifts depend only on the location of the field point relativeto the geometrical image, not where the source and interface are. Of course, ifz = r cos q − h is negative the reflected field is virtual rather than real.

The integrals I and K are oscillatory, but not difficult to evaluate numerically,even up to kr ∼ 1000. Figure 4 shows the lateral and transverse shifts forfour representative polarizations that will be used in all subsequent figures,corresponding to TE polarization (a1 = 1, a2 = 0), TM polarization (a1 = 0,a2 = 1), polarization at 45◦ to the plane of incidence (a1 = 1/

√2, a2 = 1/

√2)

and circular polarization (a1 = 1/√

2, a2 = i/√

2). The main features, to beexplained in subsequent sections, are: strong oscillations (§7) about a non-zero mean (§5) for q > qc, increasing in amplitude towards finite values atq = qc (§8); very small values of the lateral shifts for q < qc (§5), except fora large peak at q = qB for TM polarization (§6). (The increasing strengths ofthe transverse shifts as q approaches zero for polarizations that are neither

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–0.06

–0.05

–0.04

–0.03

–0.02

–0.01

0(a) (b)

(c) (d)

–0.05

0

0.05

0.10

0.15

0 20 40 60

°

80 0 20 40 60 80

–0.08

0.06

–0.04

–0.02

0

0.02

–0.05

0

0.05

sφsθ

sθ

sθ

s

θ

sφ

sφ

sφ

sθ

Figure 4. Lateral direction shift sq (blue) and transverse shift s4 (red) for n = 2/3 and kr = 100,as functions of observation angle, computed exactly from (4.10), for (a) TE polarization (a1 = 1,a2 = 0); (b) TM polarization (a1 = 0, a2 = 1), (c) 45◦ linear polarization (a1 = 1/

√2, a2 = 1/

√2),

and (d) circular polarization (a1 = 1/√

2, a2 = i/√

2). (Online version in colour.)

TE nor TM, and which occur even for the unreflected dipole field (cf. §3), areassociated with very small values of the fields and therefore of the unnormalizedenergy flow S.)

5. Geometrical optics

For kr � 1, the Hankel function in equation (4.7) can be replaced by its large-argument asymptotic approximation, leading to the wave function

ji ≈ k3/2 exp((1/4)ip)√2pr sin q

∫C

dq′s′3/2Ri(q′) exp{ikr cos(q′ − q)}. (5.1)

The main contribution to this integral comes from the saddle-point at q′ = q, andstraightforward application of the method of stationary phase (Wong 1989) gives

ji,sp = k sin qRi(q)exp(ikr)

r. (5.2)

This is the crudest approximation, giving the reflected field as the incident fieldmodulated by the reflection coefficient.

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For the Poynting vector, equation (2.10) now gives

Sr ,sp =(

k2 sin2 q

cr2

)(|a1|2|R1|2 + |a2|2|R2|2)

Sq,sp =(

k2 sin2 q

cr2

)1kr

(|a1|2vqm1 + |a2|2vqm2)Q(q − qc)

and Sf,sp =(

k2 sin2 q

cr2

)|R1R2|

kr[4 cot qIm(a∗

1a2 exp{i(m2 − m1)})

− Q(q − qc)Re(a∗1a2 exp{i(m2 − m1)})(vqm2 − vqm1)]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (5.3)

in which (here and hereafter) Q denotes the unit step. Normalizing, we obtainthe shifts:

sr ,sp ≈ 1, sq,sp = 1kr

(|a1|2vqm1 + |a2|2vqm2)Q(q − qc)

and sf,sp = 1kr

Q(qc − q)4 cot qR1R2Im(a∗

1a2)(|a1|2|R1|2 + |a2|2|R2|2)

+ 1kr

Q(q − qc)[4 cot qIm(a∗1a2 exp{i(m2 − m1)})

− Re(a∗1a2 exp{i(m2 − m1)})(vqm2 − vqm1)]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (5.4)

At this level of approximation, similar shifts have been found before for bounded(e.g. Gaussian) beams. The expression for sq,sp, giving the lateral shift in termsof the derivatives of the phases of the reflection coefficients, is familiar as theArtmann formula (Schilling 1965; Lotsch 1970) for the Goos–Hänchen shift, andthe transverse shift sf,sp has also been calculated by several authors (Schilling1965; Bliokh & Bliokh 2006, 2007).

Figure 5 shows that these geometrical shifts reproduce the mean of theoscillations in the exactly computed shifts for q > qc, and also the transverseshift for circular polarization. But they fail to give the oscillations, the smoothbehaviour at q = qc or the peak at q = qB for TM polarization.

6. Brewster peak

In passing from S in equation (5.3) to the normalized s in equation (5.4),one point was glossed over. If a1 = 0, that is for TM polarization, the zeroin R2 at q = qB implies a zero in the geometrical approximation to Sr , so thegeometrical approximation to Sq is undefined (0/0). To resolve the behaviournear the Brewster angle, it is necessary to examine R2 more closely near qB.

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–0.06

–0.04

–0.02

0(a) (b)

(c) (d)

–0.05

0

0.05

0.10

0.15

0 20 40 60°

80 0 20 40 60 80–0.08

–0.04

0

0.02

0.10

–0.05

0

0.05

S

θ

25 30 35 40 45

0

0.05

0.10

0.15

Figure 5. As figure 4, but including the approximations of §§5–7. Thin dashed curves: geometricaloptics (saddle-point approximation) (5.4); thick dashed curves: geometrical optics + lateral wave(end-point approximation) (7.5); dotted black curve in (b): Brewster peak approximation (6.5),with inset illustrating the better fit for kr = 400. (Online version in colour.)

The required local approximation is

R2(q) = A(q − qB) + B(q − qB)2 + . . . , where A = 1 − n4

2n3

and B = 2 + n2 − n4 − n6 − n8

4n6. (6.1)

When inserted into equation (5.1), this gives, when expanded for q near qB,

j2,B = k3/2 exp((1/4) ip)√2pr sin q

∫∞

−∞da sin3/2(q + a)R2(q + a) exp{ikr cos a}

≈ k3/2 exp(

14ip

)exp(ikr)

×√

sin q

2pr

∫∞

−∞da[A(q − qB) + Aa + Ca2] exp

{−1

2ikra2

},

where C = B + 32A cot qB = 2 + 4n2 − n4 − 4n6 − n8

4n6. (6.2)

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The integrals are elementary, leading to

j2,B = k sin qexp(ikr)

r

[A(q − qB) − iC

kr

]. (6.3)

The correction proportional to C resolves the normalization indeterminacy,leading to the shifts

sq,B(q) = A√A2 + C 2 + 2A2(kr(q − qB))2 + (A4/C 2)(kr(q − qB))4

and s4,B(q) = 0

⎫⎪⎬⎪⎭. (6.4)

This shows that the Brewster peak in the lateral shift resembles the square rootof a Lorentzian function. As figure 5b shows, this is an accurate approximationto the exactly computed Brewster peak.

The height of the peak at q = qB is

sq,B(qB) = A√A2 + C 2

= 2n3

√4 + 16n2 + 20n4 + 12n6 + n8

. (6.5)

This is independent of kr, corresponding to a deflection angle that remainsconstant far from the surface, or, equivalently, to a lateral apparent imagedisplacement that increases as kr. For n = 2/3, sq,B(qB) = 24/

√26497 = 0.14739,

corresponding to a deflection of about 8.5◦. The 1/√

2 width of the peak, definedby angles qw at which

sq(qw) = 1√2sq(qB), (6.6)

is

qw − qB = ± 1krC

√C (

√2C 2 + A2 − C ). (6.7)

For n = 2/3, qw − qB = ±4.35809/kr . The 1/kr dependence means that the spatialrange over which the large lateral deviation occurs is proportional to thewavelength, independent of r .

There is also a Brewster peak in the otherwise negligible lateral shift forincidence from the less-dense medium, where of course there is no critical angle;this peak is also described by equation (6.4), with n > 1. Its height is given byequation (6.5); for n = 3/2, representing, glass, sq,B(qB) = 108/

√77713 = 0.38742,

corresponding to a deflection of about 22◦.

7. Lateral wave

For kr � 1, there is a contribution from the branch-point in the integral (equation(5.1)) as well as the saddle. In this section, we consider observation angles q thatare not close to qc, so the branch-point is well separated from the saddle-point

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cut

Re a

Im a

Clat

Ccrit

Figure 6. Integration contours for equations (7.2) and (8.1).

at q′ = q. For the branch-point contribution, we need the local approximation tothe reflection coefficient:

Ri(qc + a) ≈ 1 − 2√−2na

(1 − n2)1/4gi

(g1 = 1, g2 = 1

n2

). (7.1)

Substituting into equation (5.1), with q′ = qc + a gives the contribution from thebranch-point as

ji,lat = −2k3/2n2 exp((1/4)ip)√pr sin q(1 − n2)1/4

gi exp{ikr cos(q − qc)}

×∫Clat

da

√a exp(ip) exp{ikra sin(q − qc)}, (7.2)

with the contour shown in figure 6.This integral represents the lateral wave: as is well known (Brekhovskikh 1960),

the phase krcos(q − qc) can be interpreted in terms of a ray issuing from thesource in the critical direction, travelling in the rarer medium just below thesurface (figure 1), and then leaving, again in the critical direction, to reach thefield point (r , q). (In the three-leg calculation, the height h, representing thelocation of the surface if the source and image are fixed, cancels, as it does in thegeneral formulae of §4, leaving only the distance r from the geometrical image tothe field point.)

The contribution (7.2) is the lateral wave. For q < qc, deformation of thecontour towards a = −i∞ shows that the integral vanishes. Therefore, the lateralwave contributes only for q > qc, when use of∫

Clat

du√

u exp(ip) exp(iu) = −√p exp

(14ip

), (7.3)

gives

ji,lat = 2in2 exp(−(1/4)ip) exp{ikr cos(q − qc)}gi√sin q(1 − n2)1/4r2 sin3/2(q − qc)

Q(q − qc). (7.4)

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This is smaller than the geometrical wave (5.2), from the saddle-point,by a factor 1/kr. However, the oscillatory factor exp{ikr cos(q − qc)}, and thederivatives involved in the calculation of S, combine to amplify the lateral wave,leading to shifts of the same order as the geometrical:

sq,lat = 2Q(q − qc)

kr√

sin5 q sin(q − qc)(1 − n2)1/4

× Im[exp

{−2ikr sin2 1

2(q − qc)

}

× (|a1|2n2 exp(−im1) + |a2|2 exp(−im2))]

and sf,lat = 2√

sin q Q(q − qc)

kr√

sin3 q sin(q − qc)(1 − n2)1/4

× Im[

a∗1a2

(n2 exp

{2ikr sin2 1

2(q − qc) + im1

})

+ exp{−2ikr sin2 1

2(q − qc) − im2

}]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (7.5)

As figure 5 illustrates, these expressions fit the exactly computed oscillationsrather accurately. The fit appears less good for the 45◦ linear polarization and sf

(figure 5c), but it improves for larger values of kr.

8. Critical direction

At the critical angle, the lateral-wave formulae (7.4) and (7.5) diverge. This isbecause when q = qc, the saddle-point and the branch-point coincide, and theargument of §7, which depended on them being well separated, is not valid. Theversion of equation (5.1) appropriate for this case is not equation (7.2) but

ji,crit ≈ k3/2 exp{i(kr + (1/4)p)}√2prn

×∫Ccrit

da sin3/2(qc + a)Ri(qc + a) exp{−1

2ikra2

}, (8.1)

where the contour is shown in figure 6.For a reason to be explained later, the approximation (7.1) for the reflection

coefficients gives critical-angle shifts that are accurate only when kr is extremelylarge. For more moderate values, the following formula, capturing the fact that|Ri| = 1 if q > qc (total reflection) is better:

Ri(qc + a) ≈(

1 −√−2na gi

(1 − n2)1/4

) (1 +

√−2na gi

(1 − n2)1/4

)−1

. (8.2)

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Then, equation (8.1) and the required derivatives can be written as

ji(qc) ≈ kn

r√

2prexp

{i(kr + 1

4p

)}F0(ai(kr)),

1r

vr(rji(qc)) ≈ ikji(qc)

and vq(sin q ji(qc))q=qc ≈ −k2n2

√2p

exp{i(kr − 1

4p

)}

×[−i

√1 − n2

2nkrF0(ai(kr)) + 1√

krF1(ai(kr)) + 3

√1 − n2

2nkrF2(ai(kr))

]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

,

(8.3)in which

ai(kr) =√

2ng1

(kr)1/4(1 − n2)1/4

and Fm(a) =∫Ccrit

daam

(1 − a

√a exp(ip)

1 + a√

exp(ip)

)exp

(−1

2ia2

)⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

. (8.4)

Remarkably, the integrals can be evaluated analytically:

F0(a) = −√2p exp

(−1

4ip

)

− i√

2a2

exp(

− i2a4

) [p√

2(

1 + erf(

exp((3/4)ip)

a2√

2

))

+G

(−1

4, − i

2a4

)G

(54

)− G

(14, − i

2a4

)G

(34

)],

F1 (a) = 23/2

a2

√p exp

(−1

4ip

)

+ i√

2a4

exp(

− i2a4

) [p√

2(

1 + erf(

exp((3/4)ip)

a2√

2

))

+ 14

(3G

(−3

4, − i

2a4

)G

(34

)+ G

(−1

4, − i

2a4

)G

(14

))]

and F2 (a) = −23/2

a4

√p exp

(−1

4ip

)− 25/4

a3G

(34

)exp

(18ip

)

− 27/4

aG

(54

)exp

(−1

8ip

)+ √

2p exp(

14ip

)

− i√

2a6

exp(

− i2a4

) [p√

2(

1 + erf(

exp((3/4)ip)

a2√

2

))

+G

(−1

4, − i

2a4

)G

(54

)− G

(14, − i

2a4

)G

(34

)]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (8.5)

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0 100 200 300 400 500 0 100 200 300 400 500kr

scrit

sf

sq

–0.025

–0.020

–0.015

–0.010

–0.005

0(a) (b)

(c) (d)

–0.05

–0.04

–0.03

–0.02

–0.01

0

–0.010

–0.005

0

0.005

0.010

0.015

0

0.05

0.10

0.15

sf

sq

sf

sq

sf

sq

Figure 7. Critical-angle lateral and transverse shifts sq (blue), sf (red), as functions of distance krfrom the geometrical image, for (a) TE polarization (a1 = 1, a2 = 0); (b) TM polarization (a1 = 0,a2 = 1), (c) 45◦ linear polarization (a1 = 1/

√2, a2 = 1/

√2) and (d) circular polarization (a1 =

1/√

2, a2 = i/√

2). Full curves: shifts computed exactly from equation (4.10); dotted curves: shiftscomputed approximately from equations (8.3)–(8.5). (Online version in colour.)

Using equation (2.10), the Poynting vectors and lateral and transverse shifts caneasily be calculated and, as figure 7 illustrates, they agree very well with the shiftscomputed exactly (equation (4.10)).

From equation (8.4), the limit kr → ∞ corresponds to a → 0. Therefore, it istempting to use not the exact expressions (8.5) for the integrals Fm(a), but theirsmall-a two-term approximations

F0(a) = √2p exp

(−1

4ip

)− 25/4G

(34

)exp

(−1

8ip

)a + O(a2)

F1(a) = −2−1/4G

(14

)exp

(18ip

)a − 2

√2p exp

(14ip

)a2 + O(a3)

and F2(a) = −√2p exp

(14ip

)+ 21/43G

(34

)exp

(38ip

)a + O(a2)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

. (8.6)

This would be equivalent to using the simpler approximation (7.1) for thereflection coefficient, rather than equation (8.2), leading to the shifts

sr(r , qc) ≈ N , sq(r , qc) ≈ −NC

(kr)3/4

(|a1|2 + |a2|2

n2

)

and sf(r , qc) ≈ −NC

(kr)3/4Re

[a∗

1a2

(exp((3/8)ip)

n2− exp

(−3

8ip

))]⎫⎪⎪⎪⎬⎪⎪⎪⎭

, (8.7)

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0.2 0.4 0.6 0.8 1.00

0.5

1.0

1.5

2.0

2.5

a

|Fm|

1

2

0

Figure 8. Full curves: exact integrals Fm(a) (equations (8.5)); dotted curves: approximations (8.6),for the indicated values m = 0, 1, 2.

where the normalization N is given by

1N 2

= 1 + C 2

(kr)3/2

(|a1|2 + |a2|2

n2

)2

+ C 2

(kr)3/2Re2

[a∗

1a2

(exp((3/8)ip)

n2− exp

(−3

8ip

))]. (8.8)

But although these formulae do give the leading-order shifts for kr � 1, thelimiting regime is approached extremely slowly. To see why, notice that inequation (8.6), the second terms are smaller than the first if,

a √

p

23/4G(3/4)= 0.86 . . . (F0), a G(1/4)

27/4√

p= 0.61 . . . (F1)

and a 21/4√p

3G(3/4)= 0.57 . . . (F2)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

, (8.9)

consistent with graphs of the exact and approximate Fm(a) (figure 8). Theseinequalities imply that the condition

(kr)i � 64n2gi

(1 − n2), (8.10)

must be satisfied if the leading-order approximations (8.7) are to accuratelyrepresent the shifts. For n = 2/3, this gives, for the two polarizations, (kr)1 � 51and (kr)2 � 1312. Computer explorations (not shown) confirm that only for pureTE polarization does the leading-order formula accurately represent the lateralshift (there is no transverse shift for this special case) for kr < 2000—the largestvalue for which direct numerical integration is unproblematic.

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9. Concluding remarks

It is becoming clear (Angelsky et al. 2011) that the Poynting vector can beexplored experimentally for complicated fields with general polarization, raisingthe possibility of observing the shift phenomena identified here—if not in opticsthen surely for microwaves. These phenomena include: strong oscillations fromthe interference of the lateral wave with the geometrical wave; the Brewster peakfor TM polarization, also for incidence from the rarer medium; existence of atransverse shift for linear polarizations that are not pure TE or TM; and thequantitative dependences of the shifts on kr : 1/kr away from the critical angle,1/(kr)3/4 at the critical angle.

The large-kr approximations obtained here could be unified into a uniformrepresentation for the shifts, valid from the oscillatory regime q > qc, throughthe critical angle q = qc itself and into the partial reflection regime q < qc. Thiswould be an application to general shifts of the uniform approximations alreadyobtained (Brekhovskikh 1960; Lai et al. 1986) for the scalar wave function and theGoos–Hänchen shift for beams. It has not been presented here because it would bebased on the simple approximation (7.1) to the reflection coefficients, not the moresophisticated approximation (8.2). Therefore, as the argument of §8 (supportedby numerics) indicates, such a uniform approximation would be accurate only forextremely large kr, except for the special case of TE polarization.

Several extensions come to mind. (i) A parallel discussion of the transmittedwave would be fairly straightforward, (ii) complexifying the position of thesource would be a model for shifts from Gaussian beams (Deschamps 1971),complementing existing treatments but not restricted to paraxial fields, and (iii)the relative refractive index n could be complex or negative, leading to treatmentof shifts from diverging waves reflected from absorbing or left-handed materials.

I thank Dr Mark Dennis, Dr Jörg Götte, Prof. John Hannay and two referees for helpful commentsand suggestions.

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