Download Polarization-dependent Goos-Hänchen shift at a graded dielectric

Polarization-dependent Goos-Hänchen shift at a graded dielectric interface
W. Löfflera , M. P. van Extera , G. W. ’t Hoofta , E. R. Eliela , K. Hermansb , D. J. Broerb , J. P. Woerdmana
a Huygens
Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
of Technology Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands
b University
Abstract
We examine the polarization differential Goos-Hänchen beam shift upon total internal reflection, for a graded-index
dielectric interface. We find a generic scaling law where the magnitude of this shift depends solely on the product of
wavelength and gradient steepness. The analytic results are extended using transmission matrix calculations in cases
where the assumptions made to allow analytical treatment might become questionable. Two important cases in this
category are: (i) incident angle close to the critical angle and (ii) gradients with an overall thickness of the order of a
wavelength. We demonstrate this effect experimentally using a polymer-blend sample with a gradual refractive-index
transition induced by diffusion.
Keywords: graded-index (GRIN) optics, Goos-Hänchen (GH) shift, polarization, inhomogeneous media
1. Introduction
When a light beam undergoes total internal reflection
100% of its energy is reflected. Because evanescent
fields penetrate into the lower-index medium, the reflected beam does not follow the geometrically expected
path, but is translated a little, the amount depending on
the polarization of the incident light. This translation,
known as the Goos-Hänchen (GH) shift [1], is a correction to geometric optics. It arises because the planewave components of the incident beam pick up a phase
jump upon reflection that depends on the incident angle.
The GH effect has been studied for interfaces separating homogeneous dielectric media [1] and metamaterials [2], as well as for metallic [3] and absorbing
[4, 5] layers. In inhomogeneous media, the polarizationdependent GH effect is unexplored yet; since abrupt interfaces are a special case of graded transitions, new
insight can be obtained by comparing these two cases
(See, e.g., [6, 7]). The characteristic geometric parameter of total internal reflection, i.e. the critical angle, is
the same for gradual and abrupt transitions but the beam
is not abruptly but gradually steered into the reflected
direction. This brings us to the question that we address
here: How does the (polarization-dependent) GH beam
shift depend on the refractive index profile?
Email address: [email protected]
(W. Löffler)
Preprint submitted to Optics Communications
Since graded-index (GRIN) dielectrics are very important for waveguide optics, the influence of polarization on the phase of a propagating mode has been investigated theoretically in a cylindrical geometry [8].
However no analytical results or experimental tests for
the polarization differential GH beam displacement in
GRIN media have been reported. The GH shift in 1D
GRIN media has only been studied theoretically, for
perpendicular s (TE) polarization [9]. A related effect,
the transverse shift of a circularly polarized beam in a
GRIN medium can be described within the Berry-phase
framework [10]. Recently, Bliokh et al. [11] developed
an analytic theory describing ray paths in general inhomogeneous media emphasizing geometric phases. In
principle the analytic part of our investigations is contained in this theory, we however find direct analytic and
numeric results for the special, experimentally attractive
case of a planar (1D) structure, which was not discussed
before [10, 11]. Here we discuss only the case of total
reflection, tunneling of light in GRIN media has been
studied in [12, 13]. The 1D GRIN medium is also the
simplest case in the very young field of transformation
optics [14, 15]: There, the full potential of inhomogeneous media is demonstrated, concealing objects and
allowing omnidirectional retro-reflection to name a few
examples. The GH shift that we discuss here marks a
general limit of such devices.
In this paper we first give an analytical theory of the
polarization-dependent GH beam shift for a graded inApril 21, 2010
[17] Hy (x) = n(x) g(x), which results in
h
i
∂2x g + n2 k02 − β2 + δV(x) g = 0
δV(x) =
with
(3)
2
1 2
∂ n − 2 (∂ x n)2 .
n x
n
(4)
For sufficiently smooth GRIN media we can treat δV as
a small perturbation. This allows us to approximate the
propagation constant in the x-direction, k x,p , as
Figure 1: (Color online) Scheme of the experiment: A light beam impinges from the high-refractive index (n1 ) material upon the gradedindex (GRIN) transition layer where the refractive index gradually
reaches n2 . Right: Exemplary refractive index profile. The turning
point of the beam is at x = xtp .
k2x,p
=
k2x
"
#2
δV(x)
+ δV(x) ≈ k x +
.
2k x
(5)
Our problem is ideally suited to be solved using the
WKB method (see, e.g., [18]): We consider “smooth”
index variations where the refractive index changes only
little over a wavelength. The path of the beam is treated
ray-optically, but we keep track of the phase. The acquired phase starting at x = 0 up to the turning point
x = xtp and back (see Fig. 1) is for s polarization
Z xtp
φs = 2
k x (x0 )dx0
(6)
terface, subsequently we extend the results using numerical simulation for cases where the assumptions
made for analytical treatment are questionable. Finally,
we give a first experimental demonstration of the effect.
2. Theory and numerical simulations
0
As a starting point of our analytic description, we take
a linearly polarized Gaussian light beam that undergoes
total internal reflection in a 1D graded dielectric structure (see Fig. 1). Because we are interested in beam
displacements in the plane of incidence, it is sufficient
to analyze Maxwell’s equation in two dimensions.
For simplicity we assume an infinitely extended
refractive-index gradient. The gradient is along the x
axis and n(x) is decreasing with positive x; the whole
system is translationally invariant in the y-direction. We
assume n to be real, i.e. neglect absorption. The fields
can be separated like E(r) = E(x) exp(iβz) for the electric field and H(r) = H(x) exp(iβz) for the magnetic field
(β ≡ kz is the propagation constant parallel to the interface). We only need to consider one component of the
fields E and H, Ey and Hy . For s polarization (TE, Ey (x)
component) the wave equation is (see, e.g., Ref. [8]):
(1)
∂2x Ey + n2 k02 − β2 Ey = 0
and for p polarization (TM, Hy (x) component)
∂2x Hy − ∂ x ln n2 ∂ x Hy + n2 k02 − β2 Hy = 0,
while for p polarization:
Z xtp
Z
φp = 2
k x,p (x0 )dx0 ≈ φ s +
0
0
xtp
δV(x0 ) 0
dx (7)
k x (x0 )
The absolute GH shift at a GRIN interface has little
meaning since the reflection plane is undefined. However, the (interesting) physical differences of s and p polarized beams manifests themselves in the relative beam
displacement; this is evident by comparing Eqs. 6 and 7.
Therefore we focus here on the reflection phase difference between s and p polarization:
Z xtp
δV(x0 ) 0
φ p − φs =
dx
(8)
k x (x0 )
0
For easy interpretation we introduce a scaling parameter x0 , being a measure of the width of the gradient, and
rewrite the refractive index profile as n(x) = n0 f (x/x0 )
where n0 is a constant. We introduce the incident angle
as θinc , β ≡ o−1
n0 sin θinc with on0 ≡ λ/[2π n(0)], and
x̃ ≡ x0 /x0 . Then the differential phase shift is
(2)
where ∂ x is the partial derivative with respect to the xcoordinate; the vacuum wave number is k0 = 2π/λ. We
note that Eq. 2 is not analytically solvable for a general
refractive index profile n(x) (see [16]).
The TM (p) case can be rewritten into a form that
resembles the TE (s) case by making the substitution
φ p −φ s =
on0
x0
Z
0
x̃tp
f ( x̃) f 00 ( x̃) − 2 f 0 ( x̃) 2
d x̃ (9)
q
2 2
2
f ( x̃)
f ( x̃) − sin θinc
This differential phase is dependent on on0 and therefore essentially different from the conventional phase
shift at an abrupt interface, which is independent of
2
D ps = −on0
∂ φ p − φs
∂θinc
o2n0
.
x0
Goos-Hänchen shift Dps (µm)
0 µm
3
1 µm
2 µm
3 µm
4 µm
1
2
Refractive-index
profiles
1 .5
1 .4
4
1 .3
1 .2
1 .1
1 .0
0
1
2
3
4
Position (µm)
5
λ = 675 nm
0
4 1 .5
4 2 .0
4 2 .5
4 3 .0
4 3 .5
Incident angle (°)
4 4 .0
4 4 .5
4 5 .0
Figure 2: (Color online) Transmission matrix calculation of the differential GH shift D ps as a function of incident angle. Shown are calculations for different gradient thicknesses, the inset shows the profiles.
(10)
The first index profile we discuss is a linear refractive index gradient n( x̃) = n0 (1 − x̃) in the configuration shown in Fig. 1. The resulting differential GH shift
is plotted in Fig. 2 for different gradient thicknesses.
The largest shift (black curve) depicts the well-known
case of an abrupt interface. We focus first on angles
of incidence away from the critical angle. By introducing a gradual index change the differential GH shift is
strongly reduced, already for a 1 µm thick gradient, and
more so for 2, 3 and 4 µm thick gradients. This reflects
the scaling law of Eq. 11: by increasing the gradient
width the scaling factor on0 /x0 → 0. At the critical
angle, the conventional differential GH shift (at a hard
interface) diverges, this remains true for the GRIN case.
However, for θ > θcrit , the shift is reduced more quickly
for smoother interfaces than for abrupt ones (Fig. 2).
Now we focus on the influence of the shape of the
refractive-index transition on the differential GH beam
shift. To this end we calculated the GH shift for different
profiles: linear, complementary error function (profile
achieved by many diffusion-driven processes [23]), and
hyperbolic secant (ideal profile for graded-index fibers
[8]). The refractive-index difference is kept constant
(0.5) and the profiles have a comparable width of the
index gradient (see inset Fig. 3).
From an experimental point of view, the differential
GH shift curves for different profiles show a similar reduction (compared to the abrupt interface case, black
curve in Fig. 3); this is well explained by the overall
scaling of the GH shift in GRIN media (Eq. 11). More
precisely, the GH shift follows this scaling only on average; the details of the GH shift D ps (θinc ) are indeed
Substituting Eq. 9 into Eq. 10 leads to the main result
D ps ∼
Gradient thickness
5
Refractive index
wavelength (i.e. zero-order in on0 ). The latter disappears in GRIN media and Eq. 9 gives the first-order
correction. This was also found by Liberman and
Zel’dovich [19] as well as by Bliokh and Stepanovskii
[20]; they retrieved equivalent results for the phase difference. Our expression is more straightforward owing
to the introduction of the scaling factor x0 .
The integral factor in Eq. 9 depends on how smooth
the index transition from f (0) = 1 to f ( x̃tp ) = sin θinc
is; in general the integral is negative and of order O(1).
For quadratic profiles of the form n2 (x) = n2o (1 − x2 /x02 )
the integral is approximately −π.
In contrast to [19, 20] we are interested in the differential parallel beam displacement (D ps = D p − D s ).
This is calculated from Artmann’s formula [21] and depends only on the derivative of the reflection phases
∂φ p,s /∂θinc :
(11)
This highlights the scaling of the differential GoosHänchen shift at a GRIN interface; it vanishes if the
refractive-index transition region x0 is much larger than
the (reduced) wavelength on0 . The conventional firstorder (in on0 ) differential GH shift disappears in a
graded-index medium; from Eq. 11 we learn that only
the second-order differential GH shift survives.
The analytic solution (Eq. 9 and 10) has three limitations: (i) close to the critical angle the finite spatial extent of the index gradient becomes apparent, (ii) steep
wavelength-scale gradients violate the slowly-varying
potential requirement for the WKB method, and (iii) the
role of the exact shape of the refractive-index profile remains to be quantified. These cases require numerical
calculation.
The transmission matrix method [22] is ideally suited
for numerical calculation of the reflection phase for a
1D GRIN structure. The GRIN medium is approximated as a multilayer structure with 10 nm/layer (at
λ0 = 675 nm). We verified that this is sufficiently finegrained. We obtain the reflection phase depending on
incident angle, and using Artmann’s formula (Eq. 10)
the differential GH displacement D ps of p and s polarized beams upon passage through the GRIN structure is
determined. All calculations are for the glass (n = 1.5)
– air (n = 1.0) transition.
3
erfc Refractive-index
1 .5
3
1 .4
Refractive index
Goos-Hänchen shift Dps (µm)
lin, 0 µm
4
tion. The system we have chosen is a graded polymerblend made out of ethoxylated Bisphenol-A diacrylate
(EBADA) and tetrafluoropropylmethacrylate (TFPMA)
with refractive indices of 1.54 and 1.42, respectively
and negligible absorption in the relevant spectral range.
To induce polymerization by UV light the monomers
were mixed with the photo-initiator IRGACURE 819
(EBADA with 1 %, TFPMA with 2 % by weight).
The demonstration sample was prepared directly on
a face of an equilateral BK7 prism. Below a cover
slip, an approximately 130 µm thick film of the EBADA
was deposited and (partially) polymerized initiated by a
short UV exposure, with a dose below that for full crosslinking.
Using the same technique, a layer of TFPMA was
deposited on top of this layer. During the diffusion time
T D the system was kept undisturbed to allow diffusion
of the monomer TFPMA into the partially cross-linked
matrix of EBADA. Finally, the whole sample was fully
polymerized by UV exposure. The refractive indices of
the single polymer layers have been determined to be
1.54 (EBADA) and 1.42 (TFPMA). This corresponds
to a critical angle of 67.23◦ , which also applies for the
graded interface.
Collimated light from a superluminescent diode
(λ = 677 nm, ∆λ = 9 nm) was incident on the sample in
a θ − 2θ setup. This light was modulated continuously
between s and p polarization using a photo-elastic modulator. The differential positional shift of the reflected
beams for s and p polarization was detected using a silicon quadrant photodiode as detector, together with a
lock-in amplifier [26]; correction for Fresnel refraction
at all interfaces has been done. A residual birefringence
is present in the high-index layer, this results in an offset
in the differential displacement signal. For clarity, we
took care of this by vertically shifting the experimental
curve.
The resulting differential GH shift close to the critical angle is shown in Fig. 4: We see clear indication that
the relative displacement is strongly reduced for a GRIN
interface compared to the hard interface case. The theoretically expected GH shift for a 2 µm GRIN is plotted
for comparison.
profiles
5
1 .3
1 .2
sech
1 .1
lin, 2 µm
1 .0
2
0 .0
erfc
0
1
sech
4 2 .0
0 .5
1 .0
2 .0
λ = 675 nm
lin, 2 µm
4 2 .5
1 .5
Position (µm)
4 3 .0
4 3 .5
4 4 .0
Incident angle (°)
4 4 .5
4 5 .0
Figure 3: (Color online) Transmission matrix calculations of the differential beam shift for different refractive-index profiles: linear, complementary error function, and hyperbolic secant (the black curve
shows the case of an abrupt interface for comparison). The inset
shows the GRIN profiles.
Figure 4: Experimental GH shift for the polymer-blend GRIN sample in comparison with the theoretical curve for a 2 µm thick gradient
(n = 1.54...1.42). The experimental curve is displaced vertically as explained in the text. The inset sketches the experimental configuration.
highly sensitive to the GRIN profile shape. This is because WKB scattering amplitudes are very sensitive to
the potential profile [24, 25].
4. Conclusions
In conclusion, smoothening of a refractive index gradient results in a reduction of the polarization differential Goos-Hänchen effect if the GRIN thickness is of the
order of or larger than a wavelength. This can be seen as
a manifestation of decoupling of polarization and spatial
degrees of freedom in inhomogeneous media, provided
3. Experimental demonstration
For a first experimental demonstration we require
samples with a large refractive-index difference to (i)
maximize the GH shift and to (ii) achieve an experimentally accessible angle of total internal reflec4
that the inhomogeneity on0 /x0 is small. The differential
Goos-Hänchen effect in GRIN media is a second-order
correction to geometric optics; the well-known firstorder differential Goos-Hänchen shift disappears here.
This generalizes the conventional Goos-Hänchen effect.
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This work was supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) and
by the Seventh Framework Programme for Research of
the European Commission, under the FET-Open grant
agreement HIDEAS, no. FP7-ICT-221906.
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