# Tunable giant two-dimensional optical beam shift from a tilted linear
polarizer

Niladri Modak1,∗, Sayantan Das1, Priyanuj Bordoloi1, Nirmalya Ghosh1,+
1Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata,
Mohanpur, India- 741246

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We demonstrate an intriguing giant optical beam shift in a tilted polarizer
system analogous to the Imbert-Fedorov shift in partial reflection around the
Brewster’s angle of incidence. We explain this giant shift using a generalized
theoretical treatment for beam shift in a tilted uniaxial anisotropic material
incorporating the three-dimensional orientation of its optic axis in the
laboratory frame. We further demonstrate regulated control and tunability of
both the magnitude and direction of the shifts by wisely tuning the input
polarization and the corresponding orientation angles. These findings open up
the possibility of precise micron-level beam steering using a simple linear
polarizer.

The reflection and refraction of real optical beams at interfaces deviate from
Snell’s law when its cylindrical symmetry is broken [1]. In such a scenario,
the beam, comprising of distribution of many wave vectors, exhibits
polarization-dependent longitudinal and transverse deflections in the space
domain (spatial shift) or in the momentum domain (angular shift) [1]. These
so-called Goos–Hänchen (GH) and Imbert–Federov (IF) beam shifts have evoked
recent intensive investigations [2, 3, 4, 5, 6, 7]. While the GH shift
originates from the angular gradient of the polarization-dependent
reflection/refraction coefficients of the material, the IF shift has its
origin in spin-orbit interaction of light appearing as a consequence of the
evolution of momentum or space-gradient of the geometric phase leading to
space or momentum domain beam shifts [1]. These beam shifts observed in
diverse optical systems are not only fundamentally interesting [8, 5, 9, 10,
11, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20] but also have potential
applications in metrology and have opened up novel route towards development
of spin-orbit photonic devices [21, 7, 22]. The magnitude and nature of the
beam shifts in partial and total internal reflection strongly depend on the
Fresnel reflection and transmission coefficients of the interface and on their
angular dispersion [1]. In contrast, Aiello et al. introduced a novel variant
of transverse beam shift, namely, geometrical spin hall effect of light
(SHEL), which is independent of any material parameters and is purely
geometric [20]. This was first experimentally demonstrated in a tilted linear
polarizer system and has attracted particular attentions [19]. In a subsequent
study, it was shown that such effect could be observed in a generalized tilted
homogeneous uniaxial anisotropic system having either dichroism (polarizer) or
retardance (waveplate) [9, 23]. Importantly, it was demonstrated that this
beam shift is not purely geometric; rather, it depends upon the polarization
anisotropy parameter of the sample. Understanding the nature of such
intriguing beam shifts in tilted anisotropic systems, therefore, demands
further investigations both on conceptual and practical grounds [24, 10, 23,
11].

In this paper, we experimentally demonstrate an intriguing giant optical beam
shift in a tilted polarizer system analogous to the Imbert-Fedorov shift in
partial reflection around the Brewster’s angle of incidence [13]. Complete 2D
tunability of this giant beam shift is demonstrated further by tuning the
input polarization state and the 3D orientation angles of the optic axis of
the tilted polarizer.

To model the beam shifts in a general tilted uniaxial anisotropic system, we
consider two sets of coordinate systems: one representing the lab frame
$\\{x,y,z\\}$ with unit vectors $\\{\hat{e}_{x},\hat{e}_{y},\hat{e}_{z}\\}$
and the other describing the local reference frame
$\\{\parallel,\perp,\zeta\\}$ of the uniaxial anisotropic material having unit
vectors $\\{\hat{e}_{\parallel},\hat{e}_{\perp},\hat{e}_{\zeta}\\}$ (see
Fig.1(a) and (b)). Here, $\hat{e}_{z}$ is the direction of propagation of the
input beam, and $\hat{e}_{\parallel}$ represents the direction of the optic
axis of the uniaxial system. The azimuthal orientation $\psi$ of the optic
axis in the local frame and the tilt angle $\phi$ around $y$ axis provide a
complete description of the three-dimensional orientation of the optic axis of
the uniaxial system in the lab frame (see Fig.1(a) and (b)). Using this
geometrical framework, the unit vectors $\hat{e}_{\parallel}$ and
$\hat{e}_{\perp}$ in the lab frame can be obtained as (see Sec.(S1) of
supplemental material (SM)).

$\hat{e}_{\parallel}=\begin{pmatrix}\cos{\psi}\cos{\phi}\\\ -\sin{\psi}\\\
\cos{\psi}\sin{\phi}\end{pmatrix};\
\hat{e}_{\perp}=\begin{pmatrix}\sin{\psi}\cos{\phi}\\\ \cos{\psi}\\\
\sin{\psi}\sin{\phi}\end{pmatrix}$ (1)

Accordingly, the angle between the direction of propagation of the beam and
the optic axis becomes
$\theta\equiv\cos^{-1}{(\hat{e}_{\parallel}\cdot\hat{e}_{z}})=\cos^{-1}{(\cos{\psi}\sin{\phi})}$.
When a Gaussian beam passes through such a tilted anisotropic system with
$\theta\neq 0$, the output beam experiences polarization-dependent
longitudinal and transverse shifts in the local frame [9] (see Sec.(s1) of the
SM). The evolution of the electric field of the light beam through such a
system can be modeled using the momentum domain Jones matrix $\boldsymbol{M}$
(under the first order paraxial approximation) [9].

$\boldsymbol{M}=\begin{pmatrix}T_{\parallel}(1+\Theta_{\parallel}\mathfrak{X}_{\parallel})&T_{\parallel}\Theta_{\perp}\mathfrak{Y}_{\parallel}\\\
-T_{\perp}\Theta_{\perp}\mathfrak{Y}_{\perp}&T_{\perp}(1+\Theta_{\parallel}\mathfrak{X}_{\perp})\end{pmatrix}$
(2)

Here,
$\mathfrak{X}_{\parallel/\perp}=\frac{d\ln{T_{\parallel/\perp}}}{d\theta}\
\text{and}\
\mathfrak{Y}_{\parallel/\perp}=(1-\frac{T_{\perp/\parallel}}{T_{\parallel/\perp}})\cot{\theta}$.
$T_{\parallel/\perp}$ represent complex amplitude transmission coefficients of
the anisotropic material for the two orthogonal linear polarizations along
$\hat{e}_{\parallel}/\hat{e}_{\perp}$ directions, and
$\Theta_{\parallel/\perp}$ represents angular deviation or momentum spread of
the beam along the respective directions [9]. In general,
$T_{\parallel/\perp}$ encodes both amplitude anisotropy (dichroism for a
polarizer) and phase anisotropy (retradnce for a waveplate) effects. As
evident from Eq.(2), $\boldsymbol{M}$ not only depends on the tilt angle
$\phi$ but also on the local azimuthal orientation $\psi$ of the optic axis.
For $\psi=0$, this general formalism reduces to the previously reported one
[9, 23] (see Sec.(S1) of the SM). We shall subsequently illustrate the utility
of using the two different angles in the context of controlling the magnitude
and direction of beam shifts. Using the Jones matrix $\boldsymbol{M}$,
polarization-dependent beam shifts along $\hat{e}_{\parallel}$ and
$\hat{e}_{\perp}$ directions can be determined from the respective shift
matrix operators as $\boldsymbol{G}$
$=\frac{i}{k}\begin{pmatrix}\mathfrak{X}_{\parallel}&0\\\
0&\mathfrak{X}_{\perp}\end{pmatrix}$ and $\boldsymbol{S}$
$=\frac{i\cot{\theta}}{k}\begin{pmatrix}0&\mathfrak{Y}_{\perp}\\\
-\mathfrak{Y}_{\parallel}&0\end{pmatrix}$ (see Sec.(S1) of the SM) [9]. Here,
$k$ is the magnitude of the central wave vector of the beam. The longitudinal
shift (along $\hat{e}_{\parallel}$) appears due to the dispersion of the
transmission coefficients with respect to the relative angle $\theta$ between
the optic axis and the direction of propagation of the beam. The transverse
shift (along $\hat{e}_{\perp}$), on the other hand, appears due to the
evolution of geometric phase gradient which also depends on $\theta$ [9]. The
physically observable beam shift for a given polarization $\ket{E}$ of the
illuminating beam can be obtained using the expectation value of the
corresponding shift operator
$\matrixelement{E_{i}}{\boldsymbol{G}/\boldsymbol{S}}{E_{i}}$ where
$\ket{E_{i}}$ is input polarization. Note that here, the illuminating
polarization $\ket{E}$ is modulated by the zeroth order Jones matrix
$\boldsymbol{T}$ ($=\begin{pmatrix}T_{\parallel}&0\\\
0&T_{\perp}\end{pmatrix}$) (corresponding to the central wave vector
$\vec{k}$) of the uniaxial anisotropic system as
$\ket{E_{i}}=\boldsymbol{T}\ket{E}$.

Figure 1: Schematic illustration of the orientation of uniaxial anisotropic
material. The anisotropic system ($\\{\hat{e}_{\parallel},\hat{e}_{\perp}\\}$)
is depicted as a teal blue plate. $\\{x,y,z\\}$ is the lab reference frame.
Black lines on the plate indicate the direction of the optic axis. (a) The
angle $\psi$ between $\hat{e}_{x}$ and $\hat{e}_{\parallel}$ describes
azimuthal orientation of the optic axis. (b) $\phi$ is the tilt angle around
$y$ axis. (c) Experimental setup. A 633 nm beam from a He-Ne laser source is
passed through a Glan-Thompson linear polarizer P, a biconvex lens L (focal
length 200 mm), and a sample sheet polarizer SP (LPVIS100, Thor Labs, USA)
mounted in a tilted configuration ($\psi$ and $\phi$) and detected by a CCD
camera (iKon M-912, pixel size= $24\ \mu m$, Andor, USA). The experimental
parameters in Eq.(6) are noted in Sec.(S3) of SM.

We now turn to the specific case of a tilted polarizer, for which
$T_{\parallel/\perp}$ are real. For an ideal polarizer, the angular
dispersions of $T_{\parallel/\perp}$ are rather weak and accordingly the
longitudinal shift is negligible [23, 19]. Henceforth, for the experimental
perspective, we therefore consider contribution of the transverse shift
operator $\boldsymbol{S}$ only. Thus, Eq.(2) for a tilted polarizer with
$\psi$ and $\phi$-orientation angles can be simplified to [23]

$\displaystyle\boldsymbol{M}=(\mathbb{I}-ik\Theta_{\perp}\boldsymbol{S})\boldsymbol{T}=(\mathbb{I}-ik[\Theta_{x}\boldsymbol{S}_{x}+\Theta_{y}\boldsymbol{S}_{y}+\Theta_{z}\boldsymbol{S}_{z}$
$\displaystyle])\boldsymbol{T}$ $\displaystyle\text{with}\
\boldsymbol{S}_{x}=\sin{\psi}\cos{\phi}\boldsymbol{S};\boldsymbol{S}_{y}=\cos{\psi}\boldsymbol{S};\boldsymbol{S}_{z}=\sin{\psi}\sin{\phi}\boldsymbol{S}$
(3)

Clearly, the observable beam shifts along the lab $x,\ y,\ \text{and}\ z$
directions are determined by the $\psi$ and $\phi$-dependent components of the
shift matrix, $\boldsymbol{S}_{x}$, $\boldsymbol{S}_{y}$, and
$\boldsymbol{S}_{z}$ respectively. The magnitude of these shifts can be
determined from the eigenvalues $d_{\pm}$ of $\boldsymbol{S}$ with
corresponding eigenpolarizations $\ket{r_{\pm}}$.

$\displaystyle d_{\pm}$ $\displaystyle=$
$\displaystyle\mp\frac{i\cot{\theta}}{k}(\sqrt{T_{\parallel}/T_{\perp}}-\sqrt{T_{\perp}/T_{\parallel}})$
$\displaystyle\ket{r_{\pm}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{T_{\parallel}+T_{\perp}}}\begin{pmatrix}\sqrt{T_{\parallel}}\\\
\pm\sqrt{T_{\perp}}\end{pmatrix}$ (4)

As usual, the imaginary eigenvalues will be manifested as angular shift of the
output beam [13] and the corresponding eigenpolarizations are linear here (as
$T_{\parallel},\ T_{\perp}$ are real for a polarizer). Importantly, the
eigenvalues $d_{\pm}$ will become exceedingly large for an ideal polarizer
($T_{\perp}/T_{\parallel}\gg 1$). This very feature of a polarizer enables the
beam to experience a giant deflection $D=d_{+}-d_{-}$ along local transverse
direction $\hat{e}_{\perp}$ when the incident polarizations match with the
eigenpolarizations of $\boldsymbol{S}$ (see Sec.(S2) of the SM).

The above scenario of giant beam shift for incident eigenpolarization is
similar in spirit to that observed for light beam partially reflected from an
interface around the Brewster’s angle of incidence [13]. In general, the
transverse shift matrix operator in case of partial reflection can be written
as a complex combination of the Pauli matrices $\sigma_{2}$ and $\sigma_{1}$
[25, 1]. While $\sigma_{2}$ represents SHEL shift with circular
eigenpolarizations, $\sigma_{1}$ describes the IF shift with diagonal linear
polarizations as eigenstates [26, 1]. As one approaches the Brewster’s angle
of incidence from above ($>$ Brewster’s angle), the eigenvalues of the
transverse shift operator remain real yielding space domain beam shift and the
corresponding eigenpolarizations are extreme elliptical. Conversely, when
Brewster’s angle is approached from below ($<$ Brewster’s angle), the
eigenvalues become imaginary (yielding angular shift) and the corresponding
eigenpolarizations become linear. Our situation is thus in close analogy to
the latter scenario. The transverse shift here is angular in nature with
linear polarizations as eigen states [13].

Following Eq.(3), the components of $D$ in the lab frame $\\{x,y,z\\}$ become

$D_{x}=\sin{\psi}\cos{\phi}D;D_{y}=\cos{\psi}D;D=\sin{\psi}\sin{\phi}D$ (5)

The angular shifts $\Theta_{x,y,z}$ (corresponding to imaginary $D$) in the
lab frame between two eigenpolarizations $\ket{r_{\pm}}$ can be obtained from
Eq.(5). The corresponding physically observable eigenshifts $\Delta_{x,y,z}$
of the beam centroid is related as

$\Theta_{x,y,z}=\frac{1}{Z_{0}}\imaginary(D_{x,y,z});\Delta_{x,y,z}=Z\Theta_{x,y,z}$
(6)

Here, $Z$ is the propagation distance of the beam after passing through the
tilted polarizer and $Z_{0}$ is its Rayleigh range. As evident from Eq.(6),
the locally transverse $D$ imparts its counterparts in the lab $x$, $y$, and
$z$-direction, with their magnitudes determined by the $\psi$ and $\phi$
dependent pre-factors. Using our experimental embodiment with a 2D detector
(see Fig.1(c)), $\Delta_{x}$ and $\Delta_{y}$ can only be detected. In what
follows, we therefore proceed to experimentally demonstrate large
eigenpolarization beam shifts $\Delta_{x}$ and $\Delta_{y}$ and their
tunability by suitably controlling $\psi$ and $\phi$.

First, we investigate the shifts by fixing the azimuthal orientation of SP at
$\psi=0^{o}$ and varying the tilt angle $\phi$. Here, $\psi=0^{o}$ essentially
implies that only $\Delta_{y}\neq 0$. The polarization of the illuminating
beam $\ket{E}$ is selected and controlled by P, which needs to be set at the
left eigenpolarizations $\ket{l_{\pm}}$ of $\boldsymbol{S}$ to observe the
giant eigenshift (see Sec.(S2) of the SM). The results of the variation of the
$y$-shift of the beam centroid are summarized in Fig.2. $\ket{l_{\pm}}$
correspond to the angle of the input linear polarization for which the
$y$-shift of the beam centroid is observed to be maximum, and are accordingly
marked in the figure for two tilt angles, $\phi=20^{o}$ and $30^{o}$. Note,
the difference of $y$-shift between the two extreme points provides
$\Delta_{y}$. The observed large $\Delta_{y}$ of the beam centriod for the
eigenpolarization states are even visually perceptible from the beam profiles
across the white dotted reference lines for both $\phi=20^{o},\ \text{and}\
30^{o}$ (see Fig.2(b)). Fig.2(c) illustrates the dependence of the
corresponding eigenshift $\Delta_{y}$ on $\phi$. The experimental data are
fitted with $\Delta_{y}=a\tan{\phi}+b$ (obtained using Eq.(6)). In general,
the shifts are large and the fitted value $a\approx 536\mu m$ appears to be in
agreement to the exact theoretical prediction $a\approx 230\mu m$ (see
Sec.(S3) of the SM).

The above results provide experimental evidence of the giant beam shift for
eigenpolarization of light beam passing through a tilted polarizer. We
emphasize that this giant beam shift is not a manifestation of weak value
amplification [27, 28, 29, 30, 31, 18] and there is no additional post-
selection involved in our experiment unlike in [23]. Moreover, as evident from
the experimental parameters, we are not even in the weak coupling regime which
requires the difference between the eigenvalues to be much less than the
spread of the pointer [27, 28, 30, 31, 18]. On the contrary, here the
magnitudes of the eigenshifts are several times larger than the FWHM of the
Gaussian beam or the beam waist incident on the SP ($\approx$ 20 $\mu m$).
Note that similar to the Brewster’s scenario in partial reflection [13], here
also, the eigenshifts appear around the region of minimum intensity of the
transmitted beam (see Sec.(S2) of the SM).

Figure 2: Dependence of the experimental transverse ($y$) shift on the tilt
angle $\phi$ for fixed azimuthal orientation of the optic axis of the
polarizer at $\psi=0^{o}$. (a) Variation of the experimentally observed
$y$-shift with varying angle of polarization of the illuminating beam for two
representative tilt angles, $\phi=20^{o}$ (yellow dashed line) and
$\phi=30^{o}$ (brown solid line). The angles of illuminating polarization
($6.9^{o},8.0^{o}$ for $\phi=20^{o}$ and $6.9^{o},8.1^{o}$ for $\phi=30^{o}$)
corresponding to the left eigenpolarizations $\ket{l_{+,-}}$ (for eigenshift
$\Delta_{y}$) are marked by magenta, violet ellipses inside the figure. The
noted illuminating polarization angle are not the absolute angle of
polarization in the lab frame, but are the angle of the rotational mount
attached with preselecting polarizer P. (b) The intensity normalized beam
profiles generating $\ket{l_{+}}$ and $\ket{l_{-}}$ (left and right panel) for
$\phi=20^{o}$ (top panel) and $\phi=30^{o}$ (bottom panel). White scale-bar:
$1200\mu m$. (c)The variation of $\Delta_{y}$ as a function of $\phi$. Black
circle: experimental data. Blue dashed line: corresponding theoretical fit
with Eq.(6). The corresponding theoretical predictions of the exact magnitudes
of $\Delta_{y}$ are also shown additionally (solid line). Figure 3: Dependance
of eigenshifts $\Delta_{y},\Delta_{x}$ on the azimuthal orientation $\psi$ of
the optic axis of the polarizer tilted at $\phi=50^{o}$. (a) Variation of
$\Delta_{y}$ with changing $\psi$. Black circles: experimental data. Blue
dashed line: theoretical fit with Eq.(6). (b) Variation of $\Delta_{x}$ with
changing $\psi$. Black circles: experimental data. Red dashed line:
corresponding theoretical fit with Eq.(6). (c) The dependence of net beam
shift $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ on $\psi$. Black circles:
experimental data, magenta dashed line: theoretical fit. (d) The direction of
the shift $\tan^{-1}({\Delta_{y}/\Delta_{x}})$ on the $xy$ plane is plotted
with changing $\psi$. Black circles: experimental data; violet solid line:
theoretical prediction. (e) The intensity normalized beam profiles generating
two eigenpolarizations $\ket{l_{+}}$ and $\ket{l_{-}}$ (left and right panel)
for four different $\psi$ (different rows). White scale bar: $1200\mu m$.
Magenta arrows indicate the corresponding directions of shifts.

We now proceed to study the influence of the azimuthal orientation $\psi$ of
the tilted polarizer on the beam shift. It is evident from Eq.(6), for
$\psi\neq 0^{o}$, one will have contributions of both $\Delta_{x}$,
$\Delta_{y}$. We keep $\phi=50^{o}$ and vary $\psi$ over its entire range to
obtain full range of variations of $\Delta_{x}$ and $\Delta_{y}$ (see Fig.3).
Several interesting trends can be gleaned from the observed dependence of
$\Delta_{x}$, $\Delta_{y}$ on $\psi$. $\Delta_{y}$ attains its maximum around
$\psi=n\pi;n=0,\pm 1,\pm 2,\ldots$, (see Fig.3(a)). The magnitudes of the
shift are also found to be in good agreement with the corresponding
theoretical predictions. $\Delta_{x}$, on the other hand, is observed to be
much weaker as compared to $\Delta_{y}$, and it assumes both ‘$+$’ve and
‘$-$’ve values as anticipated from Eq.(6) (see Fig.3(b)). Relatively less
agreement with the theory in this case possibly arises due to the much smaller
magnitude of $\Delta_{x}$ (corresponds to only $\sim 3-4$ pixels of the CCD
camera). The maximum magnitude of the net beam shift
$\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ in the $xy$ plane is obtained at
$\psi=n\pi;n=0,\pm 1,\pm 2,\ldots$ (see Fig.3(c)). With regard to the
direction of the observed shift $\tan^{-1}{\Delta_{y}/\Delta_{x}}$ (see
Fig.3(d)), it is noted that around $\psi=(2n+1)\pi/2;\ n=0,\pm 1,\pm
2,\ldots$, both $\Delta_{x},\Delta_{y}\rightarrow 0$ and hence the direction
of the shift $\tan^{-1}{\Delta_{y}/\Delta_{x}}$ is undefined (marked with
black parallel lines in Fig.3(d)). Accurate experimental detection of the
shifts and obtaining their directions around these regions are very
challenging. On the other hand, the shifts are quite prominent around
$\psi=(2n+1)\pi/2;\ n=0,\pm 1,\pm 2,\ldots$. The observed discontinuity in
$\tan^{-1}{\Delta_{y}/\Delta_{x}}$ at $\psi=n\pi;n=0\pm 1,\pm 2,\ldots$ arises
due to the choice of the range of principle values of the $\tan^{-1}$ function
to lie within the interval ($-\pi/2,\pi/2$). These experimental results
confirm that the introduction of the local azimuthal orientation of the tilted
polarizer modifies the magnitude and the direction of locally transverse beam
shifts in the lab frame. This offers regulated 2D control of the beam shift.

It is however, apparent from Fig.3(c) and (d) that by regulating the angles
$\psi$, $\phi$, and the input polarization state, one can only obtain a
limited tuning of the beam shift on the lab $xy$ plane. The beam shift along
$x$ direction in the lab frame is not achievable by imparting the tilt around
$y$ axis. This limitation can be overcome if we introduce the tilt $\phi$
around $x$ axis separately as we demonstrate now. From Fig.4, it is evident
that the tilt around $\hat{y}$ (Fig.4(a)) complements the tilt around
$\hat{x}$ (Fig.4(b)) (see Sec.(S1) of the SM) for a fixed value of $\phi$.
These results clearly demonstrate that by introducing both the tilts and by
judiciously selecting the illuminating polarization state $\ket{E}$, in this
simple experimental embodiment, one can obtain completely tunable giant beam
shifts in the transverse $xy$ plane.

Figure 4: Polar illustration of the amplitude and direction of the net
eigenshifts in $xy$ plane from a tilted polarizer for varying azimuthal
orientation $\psi$ and tilt angle $\phi$. ‘$+$’-headed arrows: the shift for
$\phi=+50^{o}$, ‘$-$’-headed arrows: the shift for $\phi=-50^{o}$. Length of
the arrows: the magnitude of shifts, polar angle of the arrows: the direction
of the shifts. Different coloured arrows indicate eigenshifts for different
values of $\psi$. Direction and magnitude of eigenshifts for the tilt about
(a) $y$ axis and (b) $x$ axis.

In summary, we have demonstrated an intriguing giant optical beam shift in a
tilted linear polarizer system. We have understood this beam shift using a
robust theoretical framework for optical beam shift in a tilted anisotropic
system that encompasses general 3D orientation of the optic axis. Remarkably,
the experimental beam shifts are observed to be $\sim 300\mu m$, almost $10$
times the beam waist of the interacting Gaussian beam. This giant beam shift
is interpreted as an eignepolarization shift of the corresponding shift
operator for an ideal tilted linear polarizer, and an interesting analogy with
the transverse beam shift for partial reflection at Brewster’s angle of
incidence is found. We have experimentally demonstrated regulated 2D control
over such beam shift by wisely tuning the input polarization and the two
different angles describing the 3D orientation of the optic axis of the tilted
polarizer. The attainment of complete tunability of 2D giant beam shift using
a linear polarizer in a simple experimental embodiment may have potential
applications in precision beam steering [32].

## Supplemental material

### S1. Theoretical framework for polarization dependent tunable beam shifts
from a uniaxial anisotropic material

#### a.

We have represented the local co-ordinates $\hat{e}_{\parallel}$ and
$\hat{e}_{\perp}$ of the anisotropic materials in the laboratory frame
$\hat{e}_{x}$, $\hat{e}_{y}$, and $\hat{e}_{z}$ in the main text. Now we
demonstrate that how $\hat{e}_{\parallel}$ and $\hat{e}_{\perp}$ in the
laboratory frame can be obtained by consecutive operation of two rotation
matrices on $\hat{e}_{x}$ and $\hat{e}_{y}$ respectively (see Fig.5): rotation
$-\psi$ around $\zeta$ axis $R_{\zeta}(-\psi)$ and rotation $-\phi$ around $y$
axis $R_{y}(-\phi)$.

$\hat{e}_{\parallel/\perp}=R_{y}(-\phi)R_{\mathfrak{z}}(-\psi)\hat{e}_{x/y}$
(7)

Figure 5: Schematic illustration of the rotation introduced to evolve the
optic axis in three dimension when the tilt is given around $y$ axis.
$\\{x,y,z\\}$ is the laboratory reference frame, $\\{\parallel,\perp,\zeta\\}$
is the local frame of the material.

The optic axis swipes the full area of the deep green disk under a full
evolution of $\psi\ \text{from}\ 0\ \text{to}\ 2\pi$ (see Fig.5). This disk
swipes full the volume of the light green sphere when $\phi$ goes from $0$ to
$2\pi$. Thus one can locate the optic axis (along $\hat{e}_{\parallel}$) in
full three dimension using two consecutive rotation $R_{\zeta}(-\psi)$ and
$R_{y}(-\phi)$.

#### b.

We have introduced the azimuthal orientation $\psi$ in addition to the
previously reported tilt angle $\phi$ [19, 9, 23]. As Bliokh et al. predicted,
the longitudinal and transverse shift appears when there is a non-zero angle
$\theta$ between optic axis of the system and direction of propagation of the
central wave vector $\vec{k}$ of the beam [9, 23]. In our case, $\theta$
depends on both $\psi$ and $\phi$. At $\psi=0$, $\theta$ becomes $\pi/2-\phi$
which was obtained previously by Bliokh et al. [9, 23]. From Eq. (2) of the
main text, $\boldsymbol{M}$ suggests polarization dependent locally
longitudinal (along $\hat{e}_{\parallel}$) and transverse (along
$\hat{e}_{\perp}$) shift.

$\boldsymbol{M}=(\mathbb{I}-ik[\Theta_{\parallel}\boldsymbol{G}+\Theta_{\perp}\boldsymbol{S}])\boldsymbol{T}$
(8)

Where $\boldsymbol{G}$ is the shift matrix for longitudinal shift,
$\boldsymbol{S}$ is that for transverse shift, $\boldsymbol{T}$ is the zeroth
order Jones matrix of the anisotropic system under consideration. $k$ is the
central wavevector of the incident beam. As these local coordinate does not
coincide with the loboratory $\\{x,y,z\\}$ frame, both the local shifts have
their components along $\hat{e}_{x}$, $\hat{e}_{y}$, and $\hat{e}_{z}$. Both
the angles $\psi$ and $\phi$ control the magnitude and direction of the shifts
in the global frame. For the example of the tilted polarizer, the locally
transverse shift is manifested in the laboratory frame depending on the value
of $\psi$ and $\phi$ (see Eq.(5) and (6) of the main text).

Figure 6: The theoretically obtained variation of the eigen shifts with
changing azimuthal orientation $\psi$ at $\phi=50^{o}$ when the tilt is
imparted around $x$ axis. (a) Variation of $\Delta_{x}$ (red solid line) and
$\Delta_{y}$ (blue solid line) with changing $\psi$. (b) The dependence of
resultant magnitude of the shift $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$
(magenta solid line) and the direction of the shift
$\tan^{-1}{\Delta_{y}/\Delta_{x}}$ (violet solid line) on $\psi$. These plots
are obtained from Eq.(11). The parameters are taken same as mentioned in
Sec.(S3)

.

#### c.

As noted in the main text, tilt $\phi$ around $y$-axis provides only limited
tunability of the beam shift in the laboratory frame which can be overcome by
imparting the tilt $\phi$ around $x$-axis. Now we reiterate the framework for
the tilt around $x$-axis. In that case, the unit vectors of the local
reference frame of the system can be written as follows.

$\hat{e}_{\parallel/\perp}=R_{x}(-\phi)R_{\mathfrak{z}}(-\psi)\hat{e}_{x/y}$
(9)

The unit vectors take the following form in the laboratory frame.

$\hat{e}_{\parallel}=\begin{pmatrix}\cos{\psi}\\\ -\cos{\phi}\sin{\psi}\\\
\sin{\phi}\sin{\psi}\end{pmatrix};\
\hat{e}_{\perp}=\begin{pmatrix}\sin{\psi}\\\ \cos{\phi}\cos{\psi}\\\
-\cos{\psi}\sin{\phi}\end{pmatrix}$ (10)

Using Eq.(10) and following Eq.(6) of the main text, the eigenshifts in the
laboratory frame can be written as

$\Delta_{x,y,z}=\frac{Z}{Z_{0}}\imaginary(D_{x,y,z})$ (11)

Figure 7: The theoretically obtained variation of the magnitude of total eigen
shifts $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}+\Delta_{z}^{2}}$ and the magnitude
of eigen shifts in $xy$ plane $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ with
changing $\psi$ and $\phi$ for tilt around $y$ and $x$ axis. (a) Variation of
$\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}+\Delta_{z}^{2}}$ when the tilt is given
around $y$ axis. (b) The dependence of $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$
on $\psi$ and $\phi$ when the tilt is given around $y$ axis. (c), and (d)
Variation of the $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}+\Delta_{z}^{2}}$ and
$\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ with changing $\psi$ and $\phi$ when
the tilt is given around $x$ axis. The divergence in the magnitude of the
shifts can be overcome by including higher order terms of the shift matrices
[33]. The eigenshifts get maximized and minimized at the complementary values
of $\psi$ and $\phi$ for the tilt around $y$ and $x$ axis. The parameters are
taken same as mentioned in Sec.(S3).

Fig. 6(a) demonstrates the variation of eigen shifts $\Delta_{x}$ (red solid
line) and $\Delta_{y}$ (blue solid line) with changing the azimuthal
orientation $\psi$ at a fixed tilt $\phi=50^{o}$ around $x$ axis. In Fig.6(a),
the magnitude of resultant shift $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ is
plotted in magenta solid line with changing $\psi$. The direction of the shift
$\tan^{-1}({\Delta_{y}/\Delta_{x}})$ is plotted in violet solid line.
Comparing Fig.6 and Fig.3 of the main text, one can conclude that the tilt
around $x$ axis and $y$ axis are complementary to each other by means of the
direction of the eigenshifts.

We also plot the magnitude of the resultant shift in three dimension
$\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}+\Delta_{z}^{2}}$ and that in the $xy$
plane $\sqrt{\Delta_{x}^{2}+\Delta_{y}^{2}}$ with changing $\psi$ and $\phi$
for the tilt around $y$ (see Fig.7(a) and (b)) and $x$ (see Fig.7(c) and (d))
axis. Fig.7 is an evidence of complementarity of the magnitude of the eigen
shifts for the tilt around $y$ and $x$ axis. In conclusion, wisely tuning both
these tilts and selecting the input polarizations, one can steer the beam in
$xy$ plane.

### S2. Connection of the polarization of the incident beam to the eigen
polarizations of $\boldsymbol{S}$

$\boldsymbol{S}$ is a non-Hermitian matrix and hence, one can distinguish
between its left and right eigen polarizations. The right eigen polarizations
are given in Eq.(4) of the main text. The corresponding left eigen
polarizations are

$\ket{l_{\pm}}=\frac{1}{\sqrt{T_{\parallel}+T_{\perp}}}\begin{pmatrix}\sqrt{T_{\perp}}\\\
\pm\sqrt{T_{\parallel}}\end{pmatrix}$ (12)

When the illuminating polarizer P (see Fig.1 of the main text) matches with
$\ket{l_{\pm}}$, then the zeroth order Jones matrix $\boldsymbol{T}$ of the
tilted polarizer acts on it to modify it into $\ket{r_{\pm}}$ which then
interacts with the shift matrix. Hence one can differentiate between
illuminating polarization $\ket{E}$ and input polarization $\ket{E_{i}}$.

Figure 8: Brown solid line illustrates the variation of the intensity of the
output beam passing therough the tilted polarizer with changing angle of the
illuminating polarization at $\phi=30^{o}$ and $\psi=0^{o}$. The illuminating
polarization angle noted are not the angle of polarization in the laboratory
frame, but are the angle of the rotational mount attached with P.

Eq.(12) suggests that the illuminating polarization $\ket{E}$ should nearly
coincide with $\hat{\parallel}$ to achieve maximum shift from the tilted
polarizer. $\hat{\parallel}$ is the direction of the absorption axis of the
polarizer. Hence, the large shifts occur when the output beam intensity is
very low. Similar situation also arises for Brewster’s angle scenario in
transverse beam shift from a partial reflection. Fig.8 demonstrates the
variation of the intensity with changing the angle of the illuminating
polarization for tilt angle $\phi=30^{o}$. From Fig.2(a) of the main text and
Fig 8, it is clear that the maximum shift occurs around cross polarization
incidence.

### S3. Experimental parameters

We have experimentally measured the transmission coefficients $T_{\parallel}$
and $T_{\perp}$ with changing tilt angle $\phi$ at a fixed azimuthal angle
$\psi=0$. Fig.9 demonstrates the same. Fig.9 indicates that for our
experimental regime, $\frac{T_{\perp}}{T_{\parallel}}\sim 10^{2}\gg 1$. This
approximation is also supported by the previous reports [23, 19]. It is
reasonable to take $T_{\parallel}$ and $T_{\perp}$ as constant over the
experimental regime. We take $T_{\parallel}\equiv<T_{\parallel}>\approx
258.92$ and $T_{\perp}\equiv<T_{\perp}>\approx 1.98\times 10^{4}$.

Figure 9: The variation of $T_{\perp}$ (blue circled line) and $T_{\parallel}$
(red squared line) with changing tilt angle $\phi$ at azimuthal orientation
$\psi=0$.

In our experimental set up, the parameters mentioned Eq.(4) and (6) of the
maintext, take the following values. The wavelength of the input beam
$=\frac{2\pi}{k}=633\ nm$, the Rayleigh range $Z_{0}=1.37\ mm$, the
propagation distance after passing SP is $Z=18\ cm$. As mentioned, we take
$T_{\parallel}\equiv<T_{\parallel}>\approx 258.92$ and
$T_{\perp}\equiv<T_{\perp}>\approx 1.98\times 10^{4}$.

## References

  * [1] K. Y. Bliokh and A. Aiello, “Goos–hänchen and imbert–fedorov beam shifts: an overview,” Journal of Optics, vol. 15, no. 1, p. 014001, 2013\.
  * [2] M. Mazanov, O. Yermakov, A. Bogdanov, and A. Lavrinenko, “On anomalous optical beam shifts at near-normal incidence,” arXiv preprint arXiv:2107.09738, 2021.
  * [3] J. Wang, M. Zhao, W. Liu, F. Guan, X. Liu, L. Shi, C. T. Chan, and J. Zi, “Shifting beams at normal incidence via controlling momentum-space geometric phases,” Nature Communications, vol. 12, no. 1, pp. 1–7, 2021.
  * [4] K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin hall effect of light,” Science, vol. 348, no. 6242, pp. 1448–1451, 2015.
  * [5] T. Bardon-Brun, D. Delande, and N. Cherroret, “Spin hall effect of light in a random medium,” Physical review letters, vol. 123, no. 4, p. 043901, 2019\.
  * [6] K. Lekenta, M. Król, R. Mirek, K. Łempicka, D. Stephan, R. Mazur, P. Morawiak, P. Kula, W. Piecek, P. G. Lagoudakis, et al., “Tunable optical spin hall effect in a liquid crystal microcavity,” Light: Science & Applications, vol. 7, no. 1, pp. 1–6, 2018.
  * [7] B. Wang, K. Rong, E. Maguid, V. Kleiner, and E. Hasman, “Probing nanoscale fluctuation of ferromagnetic meta-atoms with a stochastic photonic spin hall effect,” Nature nanotechnology, vol. 15, no. 6, pp. 450–456, 2020.
  * [8] K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Physical review letters, vol. 101, no. 3, p. 030404, 2008.
  * [9] K. Y. Bliokh, C. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-hall effect and circular birefringence of a uniaxial crystal plate,” Optica, vol. 3, no. 10, pp. 1039–1047, 2016.
  * [10] O. Takayama and G. Puentes, “Enhanced spin hall effect of light by transmission in a polymer,” Optics letters, vol. 43, no. 6, pp. 1343–1346, 2018.
  * [11] W. Zhu, H. Zheng, Y. Zhong, J. Yu, and Z. Chen, “Wave-vector-varying pancharatnam-berry phase photonic spin hall effect,” Physical Review Letters, vol. 126, no. 8, p. 083901, 2021.
  * [12] M. Kim, D. Lee, and J. Rho, “Spin hall effect under arbitrarily polarized or unpolarized light,” Laser & Photonics Reviews, p. 2100138, 2021.
  * [13] J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Physical review letters, vol. 112, no. 23, p. 233901, 2014.
  * [14] F. Töppel, M. Ornigotti, and A. Aiello, “Goos–hänchen and imbert–fedorov shifts from a quantum-mechanical perspective,” New Journal of Physics, vol. 15, no. 11, p. 113059, 2013.
  * [15] A. Kavokin, G. Malpuech, and M. Glazov, “Optical spin hall effect,” Physical review letters, vol. 95, no. 13, p. 136601, 2005.
  * [16] M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Physical review letters, vol. 93, no. 8, p. 083901, 2004.
  * [17] C. Leyder, M. Romanelli, J. P. Karr, E. Giacobino, T. C. Liew, M. M. Glazov, A. V. Kavokin, G. Malpuech, and A. Bramati, “Observation of the optical spin hall effect,” Nature Physics, vol. 3, no. 9, pp. 628–631, 2007.
  * [18] O. Hosten and P. Kwiat, “Observation of the spin hall effect of light via weak measurements,” Science, vol. 319, no. 5864, pp. 787–790, 2008.
  * [19] J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs, “Observation of the geometric spin hall effect of light,” Physical review letters, vol. 112, no. 11, p. 113902, 2014.
  * [20] A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Physical review letters, vol. 103, no. 10, p. 100401, 2009.
  * [21] Z. Su, Y. Wang, and H. Shi, “Dynamically tunable directional subwavelength beam propagation based on photonic spin hall effect in graphene-based hyperbolic metamaterials,” Optics express, vol. 28, no. 8, pp. 11309–11318, 2020.
  * [22] X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin hall effect of light,” Reports on Progress in Physics, vol. 80, no. 6, p. 066401, 2017.
  * [23] K. Bliokh, C. Prajapati, C. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Optics letters, vol. 44, no. 19, pp. 4781–4784, 2019.
  * [24] Y. Qin, Y. Li, X. Feng, Z. Liu, H. He, Y.-F. Xiao, and Q. Gong, “Spin hall effect of reflected light at the air-uniaxial crystal interface,” Optics express, vol. 18, no. 16, pp. 16832–16839, 2010.
  * [25] G. Jayaswal, G. Mistura, and M. Merano, “Observation of the imbert–fedorov effect via weak value amplification,” Opt. Lett., vol. 39, pp. 2266–2269, Apr 2014.
  * [26] S. D. Gupta, N. Ghosh, and A. Banerjee, Wave optics: Basic concepts and contemporary trends.  CRC Press, 2015.
  * [27] Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Physical review letters, vol. 60, no. 14, p. 1351, 1988.
  * [28] I. Duck, P. M. Stevenson, and E. Sudarshan, “The sense in which a” weak measurement” of a spin-$1/2$ particle’s spin component yields a value 100,” Physical Review D, vol. 40, no. 6, p. 2112, 1989.
  * [29] N. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a “weak value”,” Physical review letters, vol. 66, no. 9, p. 1107, 1991.
  * [30] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, “Colloquium: Understanding quantum weak values: Basics and applications,” Reviews of Modern Physics, vol. 86, no. 1, p. 307, 2014.
  * [31] A. G. Kofman, S. Ashhab, and F. Nori, “Nonperturbative theory of weak pre-and post-selected measurements,” Physics Reports, vol. 520, no. 2, pp. 43–133, 2012.
  * [32] L. J. Salazar-Serrano, D. A. Guzmán, A. Valencia, and J. P. Torres, “Demonstration of a highly-sensitive tunable beam displacer with no use of beam deflection based on the concept of weak value amplification,” Optics express, vol. 23, no. 8, pp. 10097–10102, 2015.
  * [33] J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Optics letters, vol. 38, no. 13, pp. 2295–2297, 2013.