Beam deflection and negative drag in a moving nonlinear medium

Abstract

Light propagating in a moving medium is subject to light drag. While the light drag effect due to the linear refractive index is often negligibly small, the light drag can be enhanced in materials with a large group index. Here we show that the nonlinear refractive index can also play a crucial role in the propagation of light in moving media and results in a beam deflection. We perform an experiment with a rotating ruby crystal that exhibits a very large negative group index and a positive nonlinear refractive index. The negative group index drags the light opposite to the motion of the medium. However, the positive nonlinear refractive index deflects the beam along with the motion of the medium and hinders the observation of the negative drag effect. Hence, we show that it is necessary to measure not only the transverse shift of the beam but also its output angle to discriminate the light drag effect from beam deflection. Our work provides insight into applications for all-optical control of light trajectories, particularly for beam steering, mode sorting, and velocimetry.

Full text

544 Vol. 10, No. 5 / May 2023 / Optica Research Article
Beam deflection and negative drag in a moving
nonlinear medium
Ryan Hogan,1,* Akbar Safari,1Giulia Marcucci,1Boris Braverman,1AND
Robert W. Boyd1,2
1Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
2Institute of Optics, University of Rochester, Rochester, New York 14627, USA
*rhoga054@uottawa.ca
Received 20 September 2022; revised 16 March 2023; accepted 3 April 2023; published 24 April 2023
Light propagating in a moving medium is subject to light drag. While the light drag effect due to the linear refractive
index is often negligibly small, the light drag can be enhanced in materials with a large group index. Here we show that
the nonlinear refractive index can also play a crucial role in the propagation of light in moving media and results in a
beam deflection. We perform an experiment with a rotating ruby crystal that exhibits a very large negative group index
and a positive nonlinear refractive index. The negative group index drags the light opposite to the motion of the medium.
However, the positive nonlinear refractive index deflects the beam along with the motion of the medium and hinders
the observation of the negative drag effect. Hence, we show that it is necessary to measure not only the transverse shift
of the beam but also its output angle to discriminate the light drag effect from beam deflection. Our work provides
insight into applications for all-optical control of light trajectories, particularly for beam steering, mode sorting, and
velocimetry. © 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
https://doi.org/10.1364/OPTICA.476094
1. INTRODUCTION
Propagation of light in moving media has been studied for more
than two centuries [1–11]. Upon propagation, the trajectory of
light can be manipulated through self-action effects [12,13], beam
deflection [14,15], photon-drag [16–18], and many other phe-
nomena. The photon-drag effect was hypothesized by Fresnel [1],
and then experimentally observed by Fizeau [2]. Fizeau’s landmark
experiment measured the shift of interference fringes within an
interferometer containing a tube with moving water. These shifts
in the fringes supported the idea that light is dragged in moving
media. This phenomenon has gained increasing interest in the field
of optics and is indeed still investigated in modern day research
[6,7,10,19–23].
Photon-drag can be longitudinal or transverse, i.e., along or
perpendicular to the light propagation direction, respectively. This
paper focuses on transverse photon-drag [24], distinctly different
from longitudinal drag [25]. Figure 1(a) sketches the light propaga-
tion in a medium of length Lin two cases: (i) a stationary medium
and (ii) a medium moving transversely at speed vin which the
output light is subject to a transverse shift given by
1y=v
cng−1
nφL,(1)
with cthe speed of light in vacuum, and ngand nφthe group and
phase indices, respectively.
Typically phase and group indices are not large, and therefore do
not create large transverse shifts. Recent studies show larger shifts
using slow light media (i.e., large group indices) [11,23,25,26].
Group indices as large as ng=106are often achieved through
nonlinear phenomena such as coherent population oscillation
(CPO) [27] or electromagnetically induced transparency (EIT)
[28]. Therefore, utilizing these effects can significantly boost the
photon-drag effect, or allow negative photon-drag [29].
Although some nonlinear phenomena, such as EIT and CPO,
can enhance the group index, the presence of an intense beam
invites other nonlinear responses, specifically Kerr-type nonlin-
earities that will also contribute a deflection to the measurement.
In the presence of a strong saturating beam, one must consider
nonlinear deflection in a moving medium, which can be larger than
and confused with the photon-drag effect. While the enhanced
photon-drag effect depends on the group index including any
nonlinear contribution [see Eq. (1)], the nonlinear deflection
depends on the nonlinear refractive index of the medium. Thus,
it seems that one should be able to achieve a large enhancement
in the drag effect with negligible nonlinear deflection. However,
according to the Kramers–Kronig relation, a large group index
often is associated with a sluggish response [30]. Therefore, if
the large group index is achieved through a nonlinear interac-
tion, one has to be careful with the nonlinear deflection and
measure the output angle, a critical step missing in previous
works [26,31,32].
In this paper, we use a rotating ruby rod to study the nonlinear
light propagation in a moving medium. To reach high speeds in a
solid material, rotation is more feasible experimentally than linear
motion. Consider a beam incident on the medium at a distance
2334-2536/23/050544-08 Journal © 2023 Optica Publishing Group

Research Article Vol. 10, No. 5 / May 2023 / Optica 545
Fig. 1. (a) Schematic showing laser beam propagation in (i) a station-
ary medium versus (ii) a moving medium that exhibits a transverse shift
of 1y. For simplicity of illustration, we show the laser beams as pulses.
(b) The edge of a rotary ruby rod is used to achieve an approximately
linear motion in the −y(+y) direction when the crystal rotates clockwise
(counterclockwise). (c) Single frame imaged at the input face of the crystal
(z= −2 cm) showing o- and e-beams propagated through the 2 cm long
ruby crystal. (d) Diagram showing the trajectories of o- and e-beams at
different crystal orientations highlighting the change in intensity of each
beam at 45 deg intervals. The red “x” shows the center of intensity (COI)
position for different crystal orientations highlighting the emergence of
a figure-eight-like pattern, while o- and e-beams are shown by green and
blue dots, respectively, with varying transparency to signify their relative
intensities.
rfrom the center of rotation [Fig. 1(b)], and using a slow light
medium with ng1/nφ; the transverse drag can be simplified to
1y≈ngLr
c,(2)
where is the medium’s rotational speed, and the medium radius
is r.
Similar to alexandrite [27], a recent study shows that ruby
exhibits an extremely large negative group index (ng≈ −106)
at large laser intensities at 473 nm wavelength [26,33]. Hence,
according to Eq. (2), one expects to observe a large negative
photon-drag effect in which the position of the beam shifts in the
direction opposite to the motion of the medium.
Nevertheless, since ruby also exhibits nonlinear refraction, the
beam deflects toward the direction motion of the medium due to
nonlinear deflection. Furthermore, the birefringence of the crystal
splits the input beam into ordinary (o) and extraordinary (e) beams
that separate upon propagation in the crystal. The e-beam revolves
with the rotation of the ruby rod. Moreover, the propagations of
o- and e-beams are coupled through the nonlinear interaction in
ruby, which creates an attractive force between beams and further
complicates their trajectory. We study the trajectories of these
beams experimentally and simulate the propagation using non-
linear Schrodinger equations. Due to the simultaneous presence
of birefringence, intensity-dependent photon-drag, and strong
nonlinearity, ruby can serve as a solid-state platform rich in physics
with potential applications to beam steering [34,35], polariza-
tion detection [36,37], image rotation [26,32], velocimetry [38],
and potential for solitonic behavior with associated applications
[39–41].
2. METHODS
The laser source used in the experiment, as shown in Fig. 2, is a
continuous-wave (CW) diode-pumped solid-state laser operating
at 473 nm with an output power of 520 mW. We control the power
of the laser beam using a half-wave plate and polarizing beam
splitter. We use a 2 cm long ruby rod, 9 mm in diameter, with a
Cr3+doping concentration of 5%. We focus the laser beam onto
the input face of the crystal near the edge (0.1 mm away) far from
the center of rotation (r=0.35 mm). The ruby was mounted in a
hollow spindle whose rotation was controlled by a stepper motor
and belt. The output face of the crystal was imaged onto a CCD
camera using a 4-f lens system.
When linearly polarized light is shined onto the rotating
birefringent medium, the light sees two refractive indices upon
propagation, no=1.770 and ne=1.762. Without any influence
of nonlinearity or photon-drag, the two beams (o- and e-) then
propagate with a finite angular separation of γb=8 mrad, known
as a birefringent walk-off. The relative beam intensity reaches max-
ima and minima each quarter turn of the crystal (i.e., every 90 deg).
Fig. 2. 520 mW continuous-wave laser beam at 473 nm is focused
using a 100 mm focal length plano–convex lens L1to a spot size of 20 µm
onto the input face of rotating ruby rod. The rod spins around its axis
driven by a stepper motor. The laser beam at the output of the crystal is
imaged onto a CCD camera with unity magnification using a 4-f system
consisting of two lenses L2and L3of focal length f=150 mm. The CCD
camera captures the beam, with a frame rate of 1000 fps, as the stepper
motor is rotated at various speeds. An ND filter is placed between the
dielectric mirror and lens 2, L2, for nonlinear measurements, and between
L1and the ruby for linear measurements. The CCD camera images at
different zpositions using a translation stage. Measurements are taken
at z=0, z =0.762, and z=1.524 cm to measure the transverse shift,
as well as the output angle of the beam as it exits the crystal. The fluo-
rescence filter F F (high transmission near 473 nm) is used to minimize
fluorescence. The dielectric mirror D M is used as a neutral density filter
with low absorption to limit the beam intensity for high-power tests,
while also minimizing image distortions due to aberrations induced by
thermal nonlinearities in a standard neutral density filter. Input beam
power was controlled by a half-wave plate and a polarizing beam splitter
before the ruby crystal. M, mirror; HWP, half-wave plate; PBS, polarizing
beam splitter; BD, beam dump; L1, plano–convex lens [f =100 mm]; L2,
plano–convex lens [f =150 mm]; L3, plano-convex lens [f =150 mm];
FF, fluorescence filter; DM, dielectric mirror; ND, neutral density filter
[O.D. 1]; CCD, charge-coupled device.

Research Article Vol. 10, No. 5 / May 2023 / Optica 546
The beam input is aligned such that, regardless of crystal orienta-
tion, the o-beam propagates directly through the crystal, while the
e-beam revolves around the o-beam. When imaging the input face
of the crystal onto the camera, the o- and e-beams appear distinctly
on the image [Fig. 1(c)]. However, they diffract and overlap upon
propagation and become indistinguishable at the output of the
crystal, where the transverse shifts are measured. Therefore, we use
the center of intensity (COI) of the output light, represented as a
red “x” in Fig. 1(d), to track the motion of the average position of
the laser beam.
3. RESULTS
We measure the COI at z=0 for three input powers of 0.2, 100,
and 520 mW and rotational speeds of = ±50,±100,±1000,
and ±9000 deg/s in clockwise (negative) and counterclockwise
(positive) directions. Figure 3shows the COI trajectories for an
input laser power of 0.2 mW, considered as the linear regime. We
observe that all speeds trace out figure-eight-like trajectories and
do not show a transverse shift. The figure-eight trajectory is a result
of the e-beam revolving around the o-beam in the rotating crystal.
Note that in the linear regime, the group index is of the order of
unity, and the transverse drag effect is negligibly small.
Figure 4shows the COI trajectories in the nonlinear regime. At
low speeds (≤100 deg/s), the o- and e- beams couple to each
other causing significant variation in the traces of the COI upon
rotation. At high speeds, the deviations from a figure-eight-like
pattern start to average out and resemble those of the linear results.
However, in contrast to the linear regime, a transverse shift in the
COI is very clear.
The deviations from a figure-eight pattern are more apparent in
the highly nonlinear regime (Fig. 5) and where the trajectories are
noisier and do not average out at high speeds. This noise is likely
due to a large thermal effect that locally affects the beam frame by
frame. With lower speeds, we observe that the figure-eight-like
COI trajectories knot near the center as a result of the nonlinear
coupling of o- and e- beams. Simulations are compared showing
agreement in the traced patterns, and magnitudes of the transverse
shift. A noticeable discrepancy between measurements and sim-
ulations is that the experimental trajectories do not close near the
center of the figure-eight. We attribute this disagreement to the
assumption that rotational movement is approximately transla-
tional along the ydirection. However, the beam is not infinitely
far from the center of rotation and therefore would experience a
small amount of drag in the xdirection. Another reason might arise
from the imperfect Hpolarization caused by the optical elements.
Moreover, if the crystal faces are slightly non-parallel, this also
could cause a difference in the output angle of the light depending
on the crystal orientation in addition to the birefringence effects.
Fig. 3. (a) Experimentally measured COI trajectories in the linear regime. (b) Simulated COI trajectories in the linear regime. The color scheme of
the legend in (a) is the same for the simulated curves shown in (b). COI trajectories are plotted for rotation speeds of = ±50,±100,±1000, and
±9000 deg/s. Here, clockwise and counterclockwise rotation (looking into the beam) correspond to positive and negative rotation speeds, respectively.
There is not any significant shift between the figure-eight-like trajectories at different speeds, as the group index and nonlinear refraction are negligibly small
in the linear regime. The figure-eight-like pattern does not close in the center for the experimental results due to the polarization impurity of incident light
in the low-power regime.
Fig. 4. (a) Experimentally measured and (b) simulated COI trajectories in the nonlinear regime (input laser power of 100 mW). COI trajectories of
o- and e- beams [schematically shown in Fig. 1(d)] are plotted for an input laser power of 100 mW, considered as the nonlinear regime for different rota-
tional speeds () in units of deg/s. At low speeds, o- and e-beams couple to each other, causing significant variation in the traces of the COI upon rotation.
Therefore, these patterns are different from the figure-eight-like patterns seen in the linear regime. At high speeds, the deviations from a figure-eight pattern
start to average out. All that remains in the high-speed limit is that the figure-eight patterns are shifted from one another for positive and negative rotation
speeds as a result of nonlinear deflection.

Research Article Vol. 10, No. 5 / May 2023 / Optica 547
Fig. 5. (a) Experimentally measured and (b) simulated COI trajectories in the nonlinear regime (input laser power of 520 mW) for different rotational
speeds () in units of deg/s. At low speeds, trajectories are significantly distorted and have paths similar to the 100 mW results, but with more distortion
due to stronger nonlinear coupling between beams. At high speeds, the coupling between beams is weaker due to the finite response time of the medium.
For slow speeds ≤100, the trajectories are very noisy, and no discernable pattern is easily observed. This behavior is mainly due to the thermal gradient
impressed on the crystal by intense illumination, and therefore, the transverse beam shape is drastically modified.
Fig. 6. Experimental and simulated amount of shift in the beam’s
transverse position at the end of the crystal for 0.2, 100, and 520 mW
input beam laser power. The measured shift for the linear regime
(i.e., P0=0.2 mW) for both experiment and simulations is multi-
plied by a factor of 10, showing that there is no discernible deviation from
zero shift. The magnitude of the transverse shift is shown against the mag-
nitude of the rotation speed. This shift is calculated between the position
with no rotation, i.e., =0 deg/s, and the respective transversely shifted
position. Simulations are plotted using dotted lines in green and red for
nonlinear and highly nonlinear regimes, respectively, for better compari-
son to experimental data. The fits were based on a phenomenological
exponential function in Eq. (2). The fit is not a perfect match due to the
simulated nonlinear response of the material acting on the beams upon
propagation through the crystal.
We extract the average position of these COI trajectories over an
integer number of full rotations. Figure 6shows the rotation speed
dependence of the extracted transverse shift at z=0 (crystal’s
output face) for linear, nonlinear, and highly nonlinear regimes.
While there is no clear transverse shift in the linear regime, the
nonlinear regimes show a trend similar to that of a log-normal
function centered around =100 deg/s. Therefore, the observed
transverse shift is a purely nonlinear effect.
We note that the transverse spatial shift can, in principle, com-
prise nonlinear photon-drag and nonlinear deflection. While the
photon-drag transversely shifts the beam with an output parallel to
the input beam, the nonlinear deflection deflects the output beam
at an angle with respect to the input. Therefore, to discriminate
these two effects, we measure the transverse shifts at z=0 and
two other locations after the crystal to find the output angle. It is
important to note that we cannot subtract out the deflection to get
Fig. 7. (a) Schematic showing the output beam’s angle after leaving
the crystal. The nonlinear response of the crystal changes the angle at the
interface of the crystal output face and therefore changes the propagation
pathway. (b) The output angle and its uncertainty are calculated from
the beams’ transverse positions measured at three points along the zaxis
(z=0, z =0.762, and z=1.524 cm). As the laser’s power increases, the
output angle increases as expected from nonlinear deflection.
the true shift due to the photon-drag effect. The changes in the tra-
jectory inside the crystal by nonlinear deflection cannot be imaged.
As expected from nonlinear deflection, this angle is nonzero and
intensity and rotation speed dependent (Fig. 7).
4. DISCUSSION
A. Nonlinear Refraction
Intense linearly polarized light in a rotating birefringent medium
causes o- and e-beams to both experience nonlinear refraction,
as the maximum intensity continuously moves between them
creating a moving index gradient. The gradient leads to nonlinear
coupling between beams, where the local index variation pulls one
beam toward the other with the higher refractive index, locally
distorting the figure-eight-like COI trajectory. The magnitude of

Research Article Vol. 10, No. 5 / May 2023 / Optica 548
the distortions is dictated by the rotation speed, where the speed
controls the amount of time that the beam imprints an index
gradient on the crystal. The maximum strength of beam coupling
is observed at low speeds when the beams have sufficient time to
imprint the maximum nonlinear index. On the contrary, higher
rotation speeds imprint less gradient, blurring the effect of nonlin-
ear refraction, and non-distorted figure-eight-like trajectories are
recovered.
The speed regimes highlight different interaction time scales
of the nonlinear response. Both optical and thermal processes are
relevant; however, thermal processes dominate at slow speeds and
optical at high speeds. Since the time scale of thermal processes
is of the order of several hundred microseconds [42], this would
have a greater effect locally with slower rotation speeds. However,
we examine the effects over a complete cycle, and therefore, high
rotation speeds are affected more by optical time scales, which in
our case are of the order of 3–5 ms.
We model the temporal dynamics in these two regimes using
a phenomenological fit consisting of two decaying exponentials
discussed later in this paper where we take an analogy to spatial
self-steepening [43,44]. That is, the beam is shifted due to the
group index, and therefore the group velocity, which is intensity
and rotation speed dependent. This rotation speed dependence
therefore samples the dynamics representing a non-instantaneous
temporal response of the system.
This behavior could be considered an effective time-varying
response. Time-varying media often rely on highly nonlinear
materials, such as epsilon-near-zero materials [45], that change
the refractive index in time [46,47], inviting optical effects such
as non-reciprocity [48,49]. The strong nonlinear optical response
of ruby could perhaps exhibit non-reciprocity due to an effec-
tive time-varying effect, but further work needs to be done. The
magnitude of the index gradient induced by nonlinear refraction
(1n=3×10−3) is shown in Supplement 1, as well as its use in
the formation of a Townes profile in the steady-state, stationary
medium case.
B. Simulations
To better understand the experimental results, we model and sim-
ulate nonlinear propagation of linearly polarized light through a
2 cm long rotating birefringent ruby rod, where o- and e-beams
are created and vary in relative intensity upon rotation. Due to
the weak birefringence typically associated with ruby and intense
illumination, these two beams couple to each other upon rotation.
Both beams are modeled using the nonlinear Schrödinger equa-
tion where we apply a split-step Fourier method and propagate
the two with a coupling term containing a nonlinear response
function. Following the derivation of Marcucci et al. [50], we write
wave equations for the medium using a Kerr-type nonlinearity of
thermal origin. Rotation and birefringence are also included [51].
Furthermore, a term for the effective group index is incorporated
into the coupled equations, which are intensity and rotation speed
dependent.
Using our theoretical framework, including the phenomeno-
logical fit convoluted with nonlinear propagation, we were able
to accurately simulate the amount of transverse shift, and the
transverse movement of the COI of the beams observed in the
experiment. We develop a set of generalized coupled nonlinear
Schrodinger equations, written as follows:
−∂Eo
∂z+i
2ko
∇2
⊥Eo+neff
g
∂Eo
∂y+iko
no
1nNL Eo=0,(3)
−∂Ee
∂z+i
2kecos2(γ ) ∇2
⊥Ee
+ike
necos2(γ ) 1nNL Ee+neff
g
∂Ee
∂y
+2 tan(γ ) cos(t)∂Ee
∂x+sin(t)∂Ee
∂y=0,(4)
where fields Eoand Eerepresent o- and e- beams, ko,eare o- and
e-beam wave vectors, and no,eare o- and e-beam refractive indices,
respectively. Furthermore, γis the tilt angle, ∇2
⊥=∂2
∂x2+∂2
∂y2is the
transverse Laplacian operator, and 1nNL is the nonlocal Kerr-type
nonlinearity that contains the coupling term in the kernel function
[50]. We also highlight that the nonlinear deflection term works on
the derivative of the field rather than the field directly like that of
nonlinear refraction. Dispersion is also included in the nonlinear
Schrodinger equation; however, this effect is written as neff
g
∂E
∂y
rather than neff
g
∂E
∂tc−1, when considering a CW laser rather than
pulses. This term will drive the nonlinear deflection measured at
the crystal output face, where neff
g:= neff
g(, I), as in Eq. (7).
The magnitude of the nonlinear deflection is proportional
to the magnitude of the effective group index, controlled by the
intensity and rotation speed. The rotation speed changes the con-
ditions for how quickly heat dissipates through the crystal, and
thus the magnitude of the index gradient. If the speeds are suffi-
ciently slow, the index gradient stays relatively constant and causes
an increasing amount of transverse shift. Typically, the time scale
needed to deflect the beam is always very short [i.e., 2 cm/(c/ng)];
however, once the maximum amount of transverse shift is met,
i.e., ≈100 deg/s, the crystal starts rotating faster than the time
scale needed to form the index gradient. As we increase the rotation
speed, the beam sees less index gradient and thus less transverse
shift. The curve associated with the transverse shift versus rotation
speed, seen in Fig. 6, comprises two decaying exponentials centered
about =100 deg/s. The decay rates of these two exponentials
give rise to an asymmetric distribution about =100 deg/s.
For slow speeds, the index gradient decays slower and does not
blur. At higher speeds, the beam samples only some of the index
gradient due to the shorter time scale of the rotational motion and
is therefore smaller in magnitude. This is shown in Fig. 6, where the
behavior is not symmetric about =100 deg/s.
We draw an analogy to a self-steepened pulse to explain the
asymmetry of the transverse shift versus rotation speed. In a self-
steepened pulse, the group velocity travels at different speeds
dependent on the intensity. As such, different parts of the pulse
travel at speeds according to the group velocity. Higher intensities
seen at the peak of the pulse are associated with a large group index
and thus slower group velocity. At the wings of the pulse, the inten-
sity is lower and the group velocity is larger. This causes the pulse to
become asymmetric in time, and thus the material response will be
temporally asymmetric. The rotation speed controls the amount
of time that maximum intensity is in a given area, and therefore,
the index gradient will be temporally asymmetric. As a result, the
amount of transverse shift will also change asymmetrically.
A phenomenological fit for the transverse shift was cre-
ated using the experimental data in Fig. 6with the form of

Research Article Vol. 10, No. 5 / May 2023 / Optica 549
a decaying exponential (see Supplement 1). Thus, we can
represent the maximum imprinted nonlinear group index as
1ng=ng
2Imax(t)=ng
2I0exp(−t/τc), where I0is the input inten-
sity, and τcis a characteristic decay time of the nonlinear response,
which we ascribe to thermal diffusion as the dominant thermal
contribution to the nonlinear response. Let us write the time in
terms of the rotation speed as t=τc/c, where is the rotation
speed, cis a characteristic rotation speed, and we rewrite the
transverse shift to be
1y≈rLneff
g
c,(5)
where neff
gis the effective group index written as
neff
g=n0
g+ng
2I=n0
g+ng
2Ioe−/c. (6)
We can look at the time-average response of the effective
group index for a given speed. For a given speed, average tempo-
ral response can be broken up into a fast and slow contribution
written as
neff
g=n0
g+ng
2Io1
fs
e−/s−ffe−/ f,(7)
where ng
2Io=107, and fsand ffare scaling factors equal to 0.97
and 0.94, respectively. These values are similar to those when
considering the peak power of a Gaussian pulse. sand frefer
to the slow and fast inverse time scales, where s,f=1/(2πτs,f),
and the slow and fast time scales are τs=3.5 ms and τf=175 µs,
respectively. The slow (optical) time scale is of the order of the
excited ion lifetime, typically 3 to 5 ms [26], and the fast (thermal)
time scale is of the order of thermal diffusion (≈200 µs) [42].
Equation (7) can be seen plotted in Supplement 1.
Fast and slow time scales modify the magnitude of the effective
group index, representing an approximate non-instantaneous
response. The nonlinear response of the medium is indeed
non-instantaneous, but this approach represents, to a good
approximation, the dynamics of the system while alleviating
computational expense when one includes a non-instantaneous
response in the simulations. Equation (7) is then introduced into a
generalized nonlinear Schrödinger equation, and nonlinear propa-
gation is simulated to investigate the transverse COI trajectories
and extract the amount of transverse shift. These results are shown
in Figs. 3–5, achieving good agreement among the trajectories and
the transverse shift due to photon-drag and nonlinear deflection.
From Eqs. (3) and (4), we represent the static nonlinear refrac-
tion as an index gradient of the form
1nNL(x,y, t, γ )
=n2ZZ d˜xd˜y Kγ(1x, 1y, t)I(˜x,˜y)−no,e,(8)
where 1x=x− ˜x, and 1y=y− ˜y. Here, ˜xand ˜yare Cartesian
coordinates of an arbitrary position within the space where the
nonlinear kernel function acts. The nonlinear potential in the
laboratory’s frame depends on the crystal’s response function K0
[50], which is given by the thermal properties of the material:
Kγ(x,y, t)/K0
=[cos(γ ) cos(t)x+cos(γ ) sin(t)y,−sin(t)x+cos(t)y],
(9)
and I= |Eo|2+ | Ee|2.
A further explanation of the theoretical modeling of light
propagation through a moving nonlinear medium is discussed
by Hogan et al. [52]. Upon propagating two beams through the
crystal, we extract two main parameters as a function of the input
intensity and rotation speed: (1) the position of the COI in the
transverse plane and (2) the transverse shift (overall average posi-
tion of the COI trajectories) experienced by the beams at the
output face of the crystal. The COI trajectories and the values of
the transverse shift are determined for a variety of crystal rotation
speeds, and the three powers used in the experiment with the addi-
tion of the phenomenological fit in the drift term as a modified
effective group index. Results of the simulations are presented in
Figs. 3–5for direct comparison to experimental data.
It is important to note that by moving the camera closer to the
crystal, one can image the input face of the crystal at which a seem-
ingly negative drag is observed, i.e., the beams appear to be shifted
in directions opposite to the motion of the crystal. However, we
highlight that such a measurement simply extrapolates the output
beams toward the input face of the crystal and leads to a seemingly
negative drag effect as a consequence of the large output angle due
to nonlinear deflection.
When the medium response is not instantaneous, the imprinted
refractive index profile is dragged along with the medium motion.
The light then interacts with this moving index gradient and is
therefore deflected at an angle. For example, in a typical nonlinear
interaction with a positive nonlinear refractive index, i.e., self-
focusing, the beam deflects toward the direction of medium
motion and thus resembles a positive photon-drag effect. However,
the output beam leaves the moving medium at an angle with
respect to the input beam. Therefore, in any measurement of the
transverse drag effect, it is crucial to measure the output angle to
discriminate the photon-drag effect from any beam deflection.
5. CONCLUSION
We have demonstrated experimentally and through simulation
that a 2 cm long rotating ruby crystal illuminated with 473 nm
light produces a transverse shift as a result of nonlinear photon-
drag and nonlinear deflection. In rotating saturable media with
self-focusing nonlinear refraction, one must measure the output
angle to distinguish nonlinear deflection and transverse photon-
drag. We note that even if the medium presents large negative
group indices, nonlinear deflection can dominate negative drag
when nonlinear refraction is large and positive. The maximum
transverse shift is found to be 1y= +300 µm, and the maximum
angular shift is found to be θ=13 mrad at the output face of the
crystal (z=0). Moreover, exotic trajectories were observed exper-
imentally for the COI of the beam in the transverse plane at the
crystal output face and reproduced in simulation with good agree-
ment. Since the position of the transverse profile of the beam is
controllable by the rotation speed of the crystal and input intensity
of the beam, one can imagine applications in beam steering and
image rotation, velocimetry, as well as understanding the resilience
of the state of polarization to the motion of the medium.
Funding. Natural Sciences and Engineering Research Council of Canada
(PDF-546105-2020, RGPIN/2017-06880, Banting Postdoctoral Fellowship);
Canada Research Chairs (950-231657); Canada First Research Excellence
Fund (072623); Office of Naval Research (N00014-19-1-2247); North Shore
Micmac District Council’s (NSMDC) Post-Secondary Education Program;
Multidisciplinary University Research Initiative(N00014-20-1-2558).

Research Article Vol. 10, No. 5 / May 2023 / Optica 550
Acknowledgment. The authors thank Xiaoqin Gao for valuable advice on
making figures and the structural formatting of the manuscript. The authors also
thank Orad Reshef for his insightful comments on the manuscript. R.W.B. and
R.H. acknowledge support through the NSERC, the Canada Research Chairs
program, and the Canada First Research Excellence Fund on Transformative
Quantum Technologies. In addition, R.W.B. acknowledges support through the
U.S. Office of Naval Research. R.H. acknowledges support through Indigenous
Affairs at the North Shore Micmac District Council. A.S. acknowledges the sup-
port of NSERC. B.B. also acknowledges the support of the Banting postdoctoral
fellowship of NSERC. A.S. conceived the experiment, R.H. and A.S. conducted
the experiment, R.H. and B.B. analyzed the results, G.M. and R.H. developed the
theory and conducted simulations, R.H. wrote the article and R.W.B. supervised
this work. All authors reviewed the manuscript.
Disclosures. The authors declare no conflicts of interest.
Data availability. Readers may email the corresponding author, Ryan
Hogan, for any code that may be needed for their purpose. He can be contacted at
rhoga054@uottawa.ca. We will respond to reasonable requests for the code.
Supplemental document. See Supplement 1 for supporting content.
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