Beam deﬂection and negative drag in a moving nonlinear medium Ryan Hogan1Akbar Safari1Giulia Marcucci1Boris Braverman1and Robert W. Boyd1 2 1Department of Physics University of Ottawa Ottawa ON K1N 6N5 Canada

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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, ON K1N 6N5, Canada

2Institute of Optics, University of Rochester, Rochester, NY 14627, USA

(Dated: October 5, 2022)

Light propagating in a moving medium with refractive index other than unity is subject to light

drag. While the light drag eﬀect due to the linear refractive index is often negligibly small, it 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 propagation of light in moving media and results in a beam deﬂection

that might be confused with the transverse drag eﬀect. We perform an experiment with a rotating

ruby crystal which 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 towards 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 — a crucial step missing in earlier experiments.

INTRODUCTION

Propagation of light in moving media has been stud-

ied for more than two centuries [1–11]. Upon propaga-

tion, the trajectory of light can be manipulated through

self-action eﬀects [12,13], beam deﬂection [14,15], pho-

ton drag [16–18] and many other phenomena. The pho-

ton 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 phe-

nomenon has gained increasing interest in the ﬁeld of

optics and is indeed still investigated in modern day re-

search [6,7,10,19–23]. Photon drag can be longitudinal

or transverse, i.e., along or perpendicular to the light

propagation direction respectively. This article focuses

on transverse rotary photon drag [24], distinctly diﬀer-

ent than longitudinal drag, given by

∆y=v

c ng−1

nφ!L, (1)

with vthe speed of the medium, cthe speed of light

in vacuum, Lthe length of the medium, ngand nφthe

group and phase indices, respectively. Photon drag scales

linearly with group index. Typically phase and group

indices are not large, and therefore do not create large

transverse shifts. Recent studies show larger shifts us-

ing slow light media(i.e. large group indices) [11,23–25].

Figure 1a) sketches the light propagation in a medium of

length Lin two cases, a) a stationary medium, and b) a

medium moving transversely with speed v. Experimen-

tally, rotation is more feasible than linear motion. The

beam is incident on the medium at a distance rfrom the

∗rhoga054@uottawa.ca

center of rotation, and using a slow light medium with

ng1/nφ, the transverse drag can be simpliﬁed to

∆y≈ngLrΩ

c,(2)

where Ωis the rotational speed of the medium. Note

that the beam size is much smaller than the medium

radius, r. Large group indices are often achieved by em-

ploying a nonlinear phenomena such as coherent pop-

ulation oscillations (CPO) and electromagnetically in-

duced transparency (EIT), that produce ng= 106or

even larger. However, as we show below, 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. In a nonlinear

medium, the impinging light can saturate the transition

and locally change the refractive index of the medium.

When the response of the medium is not instantaneous,

as the medium moves in the transverse direction, the im-

printed refractive index proﬁle is dragged along with the

motion of the medium Therefore, in a moving nonlinear

medium, location of peak index change is shifted with

respect to the center of the impinging light. Thus, the

light sees a gradient in the refractive index and deﬂects

at an angle. The sign of this nonlinear deﬂection de-

pends on the sign of the nonlinear refractive index and

the direction of motion of the medium. In a typical non-

linear interaction with positive nonlinear refractive in-

dex, where self-focusing is observed, the beam deﬂects

towards the motion of the medium and thus resembles a

positive photon-drag eﬀect. In nonlinear deﬂection the

output beam leaves the moving medium at an angle with

respect to the input beam, while in the photon-drag eﬀect

the output beam is in parallel to the input beam. There-

fore, one can distinguish the nonlinear deﬂection from the

drag eﬀect by measuring the output angle of the beam.

While the enhanced photon-drag eﬀect depends on the

group index including any nonlinear contribution (see

Eq. (2)), the nonlinear deﬂection depends on the nonlin-

ear refractive index of the medium. Thus, it seems that

arXiv:2210.01716v1 [physics.optics] 4 Oct 2022

(b)

(Ω < 0)&

Δ𝑦

(Ω > 0)&

Δ𝑦

Centre

rotation

Input beam

Δ𝑦

(z = −2𝑐𝑚)&

Ordinary Beam

𝑧 = −2'𝑐𝑚

(c)

10𝜇𝑚

𝜃 ≈ 225∘

(a)

𝐿

𝑣

⃗

ii)

𝑡 = Δ𝜏

𝑡 = 0

Input

Output

z=0

z=-2 cm

Extraordinary Beam

(d)

o-beam

e-beam

i) ii) iii)

vii) viii) ix)

COM

𝜃 = 45∘

𝜃 = 0∘

𝜃 = 270∘𝜃 = 315∘𝜃 = 360∘

iv) v) vi)

𝜃 = 135∘𝜃 = 180∘𝜃 = 225∘

𝜃 = 90∘

FIG. 1. (a) A schematic showcasing the beam being trans-

versely shifted in a moving versus stationary medium. Al-

though we use a CW laser, for simplicity of illustration we

show the laser as pulses. Cases i) and ii) show the propagation

of the laser in a stationary and moving medium, respectively.

(b) The beam is far away from the center of rotation, which

approximates linear motion in the y direction. The beam

moves in the -y (+y) direction when the crystal rotates clock-

wise (counterclockwise) direction. (c) A single frame imaged

at the front face of the crystal (z=−2cm) that shows the two

output beams, being the o- and e-beams due to the birefrin-

gence that propagate through the 2-cm-long ruby crystal. (d)

A diagram showing the trajectories of o- and e- beams at dif-

ferent crystal orientations highlighting the change in intensity

of each beam at 45-degree intervals. The red "x" shows the

center of mass position for diﬀerent crystal orientations high-

lighting 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.

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 [26].

Therefore, if the large group index is achieved through a

nonlinear interaction, one has to be careful with the non-

linear deﬂection and measure the output angle, a criti-

cal step missing in previous works [24,27,28]. In this

article, we use a rotary ruby rod to study the nonlin-

ear light propagation in a moving medium. In a similar

fashion to alexandrite [29], ruby exhibits a large nega-

tive group index (ng≈ −106) at wavelength 473 nm[30].

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 mo-

tion of the medium.

Nevertheless, since ruby also exhibits nonlinear refrac-

tion, the beam deﬂects towards the direction motion of

the medium due to nonlinear deﬂection. Because of the

birefringence of the crystal, the input beam splits into or-

FIG. 2. A 520 mW continuous-wave laser beam at 473 nm

is focused using a 100 mm focal length plano-convex lens L1

to 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, L2for nonlinear measurements, and between L1and the

ruby for linear measurements. The CCD camera images at

diﬀerent z positions using a translation stage. Measurements

are taken at z= 0,z= 0.762 cm 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 ﬂuorescence ﬁlter F F

(high transmission near 473 nm) is used to minimize ﬂuores-

cence from the ruby rod from being collected by the CCD

camera. The dielectric mirror DM is used as a neutral den-

sity 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 stan-

dard neutral density ﬁlter. Input beam power was controlled

by a half-wave plate and polarizing beam-splitter before the

ruby crystal. (M: Mirror, HWP: Half-wave plate, PBS: Po-

larizing 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], and a

CCD: Charge-coupled device.)

dinary (o) and extra-ordinary (e) beams which separate

upon propagation in the crystal. The e-beam revolves

with the rotation of the ruby rod. Moreover, the propaga-

tion of the o- and e-beam are coupled through the nonlin-

ear interaction in ruby which creates an attractive force

between the beams and further complicates their trajec-

tory. We study this trajectory experimentally and sim-

ulate the propagation using nonlinear Schrodinger equa-

tions. Due to the simultaneous presence of birefringence,

intensity-dependent photon drag, and strong nonlinear-

ity, ruby can serve as a solid-state platform rich in physics

with potential applications to beam steering [31,32], po-

-50 0 50

x-position [ m]

-60

-40

-20

y-position [ m]

Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)Experiment Linear (P0=0.2 mW)

=-9000

=-1000

=-100

=-50

=50

=100

=1000

=9000

(a)

-60 -40 -20 0 20 40 60

x-position [ m]

-60

-40

-20

y-position [ m]

Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)Simulation Linear (P0=0.2 mW)

(b)

FIG. 3. (a) Experimentally measured center of mass (COM) trajectories in the linear regime. (b) Simulated COM trajectories

in the linear regime. Color scheme in the legend inset of the experimental data in (a) correspond equivalently the rotation

speeds (Ω) in units of degs/s in the simulated data. Trajectories of COM of the o- and e- beams (schematic shown in Figure 1

d) are plotted for an input laser power of 0.2 mW, considered as the linear regime. COM trajectories are plotted for rotation

speeds of Ω = ±50,±100,±1000, and ±9000 deg/s. Here, clockwise and counterclockwise rotation (looking into the beam)

correspond to positive and negative rotation speeds, respectively. The COM for each speed follows the same ﬁgure-eight-like

trajectory, since the intensity is too low to introduce deviations in the transverse movement due to nonlinearity or photon drag.

The ﬁgure-eight-like pattern does not close in the center for the experimental results due to the polarization impurity in the

low power regime.

larization detection [33,34], image rotation[24,28] , and

potential for solitonic behaviour with associated applica-

tions [35–37].

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 us-

ing 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 front face of the crystal using a plano-convex

lens of focal length f= 100 mm, resulting in a 20 µm

beam diameter located near the edge (0.1 mm away) of

the ruby crystal face far from the center of rotation. The

ruby was mounted in a hollow spindle whose rotation was

controlled by a stepper motor and belt. The back face of

the crystal was imaged onto a CCD camera using a 4-f

lens system.

Shining linearly polarized light onto the rotary birefrin-

gent medium, the light sees two refractive indices upon

propagation, no= 1.770, and ne= 1.762, respectively.

Without any inﬂuence of nonlinearity or photon drag,

the two beams (o- and e-beams) then propagate with a

ﬁnite angle separation of γb= 8 mrad, known as birefrin-

gent walk-oﬀ. The relative beam intensity reaches max-

ima and minima each quarter turn of the crystal (i.e.,

∆θ= 90◦). The beam input is aligned such that, regard-

less of crystal orientation, the o-beam propagates directly

through the crystal, while the e-beam revolves about the

o-beam. We track the motion of the average position of

these two beams with an approach using the center of

mass (COM), represented as a red “x” in Fig. 1d. The

COM is representing the centre of intensity distribution.

This method is used since transverse beam proﬁles be-

come larger on propagation and begin to overlap. Figure

1c) shows a distinct Gaussian beam shape at z=−2cm.

In contrast, the o and e beams are mostly overlapped at

the crystal back face, z= 0 cm where transverse shifts

are measured.

RESULTS

We measure the COM at z= 0 for rotational speeds

of Ω = ±50,±100,±1000, and ±9000 deg/s in clockwise

(positive) and counterclockwise (negative) directions at

three diﬀerent input powers of 0.2 mW, 100 mW, and

520 mW, corresponding to weak, moderate, and intense

illumination, respectively. We plot the COM trajectories

for an input laser power of 0.2 mW, considered as the

linear regime in Fig. 3. We observe that all speeds trace

out ﬁgure-eights and do not drift transversely.

Figures 4and 5show COM trajectories in nonlinear

and highly nonlinear regimes. At low speeds (Ω≤100

deg/s), the o- and e- beams couple to each other caus-

ing signiﬁcant variation in the traces of the COM upon

rotation. Increasing intensity increases the thermal gra-

dient impressed on the crystal drastically modifying the

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BeamdeectionandnegativedraginamovingnonlinearmediumRyanHogan,1,AkbarSafari,1GiuliaMarcucci,1BorisBraverman,1andRobertW.Boyd1,21DepartmentofPhysics,UniversityofOttawa,Ottawa,ONK1N6N5,Canada2InstituteofOptics,UniversityofRochester,Rochester,NY14627,USA(Dated:October5,2022)Lightpropagatinginamovingme...

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Beam deﬂection and negative drag in a moving nonlinear medium Ryan Hogan1Akbar Safari1Giulia Marcucci1Boris Braverman1and Robert W. Boyd1 2 1Department of Physics University of Ottawa Ottawa ON K1N 6N5 Canada.pdf

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Beam deﬂection and negative drag in a moving nonlinear medium Ryan Hogan1Akbar Safari1Giulia Marcucci1Boris Braverman1and Robert W. Boyd1 2 1Department of Physics University of Ottawa Ottawa ON K1N 6N5 Canada

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