Download Is the speed of light in free

Is the speed of light in free-space
always c?
Miles Padgett FRS
Kelvin Chair of Natural Philosophy
1
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What’s the speed of light in free space? Always? The team
Jacqui Romero
www.physics.gla.ac.uk/Optics
Daniele Faccio
Václav Poto!ek,
Gergely Ferenczi,
Fiona Speirits,
Stephen M. Barnett
Daniel Giovannini
Miles Padgett
Phase and group velocity
Phase velocity- the speed at which a point of constant phase moves
Group velocity- the speed at which the envelope of the wave packet moves
!
vp =
k
d!
vg =
dk
http://resource.isvr.soton.ac.uk
Slowing light – three examples
Light in a hollow waveguide
Light in a hollow waveguide
The wavevector zig-zags
down the guide
Light in a hollow waveguide
The phase fronts create
nodes in the electric field
at the guide surface
Light in a hollow waveguide
k
1
k
2
The the mode is formed
as the overlap of two
plane-waves
The wavevector in 3D
Adding boundary conditions
Setting boundary conditions at the edge
The phase and Group velocities
>c
BUT in free space
x
2
c
<c
Slow light arises from transverse structure
•  This slowing of the z-component of the group
velocity occurs in hollow waveguides
•  BUT the slowing occurs because of the
transverse boundary conditions NOT because
of the waveguides specific material properties
•  We can introduce these boundary conditions
by shaping the optical beam
Rectangular to circular - > Bessel functions
Circular node
in the Bessel
field
µ-optic.com
Make Bessel beam in free space using an axicon
The speed of light?
The speed of light
Racing photons
Racing photons
Bessel beams are subtle……
WE USE THESE!!
≈ circular diffraction
grating
More
complicated…..
Refractive
Refractive c.f. diffractive
Diffractive
Prism
Lens
Axicon
Racing photons
Single-mode
fibre
Spatial light
modulators
SPDC
source
Time-correlated
photon pairs at 710nm
(10nm bandwidth)
Time-delayed
Photons ?
Racing photons
Single-mode
fibre
Spatial light
modulators
SPDC
source
Time-correlated
photon pairs at 710nm
(10nm bandwidth)
Time-delayed
Photons ?
Does anyone remember “hot wheels”
How to measure the slowing down?
s
i
$z
Coherence length of source >> λ
We use Hong-Ou-Mandel (HOM) interference in time-correlated photon pairs.
We measure the delay of single photons.
How to measure the slowing down?
“plane wave”
s
i
$z
reference
position
“plane wave”
Bessel beam
s
Delay of photons
i
$z
“plane wave”
The experiment
Time correlated photon pairs
are produced from
parametric-down conversion.
Idler photon goes through
polarisation-maintaining fibres,
onto the input port of a
beamsplitter.
Signal photons are given a
transverse structure via
spatial light modulators
(SLMs), which we use as
programmable diffractive
optical elements.
We obtain the HOM dip
position as a function of the
transverse structure of the
photon.
Results: Bessel case
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45-12-,/12/ 2561*7
A
!!
'&"
!
"&%
"
"&!
Dip position is to the right of the reference
dip position, indicating a delay.
"&#
!!"
0
Delay !(!m)"
,-.*/
B
↵ = 0 “plane wave” case
↵1 = 0.00225
↵2 = 0.00450
!$"
"
$"
#"
()*+
,-../0/12/
" 3#
Path
delay
(microns)
!#"
!"
%"
(
Delay increases with increasing inclination.
'
"
!
&
L 2
z= ↵
2
$
!
!"!!#
!"!!$
!"!!%
!)*+"
!"!!&
Axicon ≈ Lens
≈
Argument based on wavevector ≈ based on geometry
L 2
z= ↵
2
L 2
z= ↵
2
L 2
z= ↵
2
2w
More common case: Focusing
Consider a Gaussian beam in a confocal telescope,
L 2
z= ↵
2
For a Gaussian beam of beam waist w, hr2 i = w2 /2
w2
z=
2f
Results: Focusing
L=0.8 m
f=0.4 m
w=2.32±0.09 mm
zth = 6.7 ± 0.6µm
Δ
'&$
!
"
'&"
'&"
7.7 !m
45-12-,/12/ 2561*7
2w
'&$
"&%
"&%
"&!
"&!
"&#
"&#
!!"
!#"
!$"
"
$"
#"
!"
()*+delay
,-../0/12/
" 3#
Path
(microns)
zexpt = 7.7 ± 0.4µm
%"
!!"
!#"
!
The slowing depends upon the NA2
•  The larger the radii the larger the delay
•  Do the inner and out parts of the beam each
give rise to a separate delay – or does the
beam give a single shift based upon the
expectation value?
Results: Focusing, selecting with apertures
!
7.7 !m
Coincident counts
"
$"
#"
+ ,-../0/12/ " 3#
!"
%"
"
'&"
$
#
"&%
"&!
"&#
"
Full aperture
7.7±0.4 microns
'&$
Centre blocked
15.0±0.6 microns
Edges blocked
1.3±0.6 microns
!!"
!#"
!$"
"
$"
#"
!"
%"
()*+
" 3#
Path ,-../0/12/
delay (microns)
The delay is smaller when only light nearer the axis gets focused.
The delay is larger when only light farther from axis gets focused.
Conclusions
•  The optical delay associated with transverse
structuring is many wavelengths c.f. the Gouy
phase ≈ one wavelength
•  Delay exists for any form of structuring (inc
OAM)
•  The delay is proportional to the square of the
numerical aperture, therefore small at long
(low NA) range
Orbital Angular Momentum
σ = +1
–  Circular polarisation
–  σ! per photon
•  Orbital angular momentum
–  Helical phasefronts
σ = -1
–  ℓ! perAngular
photon momentum in terms of photons
Circular
Polarisation
•  Spin angular momentum
Helical Phasefronts
ℓ = -1
ℓ=0
ℓ=1
ℓ=2
ℓ=3
Orbital Angular Momentum
Poynting vector
E
B
S
•  But the Poynting vector is not a helix
•  Need to account also for divergence
•  This gives straight lines
Thank you!
Ghost imaging with correlated light
39
!
http://www.gla.ac.uk/schools/physics/research/groups/optics/
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