Light has angular momentum. Angular momentum can be classified into : spin angular momentum and orbital angular momentum. Due to the handedness of the circularly polarized spin states, they show spin Hall effect. Analogously, orbital angular momentum also exhibits Hall effect which has not yet been experimentally realized. In this work we developed a setup for observing orbital angular momentum Hall effect of light by detecting the orbital angular momentum spectrum of the light beam. The setup was tested with beams of known orbital angular momentum distributions and was also used to probe and quantify the orbit-orbit interactions due to reflection from a reflecting surface.

1. Debanuj Chatterjee
Dr. Nirmalya Ghosh
Towards realization of
Orbital angular momentum
Hall effect of light
Supervised by
BS-MS Thesis Presentation 2017

2. Angular momentum of light
L is angular momentum about an axis.
r is the radial vector from a point on the axis.
P is momentum.
For light S is the Poynting vector giving momentum.
E is the electric field vector.
B is the magnetic field vector.

3. Types of angular momentum of light
Spin Angular Momentum (SAM) Orbital Angular Momentum (OAM)
Picture courtesy : The Twist in Light’s Tail by Miles Padgett
Polarization rotates Phase front is twisted
σ = +1σ = -1
Transverse component of the Poynting vector
must be non zero

4. Electric field distribution of SAM and OAM
Spin Angular Momentum (SAM) Orbital Angular Momentum (OAM)
(Not to scale)
Laser beam
Cross section
axis

5. Types of OAM
Orbital angular momentum
Intrinsic Extrinsic
If total orbital angular momentum depends on the choice of
coordinates it is extrinsic, if not it is intrinsic

6. If ΔJz is zero it is intrinsic and if ΔJz is non-zero it is extrinsic
The total orbital angular momentum is given by :
The change in total orbital angular momentum on shifting the origin to
(r0x,r0y) is given by :
Types of OAM

7. If total orbital angular momentum depends on the choice of coordinates
it is extrinsic, if not it is intrinsic
Types of OAM
Azimuthal symmetry preserved
Azimuthal symmetry broken

8. Beams with OAM
Any beam with an azimuthal phase dependence(eg: eilφ) carries OAM.
Eg : Laguerre-Gaussian beam.
The beam at its waist can be described by :
Where m=…-2,-1,0,1,2… ; p=0,1,2…(we normally set p=0) ; Lp
|m|
are the associated Laguerre polynomials ; ω0 is the spot size at the
waist.

9. Beams with OAM
Paraxial wave equation
Hermite-Gaussian modes Laguerre-Gaussian modes
Solving in cylindrical coordinateSolving in Cartesian coordinate
(carries no OAM) (carries OAM)

10. Laguerre-Gaussian beam
L=+3 L=-3
(circularly polarized)
Picture courtesy : The Twist in Light’s Tail by Miles Padgett

11. Spin-Orbit interaction
SAM - Intrinsic OAM (symmetry preserved)
Intrinsic OAM – Extrinsic OAM conversion (symmetry broken)
SAM– Extrinsic OAM conversion (symmetry broken)
Producing OAM states using a s wave-plate.
Reflection of a beam carrying SAM (Spin Hall Effect)
1. Reflection of a beam carrying IOAM (OAM Hall Effect)
2. Truncating a beam with IOAM.

12. Spin Hall effect of light
Light beam with : Left Circular
Polarization + Right Circular Polarization
LCP RCP
Reflection from a surface
Spin Hall effect
Spatial separation

13. Geometric phase of light
Geometric phase is related to the geometry of
evolution of the beam and leads to change in
polarization state.
Spin redirectional Berry phase OAM redirectional Berry phase
Optical fiber
Optical fiber
HG mode
Polarization
θ
Picture courtesy : S. Duttagupta et al, Wave optics, CRCPress

14. Spin Hall Effect of light
kc is the central wave vector.
ν and μ are the y and x components of the k vectors respectively.
Reflection of a Gaussian beam from a surface.
Linearly polarized
Gaussian beam
(K. Y. Bliokh, A. Aiello, J Optics, Vol 15, 2013)

15. Spin Hall Effect of light
Linearly polarized
Gaussian beam
As the k-vectors have a y-spread, each k-vector
will have a different plane of incidence and hence
a different geometric phase.
(for RCP)
(for LCP)
(K. Y. Bliokh, A. Aiello, J Optics, Vol 15, 2013)

16. Spin Hall Effect of light
As linearly polarized light is a sum of RCP and LCP, there is a
spatial separation between RCP and LCP after reflection.
Linearly polarized
Gaussian beam
Fourier transform
(K. Y. Bliokh, A. Aiello, J Optics, Vol 15, 2013)
(k-space)
(position-space)
yo is the shift

17. Analogy of spin and laser modes
Just like linear and circular polarizations form different bases, for laser
modes the HG modes and the LG modes form the bases.
As the handedness of the circular polarizations led to Spin Hall Effect,
similarly we also expect OAM Hall Effect for the LG modes with their.
Spin Laser modes
Picture courtesy : S. Duttagupta et al, Wave optics, CRCPress

18. OAM Hall effect of light
Light beam with : +l and –l
orbital angular momentum
+l beam -l beam
Reflection from a surface
OAM Hall effect
Centroid of the beams get
spatially separated
IOAM
(OAM distribution
is a spike)
EOAM
(OAM distribution changes)
EOAM
(OAM distribution changes)

19. OAM Hall Effect of light
On reflection, the Gaussian part of the beam undergoes a shift. The
direction of the shift depends on the handedness of the beam.
L=+1
(Intrinsic OAM)
mirror
(Extrinsic OAM)
L= -1
(Intrinsic OAM)
mirror
(Extrinsic OAM)

20. OAM Hall Effect of light
L=+1
(Intrinsic OAM)
mirror
(Extrinsic OAM)
L= -1
(Intrinsic OAM)
mirror
(Extrinsic OAM)
(Experimental evidence)
(Thesis of Sumit Goswami, IISER Kolkata , 2015)
Reflection

21. OAM Hall Effect can be indirectly probed from the OAM distribution.
OAM distribution also gives us insight into orbit-orbit interaction of
light.
Experimental directions to observe
OAM Hall effect of light
The shift can be enhanced by choosing very large values of l.
The shift can be enhanced by the technique of weak masurements.
Generation of very high values of OAM states is difficult as we are
limited by the size of the optical components and losses.
Technique of weak measurement is not well formulated for OAM states

22. OAM Hall effect of light
(Simulation)
L=+3
(Intrinsic OAM)
mirror
(Extrinsic OAM)
mirror
(Extrinsic OAM)
L=-3
(Intrinsic OAM)

23. OAM Hall effect of light
(Simulation)
L=+3
(Intrinsic OAM)
mirror
(Extrinsic OAM)
mirror
(Extrinsic OAM)
L=-3
(Intrinsic OAM)

24. Quantitative estimation of EOAM
OAM Hall Effect leads to conversion of IOAM to EOAM while total OAM is conserved.
(K. Y. Bliokh, A. Aiello, J Optics, Vol 15, 2013)
(here we take l=-10 to l=10)

25. Conjugate variables
Angle Angular Momentum
Angle and Angular Momentum
Fourier decomposition
Position
(Padgett et al, New Journal of Physics, Vol 6, 2004)
(Like position and momentum)
a1 +a2 +a3 +…
a1 +a2 +a3 +…
LG modes
sinusoidal basis
(here we are only showing intensity)

26. OAM distribution
Simulation of Angle and Angular Momentum
OAM distribution

27. Generating beams with IOAM
S wave plate Fork grating on a SLM
Only two values of l can be generated Many values of l can be generated

28. S wave plate
Jones matrix :
Picture courtesy : Wikipedia
(input beam) (output beam)(s wave plate)
|0>
|0>

29. Fork grating
(Principle)
A fork grating of order lfg increases orbital angular momentum by lfg
Light beam with l = lbeam
Light beam with l = lfg + lbeam
Fork grating of order lfg

30. Fork grating
(setup)
Laser
Beam expander
SLM with Fork grating and lens
Screen / EM CCD
l = lfg
l = lbeam = 0
l = lfgl = -lfg
Picture courtesy : The Twist in Light’s Tail by Miles Padgett

31. OAM states generated with Fork grating
l = 1l = -1
l = 2
l = 3
l = 10
l = -2
l = -3
l = -10

32. (principle)
Light beam with l = -lbeam
Light beam with l = lfg + lbeam = 0
Fork grating of order lfg = lbeam
First order diffraction will be a Gaussian
Fork grating for detection of OAM
Intensity at the center is a measure of the mode l = lbeam

33. Generation and detection of OAM
LensPolarizer QWP S wave plate SLM
Laser EM CCD
σ=+1
l=0
σ=-1
l=+1
l=1+1=2
(l=+1)
l=1-1=0

34. Generation and detection of OAM
LensPolarizer QWP S wave plate SLM
Laser EM CCD
σ=+1
l=0
σ=-1
l=+1
l=1-1=0
l=-1
l=1+1=2

35. Generation and detection of OAM
LensPolarizer QWP S wave plate SLM
Laser EM CCD
σ=+1
l=0
σ=-1
l=+1
l=1+2=3
l=+2
l=1-2=-1

36. Generation and detection of OAM
LensPolarizer QWP S wave plate SLM
Laser EM CCD
σ=-1
l=0
σ=+1
l=-1
l=-1
l=+1
l=-1
In this case the beam does not see the SLM!
l=-1

37. Total phase of circularly polarized light for
a SLM
l=-1l=-1
There is almost no variation of phase with respect to gray level for LCP
(Mandira Pal et al.,Scientific Reports, 2016)

38. Generation and detection of OAM
LensPolarizer S wave plate SLM
Laser EM CCD
σ=0
(both LCP and RCP)
l=0
σ=0
RCP : l=+1
LCP : l=-1
l=-1
l=-1-1=-2l=-1+1=-0
(Only the RCP part of the beam sees the SLM)
(input beam) (output beam)(s wave plate)

39. (Scheme)
Detection of OAM for a truncated beam

40. Laser
Beam expander
SLM with Gaussian slit
SLM with fork grating and lens
Screen / Detector
(Experimental Setup)
Detection of OAM for a truncated beam
Picture courtesy : Modified from : The Twist in Light’s Tail by Miles Padgett

41. Projected OAM state Order of fork grating
l=1
l=2
l=10
Detection of OAM for a truncated beam
l=-1
l=-2
l=-10
(Intensity at the centre
is evaluated)

42. EOAM
EOAM
Image on SLM
Detection of OAM for a truncated beam
Larger width
Smaller width

43. l angular momentum hall effect of light
Observing orbital angular momentum hall effect of light
In future we can use the setup for observing orbital
angular momentum hall effect of light and also for quantifying
extrinsic orbital angular momentum of a beam.
We developed a setup for finding the orbital angular
momentum power spectrum of an arbitrary beam
l angular momentum hall effect of lightWe tested the setup with beams with known orbital
angular momentum distribution
Inference

44. Acknowledgement
I would like to thank Dr. Nirmalya Ghosh for
his support and encouragement to carry out
this work. Also thanks to Mandira Pal and
Athira di for their guidance and advices.
A big thanks to all the Lab members of
Bionap for their help and cooperation.

45. References

46. References

47. Thank You