Download Orbital angular momentum of light

Department of Physics
Seminar Ib - 1st year, 2nd Cycle
Orbital angular momentum of light
Author: Mateja Hrast
Mentor: doc. dr. Andrej Mihelič
Ljubljana, March 2015
Abstract
This seminar presents the concept of orbital angular momentum of light. The distinction between
spin and orbital angular momentum is described, together with their connection to polarization and
spatial distribution of a light beam. Laguerre-Gaussian beams are presented as light with well-defined
orbital angular momentum. The main techniques for generating such beams are mentioned and the
holographic technique is described as the most commonly used. Among applications of light with
orbital angular momentum this seminar centers on the change of selection rules for atomic transitions.
Other applications are also mentioned.
Contents
1 Introduction
2
2 Orbital and spin angular momentum of light
2.1 Angular momentum of electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Polarization dependence of angular momentum . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
3 Laguerre-Gaussian beams
4
4 Generation of light with OAM
7
5 Applications of light with OAM
5.1 Selection rules for atomic transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
8
10
6 Conclusion
11
1
Introduction
It is well known that light can transfer linear momentum to atoms [1]. In 1936 Beth [2] showed that light
can also exert a torque on matter and attributed it to exchange of angular momentum between light and
matter. He sent a right-handed circularly polarized beam of light through a half-wave plate suspended
by a fiber (see figure 1). The plate transformed the beam into a left-handed circularly polarized one and
rotated around the fiber in the process. The gained angular momentum of the plate was found to be 2~
times the number of photons that passed through it. What he measured was, however, only the spin
angular momentum (SAM) of light. Today we know that light can also carry orbital angular momentum
(OAM).
Figure 1: Beth’s experiment proving existence of SAM of light [2]
We associate SAM of light with its polarization, while OAM of light is a consequence of the spatial
distribution of the intensity and phase of the electromagnetic field. It, therefore, comes as no surprise,
that SAM is much better known than OAM. In contrast, this seminar presents the OAM of light, its
generation and applications.
The simplest beams which carry a well-defined angular momentum are Laguerre-Gaussian beams
[3, 4]. Unlike the wavefronts of a plane wave, which form parallel planes, the wavefronts of a LaguerreGaussian beam are helically shaped, where the axis of the helix is the beam axis. The twisting of the
helix is described by a parameter called the topological charge, which is associated with the OAM of
individual photons.
Beams carrying orbital angular momentum have been routinely produced in the visible range over
the last 15 years. The main technique used for their generation is the conversion of a Gaussian beam
to a Laguerre-Gaussian one with a holographic plate [5]. The use of such beams has widely spread in
many areas, including optical communications, microscopy and optical manipulation [6, 7, 8, 9]. On the
other hand, production of beams carrying OAM in the UV spectral region has only begun recently and
is still an active field of research [10].
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When light is used to excite atoms to higher atomic states, selection rules apply and significantly
limit the ensemble of reachable states. It has been shown [11] that for a light beam with non-zero OAM
those selection rules change, allowing to couple practicaly any combination of states at the same rate.
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2.1
Orbital and spin angular momentum of light
Angular momentum of electromagnetic field
~ r, t) and the magnetic
Light is an electromagnetic field, which is described by the electric field strength E(~
~ r, t). Its total linear momentum can be calculated from its volume density as [12]
field density B(~
Z
~ ×B
~ d3~r ,
P~ = 0 E
(1)
and its angular momentum as the cross product between position vector ~r and the linear momentum:
Z
~
~ ×B
~ d3~r .
J = 0~r × E
(2)
~ We expect J~ to consist of two parts,
Equation (2) gives the total angular momentum of light J.
~ + S,
~ where the first term represents OAM and the second term SAM. Taking B
~ = ∇×A
~ and
J~ = L
~ = 0), we obtain from (2) [12]
using the Coulomb gauge (that is ∇ · A
Z X
Z
~ ×A
~ d3~r .
J~ = 0
Ej ~r × ∇ Aj d3~r + 0
E
(3)
j=x,y,z
~ and A
~ only have components in the plane perpendicular to the
Note that in the Coulomb gauge, E
direction of propagation.
We can associate the second term in equation (3) with SAM, since the spin represents the rotation
~ and B
~ around the propagation direction and is therefore indepentent of the origin of the coordinate
of E
system. The first term is consequently associated with OAM.
In theory, OAM should consist of two parts: internal OAM, which represents the rotation of rays
around the beam axis, and external OAM, which is the rotation of the beam around the space origin.
The later can be set to 0 by choosing the origin of the coordinate system on the beam axis.
2.2
Polarization dependence of angular momentum
There is an alternative way of separating the two contributions to J~ in equation (3). Let us consider a
monochromatic field
~ = Re E~ exp(−iωt) ,
(4)
E
where E~ is the amplitude of the electromagnetic field and ω the angular frequency of light. Using
~ = − ∂ A~ and neglecting the terms oscilating with twice the frequency, we find for the second term in
E
∂t
equation (3)
~ ×A
~ ∝ E~ × E~∗ .
E
(5)
~
Introducing the polarization of the light beam as ê = |EE|
~ , we see, that this term is in fact dependent on
polarization.
To rewrite the first term in equation (3) we first introduce the paraxial approximation, which presumes that all rays in a beam travel at small angles with respect to the optical axis. This approximation
works well for a focused laser beam, whose wavefronts’ envelope varies slowly on the scale of the wavelength. Furthermore, we presume that near the beam focus the direction of polarization ê does not
change significantly with ~r. In that case we get
X
X
Ej ~r × ∇ Aj ∝
Ej ~r × ∇ Ej∗ ∝ ê · ê .
(6)
j=x,y,z
j=x,y,z
This term is therefore approximately independent of the direction of polarization. Altogether, the
z-component of equation (3) becomes
Z X Z
0
0 ∗ ∂
3
Jz =
Ej
Ej d ~r +
|EL |2 − |ER |2 d3~r .
(7)
2iω j=x,y,z
∂φ
2ω
3
Here φ is the azimuthal angle in the plane perpendicular to e~z and EL and ER are the left- and right~
handed polarized components of E.
Now we see that SAM is the polarization-dependent part of angular momentum, while OAM is
polarization-independent. For that reason spin of light is associated with its polarization. Light is
assigned 1 unit of SAM. For the right-handed circular polarization, the projection of spin on the z-axis
of each photon is ms = +~, and for the left-handed ms = −~. Photons cannot carry ms = 0 because
~ = 0, arising from the chosen gauge. Light that is not circularly
of the transversality condition ∇ · E
polarized does not have a well-defined spin. For example: linearly polarized light contains half of each
circularly polarized component, therefore it has 50% probability of having either of the spin values.
The OAM, on the other hand, is associated with the spatial distribution of the electromagnetic wave.
The simplest beams that carry well-defined OAM are Laguerre - Gaussian beams, which are discussed
in more detail in chapter 3.
In experiments we cannot distinguish between OAM and SAM being exchanged with matter. However, there are experiments that measure light-matter interaction with exchange of angular momentum.
For example: Simpson et al. [13] showed that shining a paraxial ray of light on a particle will result in
exchange of both OAM and SAM with the particle. The physical evidence of this exchange (rotation
speed) will depend on wheather the OAM and SAM terms are additive or subtractive. Nevertheless, the
same experiment showed, that OAM and SAM are equivalent when it comes to light-matter interaction.
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Laguerre-Gaussian beams
We know that the complex amplitude U (~r) of the electric and magnetic fields in an isotropic medium
satisfies the Helmholtz equation
∇2 U (~r) + k 2 U (~r) = 0 ,
(8)
where k = 2π
λ is the wavenumber and λ the wavelength of light.
We again use the paraxial approximation, so that the complex amplitude can be written as U (~r) =
u(~r)e−ikz . We assume that u(~r) varies slowly with z, so we can take (∂u/∂z)λ u and (∂ 2 u/∂z 2 )λ2 u.
Using this in the Helmholtz equation (8), we obtain the paraxial Helmholtz equation for u [4]:
∂2
∂2 ∂u
+
=0
u − 2ik
∂x2
∂y 2
∂z
(9)
Figure 2: z-dependences of the W (z), R(z) and ζ(z) parameters of a Laguerre-Gaussian beam [4]
4
Solving the above equation in cylindrical coordinates (ρ, φ, z) gives the complex amplitude of the
Laguerre-Gaussian beams [4, 3]
Up,` (ρ, φ, z) = Ap,`
W (0) ρ |`| |`| 2ρ2 ρ2 exp
−
Lp
W (z) W (z)
W 2 (z)
W 2 (z)
2
ρ
+ i |`| + 2p + 1 ζ(z) exp − i`φ ,
exp −ikz − ik
2R(z)
(10)
p
where W (z) = W0 1 + (z/z0 )2 is the z-dependent beam radius, R(z) = z(1 + (z0 /z)2 ) its wavefronts’
curvature radius and ζ(z) = arctan(z/z0 )pthe phase retardation (Gouy phase) with respect to a plane
or a spherical wave. Additionally, W0 = λz0 /π and z0 is a parameter describing the focusing of the
|`|
beam. Figure 2 shows typical z-dependencies of the W (z), R(z) and ζ(z) parameters. Functions Lp in
equation (10) are associated Laguerre polynomials. Since equation (10) was calculated from the paraxial
helmholtz equation, it is only valid when the paraxial approximation can be used.
A beam with the complex envelope Up,` from equation (10) is called Laguerre-Gaussian beam of
order (p, `), where p is a non-negative whole number and ` is any integer. A Laguerre-Gaussian beam
of order (0, 0) reduces to a simple Gaussian beam, whose complex amplitude is φ-independent.
Since the dependence on φ is only in the phase of equation (10), the complex amplitude of LaguerreGaussian beams is cilindrically symmetric about the beam axis. The intentsity distributions for the
lowest order beams are shown in figure 3. Observing the intensities it can be noted, that only beams
with ` = 0 have a non-zero intensity on the beam axis. The number of nodes in the radial direction is
equal to p + 1 for a given beam.
Figure 3: Intensity of the Laguerre-Gaussian beams of lowest orders.
A crucial part of the complex amplitude (10) is its dependence on φ in the phase factor e−i`φ . It
causes the wavefronts to tilt helically around the z-axis, producing beam shapes shown in figure 4. The
step length of the helix surface of such a beam is then given by λ/|`|, where λ is the wavelength of light
and ` is called topological charge. The topological charge describes the order of the singularity of phase
φ (see the third column of figure 4), which arises at the axis of a Laguerre-Gaussian beam with ` 6= 0.
The position of the singularity can also be seen in the electric field profiles shown in the last column
of figure 4. It has been shown [3] that such beams carry a well-defined orbital angular momentum. To
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each photon in a Laguerre-Gaussian beam of order (p, `) we assign the value of OAM L = ~|`|. The sign
of ` determines the handedness of the helix, therefore the intensities are the same for beams with ` and
−`, while their phases are opposite.
Figure 4: Helical wavefronts, intensity distributions in the plane perpendicular to propagation direction,
phase fronts and electric field profiles of the Laguerre-Gaussian beams with p = 0
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4
Generation of light with OAM
Generating light with OAM means generating LaguerreGaussian beams of different orders. This is usually done
by modulating the phase of a simple Gaussian beam,
which can be produced by a common laser.
The simplest technique used to do that are spiral phase
plates. One sends a Gaussian light beam throgh a dielectric plate of varying width, such as the one in figure
5. Parts of the beam belonging to different azimuthal
angles undergo a different phase change. Consequently,
the wavefront of the exiting light has a helical shape of a
Laguerre-Gaussian beam and a node in the center.
Figure 5: A spiral phase plate [14]
More commonly, the Gaussian beam is modulated using a holographic plate. Unlike spiral phase
plates, which modulate the phase, holographic plates modulate the amplitude of the field. Different
colors of the hologram in figure 6a represent different transmissivity. It has been shown [15] that in
order to produce helical wavefronts, the hologram should have the pattern of a pitch-fork, as shown
in figure 6a. This fact is the result of a numerical simulation of the interference between a Gaussian
and a Laguerre-Gaussian beam. It can also be confirmed experimentally by creating an image of the
mentioned interference, using the Gaussian beam as the reference beam and the Laguerre-Gaussian one
as the illumination beam. An example of such an image is shown in figure 6c. Obviously, the pattern is
different for different orders of the beam.
Figure 6: The pitch-fork pattern of the holographic plate used to create Laguerre-Gaussian beams: a)
computer generated, b) schematic of use c) experimental [16, 17]
When a Gaussian beam passes through the holographic plate it creates, in addition to the transmitted
Gaussian beam, two Laguerre-Gaussian beams of higher order with opposite phases. This is a direct
consequence of the conservation of angular momentum. Since a Gaussian beam has ` = 0, two LaguerreGaussian beams must be created with opposite `. These two beams travel at an angle with respect to
the original Gaussian beam, so they can easily be distinguished. This is schematically shown in figure
6b.
The technique with holographic plates has become widely spread, not only because such plates are
readily available, but also because the quality of the beam obtained is much better than by a spiral
phase plate. Figure 7 shows example beams from both methods.
Instead of a common hologram on a photographic plate, Matsumoto et al [5] proposed the use of
a liquid-crystal-on-silicon spatial light modulator (LCOS-SLM). Phase change is achieved by layers of
liquid crystals trapped between pixelized electrodes, which enable each layer of liquid crystals to rotate
separately. This modulator is then set between a transparent plate and a silicon plate, so that the light
is reflected back in the original direction. The structure of LCOS-SLM and its experimental setup are
shown in figure 8.
There are two important advantages of this technique compared to the others. Firstly, the resolution
is much better, which is shown in the last picture in figure 8. Secondly, if one wanted to switch between
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Figure 7: Laguerre-Gaussian beams created by a spiral phase plate (left) and a holographic plate (right)
[17]
Figure 8: LCOS-SLM structure and experimental setup (left and middle) and a beam obtained by it
(right) [5]
different Laguerre-Gaussian modes, all other techniques would require them to physically switch the
plates. Besides, for each mode a new plate must be produced. With LCOS-SLM, the mode can be
changed by changing the voltage on the electrodes and thus rotating the liquid crystals. This means
that the time needed to switch between different modes is only approximately the response time of liquid
crystals to the change of electric field.
Theoretically, one could also obtain Laguerre-Gaussian beams by superposition of Hermite-Gaussian
beams. These are beams with complex amplitudes obtained if the Helmholtz equation is solved in
cartesian coordinates [4]. Since both Laguerre-Gaussian and Hermite-Gaussian beams form a complete
set of solutions for the Helmholtz equation, they are expressible with one another. For example, the
Laguerre-Gaussian beam of order (0, 1) can be constructed from the Hermite-Gaussian beams of orders
(1, 0) and (0, 1), as shown in figure 9. However, as Hermite-Gaussian beams of high orders are also
relatively hard to obtain, this technique is not widely in use.
Figure 9: Conversion of two Hermite-Gaussian beams into a Laguerre-Gaussian beam
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5.1
Applications of light with OAM
Selection rules for atomic transitions
When considering photon-induced transitions between atomic states (or excited states of a nucleus),
one needs to apply selection rules, which limit the possible outcomes. Let us, for example, consider the
hydrogen atom. Its interaction with the electromagnetic field is described by the following Hamiltonian
operator
1
~ 2 + V (r) ≈ 1 p~2 + e A
~ · p~ + V (r) ,
H=
p~ + eA
(11)
2m
2m
m
8
where V (r) = −e2 /(4π0 r) is the attractive electrostatic potential between the electron and the nucleus.
~
We approximated the Hamiltonian by only taking the first order of expansion of the vector potential A.
~
~
Taking A = A0 ê exp(ik · ~r) we can also use the following expansion in the Hamiltonian
exp i~k · ~r = 1 + ~k · ~r + · · · .
(12)
Usually, only the lowest order is retained. This represents electric dipole transitions, where selection
rules are as follows:
• |∆L| ≤ 1
• ∆M ∈ {0, 1, −1}
• parity changes
• transition 0 → 0 is forbidden
Here, ∆L is the change of the total orbital angular momentum and ∆M is the change of the projection
of the orbital angular momentum on the z-axis.
Considering higher orders of expansion, selection rules for electric quadrupole transitions, magnetic
dipole transitions, etc. are obtained. The transfer of angular momentum by a single photon in these
can be larger than ~. For example, for the electric quadrupole transition:
• |∆L| ≤ 2
• ∆M ∈ {0, 1, −1, 2, −2}
• parity is conserved
• transitions 0 → 0 and
1
2
→
1
2
are forbidden
All these transitions can be seen in experiments with plane waves with short wavelenghts (up to
nanometers), but their probability decreases very fast with higher orders.
~ in terms of the Laguerre-Gaussian beam modes. It
Alternatively, we can expand vector potential A
has been shown [11], that such an expansion gives rise to another set of selection rules. For a LaguerreGaussian beam of order `, selection rules for transitions in one-electron atoms are [11]:
• |∆L| ≤ |`| + 1
• ∆M ∈ {` + 1, ` − 1}
• ∆L + |`| + 1 is even
Factor 1 in all of the above rules comes from the spin, as we cannot distinguish between the transfer of
the SAM and OAM. When ` = 0 is taken, these new rules reduce to the electric dipole transition rules,
as expected.
Let us now consider a hydrogen atom in its ground state. Angular dependece of its wave function
is given by the lowest spherical harmonic Y0,0 , which is assigned orbital angular momentum L = 0 and
projection M = 0. When exposed to a linearly polarized Laguerre-Gaussian beam of order (0, 1), we
expect the following transitions, accoring to the new selection rules:
• |∆L| ≤ 2
• ∆M ∈ {0, 2}
• ∆L is even
When the intensity of the incident light is high, the already excited atoms are still exposed to the
beam, so we must take into account two-photon absorbtion. That means we apply the above selection
rules twice and find to expect the following states after the experiment:
Y0,0 , Y2,0 , Y2,2 , Y4,0 , Y4,2 , Y4,4
Picón et al. simulated the experiment and confirmed this prediction in 2010. The outcome of their
simulation is shown in figure 10.
They also considered the case of circularly instead of linearly polarized light. The outcome was
somewhat different, because we have to take into account that the SAM is now well-defined - in the
mentioned simulation ms = −1. The selection rule for ∆M therefore reads:
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Figure 10: Electron states obtained by exposing ground state hydrogen atom to a linearly polarized
Laguerre-Gaussian beam of order (0, 1) [11]
• ∆M = |`| − 1 = 0
The only possible states after the experiment are now
Y0,0 , Y2,0 , Y4,0 ,
which was also confirmed by Picón et al.’s simuation. These results are shown in figure 11.
Figure 11: Electron states obtained by exposing ground state hydrogen atom to a circularly polarized
Laguerre-Gaussian beam of order (0, 1) [11]
5.2
Other applications
A lot of research dealing with light carrying OAM is centered around the transfer of quantum information.
Until now, information has been encoded into the spin of light. However, as there are only 2 linearly
independent values of spin projection per photon, the amount of information transfered was very limited.
If the information were to be encoded into the OAM of light, there are theoretically infinitely many
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possible states, so the information could be transfered at a much higher rate (about Tb/s) [7]. The
transfer of information thrugh the OAM of light is also much safer than any other technique used [6].
Light with OAM can also be used in particle manipulation. By transfering the OAM of a light beam
to a particle it can be made to rotate. Some successful attempts have already been made in controlling
this motion. In the experiment [18] presented in figure 12, they observed two different ways a particle
can be made to rotate. In the case presented in the left pictures, the particle rotated around the beam
axis. They assigned this rotation to the transfer of mostly OAM. In the right pictures the particle
rotated around its own axis, which was contributed manly to the exchange of SAM. However, as the
OAM of the beams currently produced is relatively low, this has been one of the very few experiments
with any results, so far. [8]
Figure 12: Two different ways of rotation caused by mostly the exchange of OAM (left) and SAM (right)
[18]
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Conclusion
It is clear that angular momentum of light can be divided into spin and orbital angular momentum.
However, there are still many open questions about the orbital part. Ever since Laguerre-Gaussian
beams were shown to carry a well-defined orbital angular momentum, scientists have been trying to find
an efficient way of generating them, as well as exploiting their properties in practice.
Generation in the visible range has become widely spread with the use of spiral phase and holographic
plates. Newer techniques, such as generation using liquid crystal phase modulators, are also becoming
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popular. In the UV and XUV spectral region, Laguerre-Gaussian beams could be produced using pulsed
tunable sources, such as high harmonic generation [10] or free electron laser sources [19], which allow
for a detailed structural analysis and studies of fast processes in matter.
Most of the applications of light with OAM have only been discovered recently. The most promising
ones are in quantum information, where light with OAM is used for a faster and safer information
transfer. In atomic physics, the interaction of these helically shaped beams with atoms are studied. It
has been shown that the selection rules for atomic transitions change and therefore allow a more efficient
way of reaching those states that remain inaccessible by dipole excitation.
However, the study of orbital angular momentum is not limited to light beams. One can also produce
helically shaped electron beams carrying high orbital angular momentum. Among others, a possible way
of generating such beams are spiral phase plates, very similar to the ones used to generate light beams.
Applications of electron beams carrying orbital angular momentum are also being explored and seem
equally promising [20].
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