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Just Staring at Nothing –
The Joys of Singular Optics
Taco D. Visser
Free University Amsterdam
Presentation based on Opt. Commun. 281, pp. 1-6 (2008).
What is Singular Optics about?
• Singular Optics [1,2] deals with the behavior of wavefields in
the vicinity of points where certain parameters are
undefined or singular.
• But such singularities occur not only in Optics. At certain
places time is singular!
References:
[1] J.F. Nye, Natural Focusing and Fine Structure of Light
(IOP Publishing, Bristol, 1999).
[2] M.S. Soskin, M.V. Vasnetsov, in: E. Wolf (Ed.), Progress in Optics,
vol. 42 (Elsevier, Amsterdam, 2001).
Time Singularities
Time is undefined, or singular, at the two Poles
where different time zones intersect.
Phase Singularities
• A phase singularities occurs
at points where the wave field amplitude
is zero, the phase f at such points is
undefined or singular.
• Topological charge
• Singularities of the tides in the North Sea
were observed in the 1830s [3].
[3] T.D. Visser, Physics World, February 2004.
Phase Singularities occur in all sorts of physical systems:
•
•
•
•
•
•
•
The Quantum Mechanical wavefunction
Bose-Einstein condensates
Light that is scattered
The field radiated by antennas
The focal region of lenses
Two-point correlation functions
The state of polarization of light
In this presentation we concentrate on the different types
of singularities that are found in optical fields.
Zooming in on light
• Geometrical optics studies rays and their reflection and
refraction. It describes the macroscopic behavior of light.
• Physical optics deals with scalar or electromagnetic
waves. It analyses intensity variations on the wavelength scale.
• Singular optics analyzes topological structures that can be
arbitrarily close to each other. Their dimensions are much
smaller than the wavelength.
Singular optics deals with the fine-structure of light.
Many contributions by Michael Berry & John Nye.
Phase Singularities in Optics:
Vortex (‘Dark Core’) Beams
Cross-section of a Laguerre-Gauss beam
of radial order m = 0, azimuthal order n = 1.
Extraordinary Optical Transmission
First observed by
Ebbesen et al. [4]
Nano-apertures (size < l) in thin metal films can transmit much more
light than expected!
Strong dependence on wavelength, polarization, aperture size,
aperture shape and array pattern.
[4] T.W. Ebbesen et al., Nature 391, pp. 667-669 (1998)
Early Explanations
Waveguide modes of slit
Perfect conductor
Incident plane wave
Early models give information on the role of polarization and wavelength.
But they do not not take into account the type of metal or surface
modes (surface plasmons).
And: Is the field near the slit really a plane wave?
A Better Explanation
• Solve Maxwell’s Eqs. for a ‘real’ metal film with finite
conductivity and finite thickness
• Study the Poynting vector (power flow) near aperture
• Analyse surface modes on metal film (plasmons)
(can different apertures ‘talk’ to each other?)
• Use a rigorous Green’s tensor formalism
• Such an approach [5,6] reveals phase singularities and phase
saddles of the Poynting vector that can annihilate when a
parameter such as the slit width or the wavelength is varied
in a continuous manner.
[5] H.F. Schouten et al., Phys. Rev. E 67, 036608 (2003)
[6] H.F. Schouten et al., Optics Express 11, p. 371 (2003).
Optical Vortices and Saddles
• The next slide shows the power flow of light incident (from
below) on a slit in a thin silver film. Phase singularities
(‘optical vortices’) and saddles of the Poynting vector can be
seen. The vortices within the film correspond to a power
flow through the slit that is larger than expected according
to geometrical optics.
• The movie that comes after the next slide shows what
happens when the slit width is slightly increased: the two
vortices and the two saddles below the slit move closer to
each other and eventually annihilate. After the annihilation
the flow of power is much smoother, this corresponds to an
increased transmission as compared to before this event.
Power flow for Silver
w=
T=
Annihilation
w=
T=
Phase Singularities of a 2-Dimensional
Poynting Vector Field
source
sink
vortex
saddle
saddle
dipole
• List
location or contactmonkey
for specification
(or other related
documents)
herephase, or direction of the Poynting vector is singular
The
at points where its modulus |S| = 0.
Topological Charge and Index
Monochromatic field
At points where the amplitude A = 0, the phase f is undefined or singular.
The topological charge of grad f = topological index t
Phase
Grad f
Phase
Grad f
Positive
Saddle
Vortex
Both topological charge and topological index are conserved quantities!
Under smooth changes of a system parameter, such as l,
t = -1
s=1
=1
phase singularities
can be tcreated
or annihilated.s = 0
Two vortices of opposite sign can only annihilate if two saddles
Maximum
Negative
are also involved (as in the movie you just saw).
Vortex
s = -1
t=1
s=0
t=1
A New Transmission Anomaly
Just as vortices within a silver film correspond to an increased
transmission, a reversal of their handedness goes together
with an unexpectedly low transmission. We predict that this
occurs for, e.g., a slit in a thin silicon film [7]. For such a
system our calculations show a change in handedness
accompanied by a transmission that is lower than predicted by
geometrical optics:
[7] H.F. Schouten et al., J. of Optics A 6, S277 (2004).
Changing the handedness of the
vortices: Silicon
• A narrow slit in a silicon plate acts as a back-scatterer: T< 0
• This anomalously low transmission coincides with a
change in handedness of the optical vortices.
• Note the presence of two ‘sink’ singularities and a `monkey saddle’.
Radially Polarized Beams
Beam axis
Linearly polarized beam
Radially polarized beam
Radially polarized beams can be created in various ways, e.g., by
combining two, mutually orthogonally polarized, Hermite-Gaussian
beams.
Focusing Radially Polarized Beams [8]
The electric field in the focal region has two components:
one radial, the other longitudinal. The polarization ellipse is
perpendicular to the focal plane.
[8] T.D. Visser & J.T. Foley, J. Opt. Soc. Am. A 22, p. 2527 (2005).
Polarization Singularities [9]
• In a monochromatic wavefield, the end-point of the electric
field vector traces out an ellipse over time.
y
Three parameters describe the ellipse:
1. Handedness
2. Eccentricity
3. Orientation angle y
There are two types of polarization singularities:
When the ellipse degenerates into a line [linear polarization]
the handedness is undetermined → L -point.
When it is a circle [circular polarization]
the orientation angle y is singular → C-point.
N.B. at a polarization singularity, the state of polarization is
well-defined!
[9] R.W. Schoonover & T.D. Visser, Optics Express 14, 5733 (2006).
Poincaré Sphere for Radial Light
• The state of polarization is represented by a point (s1,s2,s3) on
the Poincaré sphere.
• Traditionally defined for Cartesian field components Ex and Ey.
• We define an analogous geometrical picture for radially
polarized light, expressing the Stokes parameters in terms of
the radial and longitudinal components er and ez:
• Circular polarization on
North and South Pole
• Linear polarization
along the equator
Circular Polarization: C-points
The orientation of the major semi-axis of the
polarization ellipse near a C-point :
s=+½
s=-½
c.f. fingerprints
• At C-points the orientation angle y of the polarization
ellipse is singular.
• The topological index t is a half-integer, i.e., the ellipse
rotates 180o around a C-point.
Topology near a C-point
Three generic topologies can occur:
Star s = -1/2
Lemon s = 1/2
Monstar s = 1/2
• Lemons can smoothly change into a monstar, and then annihilate
with a star singularity.
• These three different types of singularities have all been
predicted to occur in focused, radially polarized fields.
• Annihilations have also been predicted:
Annihilation of C-points
When the aperture angle a of the focusing system is changed,
C-points can annihilate each other [9]:
(a) a = 52o
(b) a = 61o
(c) a = 65o
Shown here is the local orientation of the major axis of the polarization
ellipse:
(a) A lemon and a star; (b) The lemon has morphed into a monstar;
(c) The monstar and star have merged and annihilated each other.
Coherence of Wave Fields
Amplitude
Field at r1
time
Highly correlated
Field at r2
Uncorrelated
Field at r3
Correlation Functions in the
Space-Frequency Domain
• The most important two-point correlation functions are the
cross-spectral density :
W (r1 , r2 ,  ) = U * (r1 ,  ) U (r2 ,  )
and the spectral degree of coherence, the normalized version of W :
m (r1 , r2 ,  ) =
U * (r1 ,  )U (r2 ,  )
S (r1 ,  ) S (r2 ,  )
with S (r,  ) = U * (r,  )U (r,  )
• The bounds of m :
the spectral density
0  m (r1 , r2 , )  1
Unity represents complete coherence.
Zero represents complete incoherence, the phase of m is then
undefined: this is a correlation singularity in R6.
Intermediate values represent partial coherence.
Singularities of the
Spectral Degree of Coherence
• At a pair of points r1, r2 such that m (r1,r2,) = 0
the field at frequency  is completely uncorrelated, i.e.,
r1 and r2 form a correlation singularity [in R6].
• Question 1: Do correlation functions exhibit vortices?
Answer: Yes
[Opt. Commun. 222, pp. 117-125 (2003)]
• Question 2: If the source is partially coherent, is the field at all
pairs of observation points then also partially coherent?
Answer: No, there can be pairs of points at which the field is
completely incoherent.
Also, there can be pairs of points at which the field is
fully coherent.
[Opt. Lett. 28, pp. 968-970 (2003) and Opt. Lett. 30, pp. 120-122 (2005)]
A Cascade of Singularities
• We have discussed phase singularities, singularities of the
Poynting vector, polarization singularities and coherence
singularities.
• But what is the connection between these different types of
singularities?
• Can one kind transform into another?
As we will see, in Young’s Experiment three different kinds of
singularities can be created in succession [10].
[10] T.D. Visser and R.W. Schoonover, Opt. Commun. 281, pp. 1–6 (2008).
Young’s Double-Slit Experiment
`The most beautiful experiment ever…’ [11]
In this experiment correlation singularities can evolve
into phase singularities which in turn can evolve in
polarization singularities!
[11] New York Times (9-24-2002).
Increasing Spatial Coherence
Increasing the spatial coherence of incident field increases the
‘sharpness’ or ‘visibility’ of the interference fringes.
But are there any coherence singularities and what happens to them
when the spatial coherence is changed?
Young’s Experiment
Consider two observation points that are located
symmetrically with respect to the z-axis:
It is easy to show that the cross-spectral density (within the
validity of the paraxial approximation) satisfies the equation
If the term between the braces is zero, then W(r1,r2,) = 0,
and the fields at r1 and r2 are completely uncorrelated.
For partially coherent fields 0 < |m (r1,r2,)| < 1,
and hence there are always such pairs of points!
So even when the incident field is partially coherent,
there are pairs of points that are completely incoherent.
If the incident field is fully coherent and co-phasal, then
m12(inc) ( ) = 1
and the spectral density (or ‘intensity’) is zero at points
All the xn are phase singularities.
The correlation singularities (pairs of positions) are indicated by
the pairs of same-colored bars. Their positions depend on
R[m12(inc)()]. If this value is increased, i.e., if the spatial
coherence of the fields at the two pinholes is increased, the
halves of two different correlation-singularities move together
and eventually annihilate when R[m12(inc)()] = 1, creating dark lines
(phase singularities) in the process. This is illustrated by the next
movie:
Annihilation of Correlation Singularities
Halves of different correlation singularities move together
when R[m] is increased. When the field at the two pinholes
becomes fully coherent (R[m] = 1) they annihilate, and
dark lines (phase singularities) are created.
Polarization Singularities
What happens when we change the direction of polarization of
one of the two beams?
Until now, both beams were assumed to be linearly polarized, with the
two directions of polarization being parallel. Therefore a scalar
description sufficed.
Polarization in Young’s Experiment
If the two directions of polarization make an angle a with each
other, then the fields in the two apertures are
The field on the screen is
This ‘new’ component destroys
phase singularities!
with K1 and K2 the usual propagators.
Next we calculate the Stokes parameters on the screen.
For a = 0, the field
everywhere is linearly
polarized along the x-axis:
s1 = 1, and s3 = 0.
A phase singularity occurs
at x = 0.317 mm (arrow).
When a = 0.005, the field
at the phase singularity is
non-zero because of
the new, non-vanishing
y-component.
Also, s1 changes from +1 to -1: x-polarization turns into y-polarization:
An L-line. Left of the L-line: s3 = +1; right of it s3 = -1: these are
2 C-lines with opposite handedness.
By slightly tilting one direction of polarization, a single phase
singularity unfolds into a triplet of polarization singularities.
Conclusions
Four types of singularities are ubiquitous in optical fields:
1. Phase singularities
2. Poynting vector singularities
3. Correlation singularities
4. Polarization singularities
In Young’s interference experiment all such singularities can appear:
1.
If the two beams are partially coherent, there are always pairs
of points that are completely uncorrelated.
2. Such correlation singularities evolve into phase singularities
when the field becomes fully coherent.
3. By slightly changing one direction of polarization, each phase
singularity decays into a triplet of polarization singularities,
namely an L-point, and two C-points of opposite handedness.
New York Times
(9/24/2002)
According to a survey held among scientists, the ten most beautiful
experiments ever done in physics are:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Young's experiment with electrons (1961)
Galileo's experiment with falling bodies (±1600)
Millikan's oil-drop experiment (1910s)
Newton's decomposition of sunlight (1665)
Young's interference experiment with light (1801)
Cavendish's torsion-bar experiment (1798)
Eratosthenes measuring the Earth's circumference (250 BC)
Galileo lets balls roll down inclined planes (±1600)
Rutherford's discovery of the nucleus (1911)
Foucault's pendulum demonstrates the Earth’s rotation (1851)