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

undefined

f at such points is 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 in a continuous manner. annihilate when a parameter such as the slit width or the wavelength is varied [5] H.F. Schouten et al.,

Phys. Rev. E

67

, 036608 (2003) [6] H.F. Schouten et al., O

ptics Express

11

, p. 371 (2003).

Optical Vortices and Saddles

• The next slide shows the to geometrical optics.

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 • 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

here phase , or direction of the Poynting vector is singular at points where its modulus |S| = 0.

Topological Charge and Index

Monochromatic field At points where the amplitude

A =

0 , the phase f The topological charge of grad f is undefined = topological index

t

or singular.

Phase Grad f Phase Grad f Positive Vortex Saddle conserved quantities!

Under smooth changes of a system parameter, such as l, phase singularities can be

t

created or annihilated .

s

= 0

t

= 1 Two vortices Negative Vortex of opposite sign can Maximum two saddles

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 When it is a circle [circular polarization] the orientation angle y is singular

→

L-point.

→

C-point.

N.B.

at a polarization singularity, the state of polarization well-defined!

is [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 (s the Poincaré sphere. 1 ,s 2 ,s 3 ) on • Traditionally defined for Cartesian field components E x and E y . • We define an analogous geometrical picture for radially polarized light, expressing the Stokes parameters in terms of the radial and longitudinal components e r and e z :

•

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 ellipse is singular.

• The topological index

t

rotates 180 o is a around a C-point. y of the polarization half-integer , i.e., the ellipse

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 = 52 o (b) a = 61 o (c) a = 65 o 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.

Field at r 1

Coherence of Wave Fields

Amplitude time Highly correlated Field at r 2 Uncorrelated Field at r 3

Correlation Functions in the Space-Frequency Domain

• The most important two-point correlation functions are the cross-spectral density :

W

(

r

1 ,

r

2 ,  ) =

U

* (

r

1 ,  )

U

(

r

2 ,  ) and the

spectral degree of coherence,

the normalized version of

W

: m (

r

1 ,

r

2 ,  ) =

U

* (

r

1 ,  )

U

(

r

2 ,  )

S

(

r

1 ,  )

S

(

r

2 ,  ) with

S

(

r

,  ) =

U

* (

r

,  )

U

(

r

,  )

the spectral density

• The

bounds of

m : 0  m (

r

1 ,

r

2 ,  )  1 Unity represents complete coherence.

Zero represents complete incoherence, the phase of undefined: this is a correlation singularity in R 6 .

Intermediate values represent partial coherence.

m is then

Singularities of the Spectral Degree of Coherence

• At a pair of points r 1 , r 2 the field at frequency  such that m (r 1 ,r 2 ,  )

r

1 and r 2 form a correlation singularity [in R 6 ].

= 0 is completely uncorrelated, i.e., • 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

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

(

r

1 ,

r

2 , and the fields at r 1 and r 2 are completely uncorrelated.

For partially coherent fields 0 < | m (

r

1 ,

r

2 ,  ) | < 1, and hence there are always such pairs of points!

So even when the incident field is partially coherent,

 ) = 0,

there are pairs of points that are completely incoherent.

If the incident field is fully coherent and co-phasal, then m ( 12 inc ) (  ) = 1 and the spectral density (or ‘intensity’) is zero at points All the

x n

are phase singularities.

The correlation singularities (pairs of positions) are indicated by the pairs of same-colored bars. Their positions depend on

R

[ m 12 (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

[ movie: m 12 (inc) (  )] = 1, creating dark lines (phase singularities) in the process. This is illustrated by the next

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 This ‘new’ component destroys phase singularities! The field on the screen is with

K

1 and

K

2 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:

s

1 = 1

, and

s

3 = 0.

A

phase singularity

occurs at

x

= 0.317 mm ( arrow ).

When a = 0.005

non-zero

, the field at the phase singularity is because of the new, non-vanishing

y

-component.

Also,

s

1 changes from +1 to -1:

x

-polarization turns into

y

-polarization: An L-line. Left of the L-line:

s

3 = +1 ; right of it 2 C-lines with opposite handedness.

s

3 = -1: these are

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 singularity decays evolve into phase singularities when the field becomes fully coherent. 3. By slightly changing one direction of polarization, each phase 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. Young's experiment with electrons (1961) 2. Galileo's experiment with falling bodies (±1600) 3. Millikan's oil-drop experiment (1910s) 4. Newton's decomposition of sunlight (1665) 5. Young's interference experiment with light (1801) 6. Cavendish's torsion-bar experiment (1798) 7. Eratosthenes measuring the Earth's circumference (250 BC) 8. Galileo lets balls roll down inclined planes (±1600) 9. Rutherford's discovery of the nucleus (1911) 10. Foucault's pendulum demonstrates the Earth’s rotation (1851)