Focusing of light
Colin Sheppard
Division of Bioengineering
and
Department of Biological Sciences
National University of Singapore
E-mail: [email protected]
Tight focusing of light
•Microscopy
•Laser micromachining and microprocessing
•Optical data storage
•Optical lithography
•Laser trapping and cooling
•Physics of light/atom interactions
•Cavity QED
Overview
•Complete spherical focusing
•Bessel beams
•Gaussian beams
•High numerical aperture focusing
•Pupil masks (super-resolving filters)
•Polarization in focusing
•4Pi geometry
•Moments
Complete spherical focusing
Complete spherical (4 ) scalar focusing
Same as field of a point source and a point sink
From scalar form of Richards and Wolf
A plane-polarized wave after focusing
• px electric dipole alongx axis• my magnetic dipole alongy axis• C is nearly linear polarization• A is polarization singularity of order 2• Richards & Wolf (Ignatovsky) polarization
Direction of propagation
A plane polarized wave after focusing:
Polarization on reference sphere
direction of propagation
C
• Polarization is same as that of • px (electric dipole along x axis)• my (magnetic dipole along y axis)
• C is nearly linear polarization• Richards & Wolf polarization
Bessel beams
Bessel Beam
Annular mask(Linfoot & Wolf, 1953
Axicon (McLeod, 1954)
Diffractive axicon (Dyson, 1958)
Bessel beams
J0 beam propagates without spreading:
Also higher order beams Jn(v) exp (in ) with a phase singularity (vortex)
Sometimes called “diffraction-free” beams
Bessel-Gauss beam
= fractional Fourier (Hankel) order
transverse coordinate
Bessel-Gauss beam
a = 0.1
annular beam
Non-paraxial Bessel beam
(plane polarized illumination)
Time-averaged electric energy density:
30 90
60 120doublespot
x-polarized illumination:
Intensity along x, y axes: NA = 1.4
Circular pupil Annular pupilBroad along x axis because oflongitudinal field component
Radial polarization TM0
Annulus at high NA:
circular polarization or TM0
•High NA, circular polarized annulus: ~same width as Airy•High NA, TM0 annulus (radially polarized illumination): similar to paraxial (Dorn, Quabis and Leuchs, PRL 91, 233901, 2003)
Widths of Bessel beams
Solid immersion lens
Gaussian beams
Highly convergent Gaussian beams,
Complex source/sink theory:
Electric + magnetic dipoles at complex location
r R
Complex source point model:
Amplitude is the same as that for a source at
the point z = iz0, where z0 is the confocal parameter
Deschamps El. Lett. 7, 684 (1971)
Couture and Bélanger, Phys. Rev A 24, 355 (1981)
Intensity in waist, LP01
Time-averaged electric energy density:
•Double-spot•Caused by magnetic dipole component
Intensity and phase along axis, LP01
Gouy phase shift
Far field
Electric field:
Amplitude:
•axially symmetric
•more directional than the scalar case
Radiation pattern in far field
•Directional even for kz0 = 0
a2( )
Complex source/sink Gaussian beam,
TM01 and TE01modes
Transverse magnetic (axial electric dipole, radial illumination):
Transverse electric (axial magnetic dipole, azimuthal):
•Surface-emitting semiconductor lasers•also in gas, solid state and dye lasers•components of TEM01* (doughnut mode)
After focusing, not radial, as axial component
Intensity in waist, TM01 and
TE01modes Transverse magnetic(axial electric dipole)
Transverse electric (axial magnetic dipole)
E
H
E
Hzero on axis
non-zero on axis (longitudinal field)
Azimuthal
Radial + longitudinal component
Highly convergent focusing
Model for focusing by high numerical
aperture lens
(Debye approximation)
Equivalent refractive locus (sphere for aplanatic system)
f f
Front focal plane
E1( , )
E(r)
Black Box
Richards and Wolf, 1959 Angular spectrum of plane waves
I2: cross-polarization component (ey)I1: longitudinally-polarized component
Aplanatic factor
Ignatovsky, 1919
Focal field as integral over angular spectrum:
Aplanatic factor
I3: cross-polarization componentI1: longitudinally-polarized component
Pupil masks
Performance parameters, paraxial
t 2
Transverse gain
Axial gain (Radius of Gyration2 about Centre of Gravity)
Moments of pupil
Centre of Gravity
Radius of Gyration squared
For real-valued pupils:
(Zero order moment)2
• Calculate performance directly from pupil
High numerical aperture scalar systems
Filter performance parameters for high-aperture focusing
F =I0
2
2 Q (c)2
dc1
1
FI =2I0
2
Q (c)2
cdc
0
1
GT
=3
21
I2
I0
GA
= 3I
2
I0
I1
I0
2
GP
=1
32G
T+ G
A( ) =1I
1
I0
2
c = cosU( , z) = ikf0
P( )J0 k sin( )exp ikzcos( )sin d
Total power or integrated intensity in axial sidelobes
Integrated energy in outer rings
3D polar gain (1st moment)
Interpretation in terms of Moment of Inertia (2nd moment) of generalized 3D pupil (cap of sphere)
(zero order moment)2
Vectorial electromagnetic case
qn = Q (c)1
1
cndc
GA = 3q
0q
2q
1
2
q0
2
Gx =3
4
10q0q
13q
0q
23q
0
24q
1
2
q0
2
Gy =3
4
3q0
22q
0q
1q
0q
2
q0
2
GT =3
4
4q0q
12q
0q
22q
1
2
q0
2
GP = (Gx +Gy +GA )/ 3
GP =2q1(q0 q1)
q0
2
In =1
1
Q(c)1 c
1+ c
n / 2
Jn k 1 c2( )exp ikzc( )dc.
Q(c) = c1/ 2 (1+ c)Aplanatic (sine condition):
Richards and Wolf (equivalent form):
c= cos
Circular polarized or unpolarized:
Only zero and first moment (Centre of Gravity)
Gains for electromagnetic case, plane
polarized input
negative means double spot
Intensity at the focus
Mixed dipole apodization (p + m) gives greatest intensity at the focus
F
Electric dipole polarization
Electric dipole wave:
Ratio of focal intensity to power input
C. J. R. Sheppard and P. Török, "Electromagnetic field in the focal region of an electric dipole wave," Optik 104, 175-177 (1997).
electric dipole ED
mixed dipole(plane polarized)
TM0 (radial polarization)
ED is highest
90o 180o
Radial polarization (TM0):
polarization on reference sphere
direction of propagationred: electric field
blue: magnetic field
Radial polarization with phase mask
Wang HF, Shi LP, Luk’yanchuk B, Sheppard C, Chong CT (2008)
Creation of a needle of longitudinally polarized light in vacuum
using binary optics,
Nature Photonics, 2, 501-505, 22
Electric dipole: Polarization on
reference sphere
Mixed = ED + MD
direction of propagation
red: electric field
blue: magnetic field
Polarization on reference sphere:
TE1, TM1
Polarization of
input wave
Bessel beams: TE1 polarization Mixed
ED TE1
Mixed
ED TE1
Mixed
ED TE1
Mixed
ED TE1
30° 60°
90° 150°
High NA: Intensity at the focus for
different polarizations
1
1
p+m
TM0p
p
p+m
Gains for different polarizations
Area of focal spot
NA = 0.89NA = 0.91
Focal volume
NA = 0.98TE1 smallest
Rotationally symmetric beams
•TM0 = radial polarized input (longitudinal field in focus)
•TE0 = azimuthal polarization
•x polarized + i y polarized = circular polarized
•TE1x + i TE1y = azimuthal polarization with a phase singularity
(bright centre)
•EDx + i EDy = elliptical polarization with a phase singularity
(bright centre) (ellipticity increases with angle from axis)
•(TM1x + i TM1y = radial polarization with a phase singularity)
•Same GT as for average over
Normalized width for rotationally
symmetric
TE1 = azimuthal polarization with phase singularity (vortex)
TE annulus narrowestTE narrowest for NA<0.98
1
1
1
1
Bessel beams:
Transverse behaviour for rotationally
symmetric (also average over )
30o 60o
90omixed
mixed
MD
mixed,
TE1 narrowest
1
1
1
1
Bessel beams for rotationally
symmetric Side lobes
Eccentricity
Transverse gain
Rad
TE1 is narrowest
ED has weakest sidelobes
Rad has weakest sidelobes
NA=0.83
1
1
1
1
1
1
Points to note
•Focusing plane polarized light results in a large focal
spot
•Focusing is improved using radially polarized
illumination
-Strong longitudinal field on axis
•Electric dipole polarization gives higher electric energy
density at focus
•Transverse electric (TE1) polarization gives smallest
central lobe
(smaller than radially polarized for Bessel beam)
•TE1 is asymmetric: symmetric version is azimuthal
polarization with a phase singularity (vortex)
4 Pi microscope (Hell)
Hell, S. Europäisches Patent EP 0 491 281 B1 (18.12.1990) "Doppelkonfokales Rastermikroskop".
4Pi (a) Fluorescence Microscope
Resolution in 4 Pi
• Axial resolution is improved
• Longitudinal field components from counter-propagating beams
cancel out, so transverse resolution is also improved
Performance
parameters for
4 Pi
1.46NA
GP = 1As GT increases, GA decreases
1
1
1
1
1
1
Micron, to be published
4 Pi
transverse
axial
1
1
Micron, to be published
spherical spot
4 Pi:
Need to match electric dipole polarization
Electric field in input plane
radial
4 Pi
Table 1. Values of the parameters F, GT and M for NA = 1.46. “% increase in resolution” is for 4Pi compared with the single lens case.
1
Micron, to be published
Localization in terms of pupil moments
Localization in terms of pupil moments, μn (scalar)
Propagation of second moments (scalar)
(Transverse width)2
(Axial width)2
Axial position of centre of gravity: (Central second moment width)2
Thanks to: Silvia Ledesma, Buenos Aires, Argentina
Juan Campos, Barcelona
Juan-Carlos Escalera, Barcelona
Manuel Martinez-Corral, Valencia
Miguel Alonso, Rochester
Nicole Carlson, Rochester
Steven van Enk, Bell Labs
Gerd Leuchs, Erlangen
Susanne Quabis, Erlangen
Rolf Dorn, Erlangen
Silvania Pereira, Delft
Peter Török, Imperial College
Kieran Larkin, Sydney
Peeter Saari, Estonia
Amar Choudhury, Gauhati
Naveen Balla (SMA)Shakil Rehman (SERI)Tang Wai Teng (SMA)Elijah Yew (SMART)