Focusing of light. Colin Sheppard Division of Bioengineering and Department of Biological Sciences National University of Singapore

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1 Focusing of light Colin Sheppard Division of Bioengineering and Department of Biological Sciences National University of Singapore

2 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

3 Overview Complete spherical focusing Bessel beams Gaussian beams High numerical aperture focusing Pupil masks (super-resolving filters) Polarization in focusing 4Pi geometry Moments

4 Complete spherical focusing

5 Complete spherical (4 ) scalar focusing From scalar form of Richards and Wolf Same as field of a point source and a point sink

6 A plane-polarized wave after focusing Direction of propagation p x electric dipole along x axis m y magnetic dipole along y axis C is nearly linear polarization A is polarization singularity of order 2 Richards & Wolf (Ignatovsky) polarization

7 A plane polarized wave after focusing: Polarization on reference sphere direction of propagation C Polarization is same as that of p x (electric dipole along x axis) m y (magnetic dipole along y axis) C is nearly linear polarization Richards & Wolf polarization

8 Bessel beams

9 Bessel Beam Annular mask (Linfoot & Wolf, 1953 Axicon (McLeod, 1954) Diffractive axicon (Dyson, 1958)

$exp (in ) with a phase singularity (vortex) Sometimes called diffraction-free beams$

10 Bessel beams J 0 beam propagates without spreading: Also higher order beams J n (v) exp (in ) with a phase singularity (vortex) Sometimes called diffraction-free beams

11 Bessel-Gauss beam transverse coordinate = fractional Fourier (Hankel) order

12 Bessel-Gauss beam annular beam a = 0.1

13 Non-paraxial Bessel beam (plane polarized illumination) Time-averaged electric energy density:

14 30 90 double spot

15 x-polarized illumination: Intensity along x, y axes: NA = 1.4 Circular pupil Annular pupil Broad along x axis because of longitudinal field component

16 Radial polarization TM0

17 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, , 2003)

18 Widths of Bessel beams Solid immersion lens

19 Gaussian beams

20 Highly convergent Gaussian beams, Complex source/sink theory: Electric + magnetic dipoles at complex location Complex source point model: Amplitude is the same as that for a source at the point z = iz 0, where z 0 is the confocal parameter Deschamps El. Lett. 7, 684 (1971) Couture and Bélanger, Phys. Rev A 24, 355 (1981) r R

21 Intensity in waist, LP 01 Time-averaged electric energy density: Double-spot Caused by magnetic dipole component

22 Intensity and phase along axis, LP 01 Gouy phase shift

23 Far field Electric field: Amplitude: axially symmetric more directional than the scalar case

24 Radiation pattern in far field a 2 ( ) Directional even for kz 0 = 0

25 Complex source/sink Gaussian beam, TM 01 and TE 01 modes Transverse magnetic (axial electric dipole, radial illumination): After focusing, not radial, as axial component Transverse electric (axial magnetic dipole, azimuthal): Surface-emitting semiconductor lasers also in gas, solid state and dye lasers components of TEM 01 * (doughnut mode)

26 Intensity in waist, TM 01 and TE 01 modes non-zero on axis (longitudinal field) Transverse magnetic (axial electric dipole) Transverse electric (axial magnetic dipole) H E Azimuthal E Radial + longitudinal component H zero on axis

27 Highly convergent focusing

28 Model for focusing by high numerical aperture lens (Debye approximation) Front focal plane Black Box E 1 (, ) f f E(r) Equivalent refractive locus (sphere for aplanatic system)

29 Richards and Wolf, 1959 Angular spectrum of plane waves Aplanatic factor I 2 : cross-polarization component (e y ) I 1 : longitudinally-polarized component

30 Ignatovsky, 1919 Focal field as integral over angular spectrum: Aplanatic factor I 3 : cross-polarization component I 1 : longitudinally-polarized component

31 Pupil masks

32 Performance parameters, paraxial Calculate performance directly from pupil For real-valued pupils: (Zero order moment) 2 Transverse gain t 2 Moments of pupil Centre of Gravity Radius of Gyration squared Axial gain (Radius of Gyration 2 about Centre of Gravity)

33 High numerical aperture scalar systems U(,z) = ikf P( )J 0 ( k sin )exp( ikz cos )sin d 0 c = cos F = F I = 1 I Q(c) 2 dc I 0 2 Q(c) 2 c dc G T = I 2 I 0 G A = 3 I 2 I 2 1 I 0 I 0 G P = 1 3 2G ( T + G A )=1 I 1 3D polar gain (1st moment) (zero order moment) 2 Total power or integrated intensity in axial sidelobes Integrated energy in outer rings I 0 2 Interpretation in terms of Moment of Inertia (2nd moment) of generalized 3D pupil (cap of sphere) Filter performance parameters for high-aperture focusing

34 Vectorial electromagnetic case Richards and Wolf (equivalent form): I n = c Q(c) 1+ c n /2 Aplanatic (sine condition): J ( n k 1 c 2 ) exp( ikzc)dc. Q(c) = c 1/2 (1+ c) c = cos q n = 1 1 Q (c) c n dc G x = 3 10q 0 q 1 3q 0 q 2 3q q 1 G 2 y = 3 4 q 0 4 Circular polarized or unpolarized: 3q 0 2 2q 0 q 1 q 0 q 2 q 0 2 G A = 3 q q q G T = 3 2 4q 0 q 1 2q 0 q 2 2q 1 G 2 P = (G x +G y +G A )/3 2 4 q 0 q 0 G P = 2q 1 (q 0 q 1 ) q 0 2 Only zero and first moment (Centre of Gravity)

35 Gains for electromagnetic case, plane polarized input negative means double spot

36 Intensity at the focus F Mixed dipole apodization (p + m) gives greatest intensity at the focus

37 Electric dipole polarization

38 Electric dipole wave: Ratio of focal intensity to power input TM0 (radial polarization) electric dipole ED ED is highest mixed dipole (plane polarized) 90 o 180 o C. J. R. Sheppard and P. Török, "Electromagnetic field in the focal region of an electric dipole wave," Optik 104, (1997).

39 Radial polarization (TM0): polarization on reference sphere direction of propagation red: electric field blue: magnetic field

40 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, , 22

41 Electric dipole: Polarization on reference sphere red: electric field blue: magnetic field Mixed = ED + MD direction of propagation

42 Polarization on reference sphere: TE1, TM1

43 Polarization of input wave

44 Bessel beams: TE1 polarization Mixed Mixed ED TE1 ED TE1 Mixed Mixed ED TE1 ED TE1

45 High NA: Intensity at the focus for different polarizations TM0 p p+m 1 p p+m 1

46 Gains for different polarizations

47 Area of focal spot NA = 0.91 NA = 0.89

48 Focal volume TE1 smallest NA = 0.98

49 Rotationally symmetric beams TM0 = radial polarized input (longitudinal field in focus) TE0 = azimuthal polarization x polarized + i y polarized = circular polarized TE1 x + i TE1 y = azimuthal polarization with a phase singularity (bright centre) ED x + i ED y = elliptical polarization with a phase singularity (bright centre) (ellipticity increases with angle from axis) (TM1 x + i TM1 y = radial polarization with a phase singularity) Same G T as for average over

50 Normalized width for rotationally symmetric 1 1 TE 1 narrowest for NA<0.98 TE annulus narrowest 1 TE1 = azimuthal polarization with phase singularity (vortex)

51 Bessel beams: Transverse behaviour for rotationally symmetric (also average over ) 30 o 60 o MD 1 mixed 1 mixed, 1 TE1 narrowest mixed 90 o 1

52 Bessel beams for rotationally symmetric Transverse gain TE1 is narrowest Side lobes ED has weakest sidelobes Rad has weakest sidelobes NA=0.83 Eccentricity 1 1 Rad

53 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)

54 4 Pi microscope (Hell) Hell, S. Europäisches Patent EP B1 ( ) "Doppelkonfokales Rastermikroskop".

55 4Pi (a) Fluorescence Microscope

56 Resolution in 4 Pi Axial resolution is improved Longitudinal field components from counter-propagating beams cancel out, so transverse resolution is also improved

57 1.46NA Performance parameters for 4 Pi G P = 1 As G T increases, G A decreases 1 1 Micron, to be published

58 4 Pi 1 transverse spherical spot axial 1 Micron, to be published

59 4 Pi: Need to match electric dipole polarization

60 Electric field in input plane radial

4 Pi 1 Table 1. Values of the parameters F, G T and M for NA = 1.46.

61 4 Pi 1 Table 1. Values of the parameters F, G T and M for NA = % increase in resolution is for 4Pi compared with the single lens case. Micron, to be published

62 Localization in terms of pupil moments

63 Localization in terms of pupil moments, μ n (scalar)

64 Propagation of second moments (scalar) (Transverse width) 2 (Axial width) 2 Axial position of centre of gravity: (Central second moment width) 2

65 Thanks to: Naveen Balla (SMA) Shakil Rehman (SERI) Tang Wai Teng (SMA) Elijah Yew (SMART) 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

CREATION OF SUPER-RESOLUTION NON-DIFFRACT- ION BEAM BY MODULATING CIRCULARLY POLAR- IZED LIGHT WITH TERNARY OPTICAL ELEMENT

Progress In Electromagnetics Research, Vol. 140, 589 598, 2013 CREATION OF SUPER-RESOLUTION NON-DIFFRACT- ION BEAM BY MODULATING CIRCULARLY POLAR- IZED LIGHT WITH TERNARY OPTICAL ELEMENT Jingsong Wei 1,