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Physical Optics

Lecture 2 : Diffraction

2017-04-10

Beate Boehme

Physical Optics: Content2

No Date Subject Ref Detailed Content

1 05.04. Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation, interferometry

2 12.04. Diffraction B Slit, grating, diffraction integral, diffraction in optical systems, point spread function, aberrations

3 19.04. Fourier optics B Plane wave expansion, resolution, image formation, transfer function, phase imaging

4 26.04. Quality criteria and resolution B Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point

resolution, criteria, contrast, axial resolution, CTF

5 03.05. Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components

6 10.05. Photon optics D Energy, momentum, time-energy uncertainty, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

7 17.05. Coherence G Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle

8 24.05. Laser B Atomic transitions, principle, resonators, modes, laser types, Q-switch, pulses, power

9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations

10 07.06. Generalized beams D Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy beams, applications in superresolution microscopy

11 14.06. PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping, confocal PSF

12 21.06. Nonlinear optics D Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, CARS microscopy, 2 photon imaging

13 28.06. Scattering G Introduction, surface scattering in systems, volume scattering models, calculation schemes, tissue models, Mie Scattering

14 05.07. Miscellaneous G Coatings, diffractive optics, fibers

D = Dienerowitz B = Böhme G = Gross

Wave optics – summary 3

Electric field in a homogeneous medium without chargescompare water basin with fixed depth

Wave equation second derivative in space is proportional tosecond derivative in time

Solutions: functions of position and time = waves

Simplest solution

= plane wave: from x and y independent, in z propagating wave

Further solution

= sperical wave, where r is the distance to the center

Optical intensity

= optical power per unit area [Watt/cm²]average value over a time much longer than period

2

22

~~

tEE

2

2

2

2

2

2

2

2

tE

zE

yE

xE

)cos(22

)cos(2kzt

t

kzt

)cos(),( kztAtrE

)cos(22

)cos(2kztk

z

kzt

2

222

ck

2),(),( trEtrI

)cos(),( krtrAtrE

Compare: A = 4r²spherical surfaceIntensity reduceswith 1/r²

Wave optics – consequences 4

Linearity of wave equation (derivation is)

the sum of two solutions (waves) is also a possible solution

Principle of superposition

Sum of plane wavesarbitrary direction

Sum of spherical wavescenter r0

Kirchhoff Integral = sum of spherical waves:

the field E(r‘) at any position r‘ can be described assuperposition of spherical waves, which originateat the points of the preceding field E(r) at the surface F

)(cos),( 00

rrktrr

AtrE

)(),( kztieAtrE )cos(),( zkykxktAtrE zyx

)(),( krtierAtrE

Complex representationfor simplified calculations( cos-sums e.g. ) Ereal = Re(EC) = ½(Ec+E*c)

F

rrkiti dF

rrerAerE

')cos()(~)'(

'

5

F

rrkiti dF

rrerAerE

')cos()(~)'(

'

Diffraction at apertures

F

Kirchhoff integral Hygens Principle

Each point of a wavefront acts asstarting point of spherical wavesTheir envelope forms a new wavefrontCalculation of field at any point

assumption: Field within the apertureequals the field without it, no interaction

F

r‘

r

6

Double Slit or slit with infinite length

Intensity after double slit with distance g or one slit with width a with screen far behind

du rrE

2cos2cos~

Constructive interference on axis

Destructive for path difference /2

g2)sin( min

1 12

2)sin( min m

gm

Constructive interference for difference

mgm )sin( max

g a

Destructive for difference eg. between rim and center of aperture /2

am2

2)sin( min

Constructive for difference /2 for rims

am

m

)sin( min

am

21)sin( max

7

Double Slit and grating

Grating

Constructive interference for difference

mgm )sin( max

g

ww

w.g

irlin

clou

ds.w

orld

pres

s.co

m Ideal diffraction grating: collimatedmonochromatic incident beam isdecomposed into discrete sharp diffraction orders

Constructive interference ascontribution of all periodic cells

The number of orders depends on grating structure, for sinusoidal

structure only two orders

Arbitrary incidence angle 0

mgm )sin()sin( 0

max

8

Double Slit and grating

Grating: g

ww

w.g

irlin

clou

ds.w

orld

pres

s.co

m

)sin(sin

)sin(sin)(

2

2

g

gNI

0.+1. +2.

+3.+4.-4.

-3.-2. -1.

Number of grating periods N

Sharpness of ordersincreases with N

Grating resolution: separation of 2 spectral lines for instance Na 589 & 589,6nm is visible with

N = 1000

Number of orders m depend on grating fine structure for sinusodial structure only two orders Blaze/echelette grating has facets with finite

slope all orders but one higher suppressed Complete setup with all orders:

Overlap of spectra at higher orders possible

Nm

9

Slit and rectangular aperture a

Constructive interference fordifference /2 between rims of slit

125,0)sin( max mam

sinsin~2

axwhere

xxI

Rectangularaperture

Square aperture

Slit with infinite length

DAiry

10

Circular aperture

Slit: first destructive interference at

Circular aperture – no separation of coordinates

Finite aperture causes finite spot size = Airy-diameter DA

distance = diameter of first dark rings

a

a )sin( min

1

NADA mim

22,1)sin(

22,11

min

a 22,1)sin( min

1

More general: NA = n sin()

~ r / f a =

2r f

Ideal Imaging

11

DAiry

Geometric representation: Gaussian Collineation Physical representation: limited aperture

finite angle light cone in image space uncomplete, constructive interference spreaded image point the optical systems works as a low pass filter resolution from scalar diffraction theory

Field in the image plane ~ Fourier transformation of the

complex pupil function A(xp,yp)

objectpoint

spherical wave

truncatedspherical

wave

(Gaussian= paraxial)

imageplane

object plane

Perfect Imaging

pp

yyxxR

i

ppExP

dydxeyxAyxEpp

ExP''2

,)','(

2 Aberrations

z = 4RU

NADAiry

22.1

'sin' unNA

Lateral PSF Axial PSF

22

''sin' NA

nun

RU

Δ 4

2

4/4/sin~0,

s

ss z

zzI 2

12~,0

s

ss x

xJxI

Perfect PSF

2 Aberrations 13

Lateral PSF

w

I(w)

1

0.8

0.6

0.4

0.2

00 1 2 3-2 -1

AiryBesselGauss

FWHM

w

E(w)

1

0.8

0.6

0.4

0.2

03 41 2

AiryBesselGauss

E95%

Pupil Filling Zero-crossing & FWHM

compactness Bessel < Airy < Gauss

Energy 95%: Gauss < Airy < Bessel

Resolution @ uniform pupil @ scalar diffraction

3,3 n DAiry ~ z NA not isotropic PSF

@ axial & lateral directionSpace for optimization

Encircled Energy

Fraunhofer Point Spread Function

Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle

Fraunhofer approximation in the far fieldfor large Fresnel number

Optical systems: numerical aperture NA in image spacePupil amplitude/transmission/illumination T(xp,yp)Wave aberration W(xp,yp)complex pupil function A(xp,yp)Transition from exit pupil to image plane

Point spread function (PSF): Fourier transform of the complex pupil function

12

z

rN p

F

),(2),(),( pp yxWipppp eyxTyxA

pp

yyxxR

iyxiW

ppAP

dydxeeyxTyxEpp

APpp''2

,2,)','(

''cos'

)'()('

dydxrr

erEirE d

rrki

I

PSF by Huygens Principle

Huygens wavelets correspond to vectorial field components The phase is represented by the direction The amplitude is represented by the length Zeros in the diffraction pattern: destructive interference Aberrations from spherical wave: reduced conctructive superposition

Axial distribution of intensityCorresponds to defocus

Normalized axial coordinate

Scale for depth of focus :Rayleigh length

Zero crossing points:equidistant and symmetric,Distance zeros around image plane 4RE

22

0 4/4/sinsin)(

uuI

zzIzI o

42

2 uzNAz

22

''sin' NA

nun

RE

Perfect Axial Point Spread Function

Defocussed Perfect Psf

Perfect point spread function with defocus Representation with constant energy: extreme large dynamic changes

z = -2RE z = +2REz = -1RE z = +1RE

normalizedintensity

constantenergy

focus

Imax = 5.1% Imax = 42%Imax = 9.8%

Abbe Resolution and Assumptions

Assumption Resolution enhancement 1 Circular pupil ring pupil, dipol, quadrupole 2 Perfect correction complex pupil masks 3 homogeneous illumination dipol, quadrupole 4 Illumination incoherent partial coherent illumination 5 no polarization special radiale polarization 6 Scalar approximation 7 stationary in time scanning, moving gratings 8 quasi monochromatic 9 circular symmetry oblique illumination

10 far field conditions near field conditions 11 linear emission/excitation non linear methods

Abbe resolution with scaling to /NA: Assumptions for this estimation and possible changes

A resolution beyond the Abbe limit is only possible with violating of certain assumptions

Aberrations

19

Deblurring of PSF and Strehl Ratio

reference sphere corresponds to perfect imaging RMS deviation as an integral

measure of performance

0),(1),( dydxyxWF

yxWExP

Strehl ratio

Approximations Marechal

useful SR >0.5 Biquadratic Exponential

r

1

peak reducedStrehl ratio

distributionbroadened

ideal , withoutaberrations

real withaberrations

I(x )

0,0

0,0)(

)(

idealPSF

realPSF

IISR

2

2),(2

),(

),(

dydxyxA

dydxeyxASR

yxWi

2241 rmsM WSR

22221 rmsB WSR 224 rmsW

E eSR 2 Aberrations

20

Zernike Polynomials: Measure of Wave Aberrations

k

n

n

nmnmM cSR

1

2241

n

n

nm

mnnmZcW ),(),(

010)(cos0)(sin

)(~),(mformformmform

RZ mn

mn

Z 4,9,16

Z 2,7,14

Z 3,8,15

Peak-Valey (PV)-values at pupil rim = +1 Fringe Constant RMS-value (Orthonormal) Standard

Circular obscuration Zernike-Tatian polynomials

Rectangular pupil Legendre polynomials

describe deviation from reference sphere with orthogonal basis functions circular shape

Zernike Polynomials: conventions (amount/sign) due to Gram-Smith-Orthogonalization

2 Aberrations21

Zernike Fringe vs Zernike Standard Polynomials

Fringe coefficients: Z 4,9,16 = n² spherical azimutal order grows if

number increasesStandard coefficients - different term numbers ZN = 0.05 RMS = 0.05 SR ~ 1 – 40 ZN² = 0.9 Elimination of tiltNo Elimination of defocus @ Zemax

In radial symmetric system for y-field (meridional)sinus-terms vanish

2 Aberrations 22

Spherical Aberration

Single positive lens shorter intersection length for marginal rays “undercorrected” on axis, no field dependence, circular symmetry Increase of spherical aberration growing axial asymmetry around the nominal image plane paraxial focus perfect symmetry best image plane circle of least RMS,

best contrast at special frequency, Z4 = 0,...

Example: Focus a collimated beam with plano-convex lens n = 1.5 worse (red) vs. best (green)

orientation spherical aberration

differs by a factor of 4:

6sin

3iii 2426

22

sin233 iiiii

2 Aberrations 23

Coma: Spot Construction

Term B2 (2+cos2) 2y asymmetric ray path for non-axial object point construction of spot circles from constant pupil

zones: circle radius ~ radius at pupil² 360°@ pupil 2x360°at image

comet shape spot ray @ interval [1 3] aspect ratio 2:3 sagittal coma ~ 1/3 of tangential coma

Coma PSF correspods to spot 55% of energy in the triangle

between tip of spot and saggital coma Separation of peak and the centroid position different image position for “center of gravity” From the energetic point of view coma

induces distortion in the image

2 Aberrations 24

Term (3B3+B4) y² cos For a single positive lens:

chief ray passes surface under oblique angle projection of surface curvatures different powers in tangential and sagittal

tangential (blue) and sagittal (red) focal lines Sequence: tangential - circle of least confusion

with smallest spots (best) – sagittal – paraxial Imaging of a circular grid in different planes

Astigmatism & Petzval

Tangential ray fan

Sagittal ray fan

Tangential focus Sagittal

focus

Circle of least confusion

2 Aberrations 25

Astigmatism B3 corrected one curved image shell with Petzval curvature B4 Single positive lens image curved toward the system negative Petzval = sign convention In general Petzval image s‘ = ½ ( 3sS-sT) departs from T, S and Best, is defined @ math

TB

S par

Distortion

Pure geometric deviation without any blur

Acceptance strongly depends on kind of objects: biological samples: 10% may be sufficient geometrical shapes 1% still critical

Definition

Conventional – reference to ideal

TV distortion - reference to real y‘ at x‘=0

2-mirror scanning systems special 1-dim pincushion

Manipulation

Stop location defines the chief ray path

D > 0 pincushion stop at rear D < 0 barrel stop in front

''

'''

yy

yyyD

ideal

idealreal

2 Aberrations 26

Fresnel diffraction

27

Different ranges of edge or slit diffraction as a function of the Fresnel numberor distance:

3. Far fieldFraunhoferweak structuresmall NF ≤ 1 a << z

2. FresnelQuadratic approximationof phaseripple structuremedium NF > 1

1. Near fieldBehind slitlarge NF >> 1a >> z

Diffraction Ranges for slit

zaNF

2

Diffraction at an edge in Fresnelapproximation

Intensity distribution,Fresnel integrals C(x) and S(x)

scaled argument

Intensity:- at the geometrical shadow edge: 0.25- shadow region: smooth profile- bright region: oscillations

22

)(21)(

21

21)( tStCtI

FNxz

xzkt 22

Fresnel Edge Diffraction

t-4 -2 0 2 4 6

0

0.5

1

1.5I(t)

Typical change of the intensity profile

Normalized coordinates

Diffraction integral

30

Fresnel Diffraction

 

arr

favz

fau

;2;2

2

1

0

20

/2

02 2

2

)(2),(

devJef

EiavuEui

uafi