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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
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