Physical Optics

Lecture 2: Diffraction 2018-04-18 Herbert Gross

www.iap.uni-jena.de 2 Physical Optics: Content

No Date Subject Ref Detailed Content Complex fields, wave equation, k-vectors, interference, light propagation, 1 11.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 18.04. Diffraction G function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 25.04. Fourier optics G phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 02.05. G resolution resolution, criteria, contrast, axial resolution, CTF Energy, momentum, time-energy uncertainty, photon statistics, 5 09.05. Photon optics K fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 6 16.05. Coherence K speckle, OCT-principle Introduction, Jones formalism, Fresnel formulas, birefringence, 7 23.05. Polarization G components Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 30.05. Laser K pulses, power Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 9 06.06. Nonlinear optics K CARS microscopy, 2 photon imaging Apodization, superresolution, extended depth of focus, particle trapping, 10 13.06. PSF engineering G confocal PSF Introduction, surface scattering in systems, volume scattering models, 11 20.06. Scattering L calculation schemes, tissue models, Mie Scattering 12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 13 04.07. Generalized beams G beams, applications in superresolution microscopy 14 11.07. Miscellaneous G Coatings, diffractive optics, fibers

K = Kempe G = Gross L = Lu 3 Contents

. Introduction . Slit diffraction . Grating diffraction . Diffraction integral . Diffraction in optical systems . Point spread function . Aberrations Fraunhofer Farfield Diffraction

. Diffraction at a slit: intensity distribution

2   d sin  0. order sin   d  I()       d sin     

. Angles with side maxima: constructive interference 1st diffraction   1  order sinmax  m   d  2 

order: -3rd -2nd -1st 0th +1st +2nd +3rd

. Diffraction pattern Diffraction at a Slit

. Equation for the minima :

m = asinq

. Finite detection with a lens 4 Huygens Principle

. Every point on a wave is the origin of a new spherical wave (green)

. The envelope of all Huygens wavelets stop forms the overall wave front (red) Huygens- . Light also enters the geometrical shadow Wavelets region . The vectorial superposition principle propagation requires full coherence direction

wave fronts

geometrical shadow 7 Diffraction at apertures

Kirchhoff integral Huygens Principle eik r'r  E(r') eit  A(r)cos(q ) dF Each point of a wavefront acts as F r'r starting point of spherical waves Their envelope forms a new wavefront Calculation of field at any point

r‘

F

r F assumption: Field within the aperture equals the field without it, no interaction

Ref: B. Böhme 8 Slit and rectangular aperture

Slit with infinite length

2 sinx  a sin I ~   where x   x  

Rectangular aperture a 

Square Constructive interference for aperture difference /2 between rims of slit

 sin( max )  0,5 2m 1 m a

Ref: B. Böhme 9 Sinc-function

Sinc-function 1 . sinus cardinalis Slit 4 . „slitfunction“ sin(4x) sin(4x) 2 4x sinx 0 I ~   = [ sinc(x) ]² -1 1  x  |1/(4x)| sin(4x) [ 4  x ]²  a where x    -1

Slit 1

sinc(x)

[sinc(x)]²

Ref: B. Böhme 10 Diffraction at slit

Transmission 1

Variable x @ Object Intensity at screen 0 -2 -1 0 1

1

 r sin min   d 1 a d Variable  @ Screen 0 -2 -1 0 1 2 sin(x) The sinc-function is the cos(x)dx  a / 2 2 2  fourier-transform sin(x / 2)  I() ~ cos(x)dx ~ ex of the slit function   ex dx  a / 2   a = width of slit  = diffraction angle Ref: B. Böhme 11 Double Slit or slit with infinite length

Intensity after double slit with distance g or one slit with width a with screen far behind Constructive interference on axis  2   2  E ~ cos ru   cos rd 

     

a g

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

min  min  min  2 min m sin(1 )  sin( )  2m 1 sin( )   sin( )  2g m 2g 1 2 a m a

Constructive interference for difference  Constructive for difference /2 for rims  max max 1  sin( m )  m sin( )   g m 2 a

Ref: B. Böhme Double Slit Interference

. Interference of laser light at a double slit setup

Ref: W. Osten 13 Double Slit and grating

Grating . Ideal diffraction grating: collimated monochromatic incident beam is decomposed into discrete sharp diffraction orders 0 . Constructive interference as

contribution of all periodic cells g  www.girlinclouds.worldpress.com . The number of orders depends on grating structure, for sinusoidal structure only two orders

Constructive interference for difference  Arbitrary incidence angle 0

max  max  sin( m )  m sin( )  sin( )  m g m 0 g

Ref: B. Böhme 14 Double Slit and grating

Grating:  ~  Number of grating periods N  g sin()  sin 2   N    . Sharpness of orders I()     g sin()  increases with N sin 2       0  m N . Grating resolution: 

separation of 2 spectral lines g  www.girlinclouds.worldpress.com

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

-4. +4. -3. +3. -2. -1. +1. +2. 0.

Ref: B. Böhme 15 Diffraction at grating – Complex Field

1. Width of slits defines occurrence of orders

2. Number of slits defines fidelity

3. Period of grating defines distance of orders

1-2-3-4 1-2-3-4 5 5 Between the maxima are 5 periods – The number increases with number of periods M Ref: B. Böhme 16 Diffraction at grating – Intensity

1. Width of slits defines occurrence of orders

2. Number of slits defines fidelity

3. Period of grating defines distance of orders

1-2-3-4 1-2-3-4 5 5 Between the maxima are 5 periods – The number increases with number of periods M

Ref: B. Böhme 17 Diffraction at circular aperture

Transmission 1

Intensity at screen 0 -2 -1 0 1 Variable r @ Object Similar, but 1 about 20% more spreaded distribution non-equidistant zeros DAiry

min 1.22 sin1  NA  D 0 -2 -1 0 1 2 Variable  The Airy-distribution 2 @ Screen   2r  follows from the 2 J1 NA    2D-Fourier-transform I(r)   NA  n'sin u'  2r  of the circular aperture NA     

Ref: B. Böhme 18

Circular aperture a  DAiry

 . Slit: first destructive interference at sin( min )  1 a  . Circular aperture – no separation of coordinates sin( min ) 1,22 1 a . Finite aperture causes finite spot size = Airy-diameter DA   More general: DAmin 1,22 mim 1,22  sin(1 ) NA NA = n sin()

~ r / f 2r = a f distance = diameter of first dark rings

Ref: B. Böhme 19 Model depth of Light Propagation

. Different levels of modelling in optical propagation

. Schematical illustration (not to scale)

accuracy

rigorous waveoptic

vectorial waveoptic

scalar waveoptic (high NA)

paraxial geometrical waveoptic optic (raytrace)

paraxial optic calculation effort Ref: A. Herkommer 20 Diffraction Integral

. General diffraction integral    Follows from wave equation with     E(r')   G(r,r') Green's theorem E(r) G(r,r') E(r') dF' Green's function G not unique F  no  no

    1 ik rr ' . Choice of G with spherical waves: G(r,r')   e 4 rr' . Kirchhoff integral   i E(r')    E(r) e ikS (r ') r r '  cosq cosq dF'     i d  2 F r  r' 21 Approximation of the Diffraction Integral

   i  e ik r r ' . Kirchhoff integral E(r)  E(r')   dF   r  r' FAP

. Fresnel integral: r r'  ( x  x ')2  ( y  y ') 2  z 2 Phase quadratic approximated (x x ')22 ( y y ') (convolution expression) z   .... 22zz i ieikz  (xx')2 ( y y')2  E(x, y, z)  E(x', y',0)e z dx'dy' z 

x2 xx'''' x 2 y 2 yy y 2 . Fraunhofer integral: r r'  z       far-field approximation for large 2z z 2 z 2 z z 2 z distances or convergent waves into x'22 y ' xx ' yy' z   focal region, x << x' 2zz (Fourier integral) ii2 ieikz x''''22  y  xx  yy  E( x ', y', z ) ezz E ( x , y,0) e dx dy  z  22

Fraunhofer Point Spread Function

. Rayleigh-Sommerfeld diffraction integral,     i  eik r r ' Mathematical formulation of the Huygens-principle E (r)   E(r')   cosq dx'dy' I   r  r' d . Fraunhofer approximation in the far field r 2 N  p 1 for large Fresnel number F  z

. Optical systems:

- pupil amplitude/transmission/illumination T(xp,yp)

- wave aberration W(xp,yp) 2iW ( x p ,y p ) - complex pupil function A(xp,yp) A(xp , y p ) T (xp , y p )  e - transition from exit pupil to 2i x x' y y' image plane 2iW x ,y  p p E(x', y')  T x , y  e p p eRAP dx dy   p p  p p AP

. Point spread function (PSF): Fourier transform of the complex exit pupil function 23 Approximation of the Diffraction Integral

   i  e ik r r ' . Kirchhoff integral E(r)  E(r')   dF   r  r' FAP i ieikz  (xx')2 ( y y')2  . Fresnel integral: E(x, y, z)  E(x', y',0)e z dx'dy' - phase quadratic approximated z  - 1/z is approximated to be constant - corresponds to the paraxial approximation of a parabolic phase of the propagator only - the inital field E(x',y') can have higher order phase contributions: abberated field can be calculated . Fraunhofer integral: ii2 - far-field approximation for ieikz x''''22  y  xx  yy  large distances in case of E( x ', y', z ) ezz E ( x , y,0) e dx dy plane waves  z y - quasi far field in case of convergent fields observed near the focal plane all wavelets in phase

final plane: wave in focus front initial plane 24

Fraunhofer Point Spread Function

. Rayleigh-Sommerfeld diffraction integral,     i  eik r r ' Mathematical formulation of the Huygens-principle E (r)   E(r')   cosq dx'dy' I   r  r' d . Fraunhofer approximation in the far field r 2 N  p 1 for large Fresnel number F  z

. Optical systems: numerical aperture NA in image space

Pupil amplitude/transmission/illumination T(xp,yp)

Wave aberration W(xp,yp)

complex pupil function A(xp,yp)

Transition from exit pupil to 2i x p x' y p y' image plane 2iW x p ,y p  R E(x', y')  T x , y  e e AP dx dy   p p  p p AP

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

2iW ( x p ,y p ) A(xp , y p ) T (xp , y p )  e

12 25 Angular Spectrum

. Plane wave  x

i kx x  k y y  k z z E(x , y , z ) A ( x , y , z ) e x n/  . Wave number    z 2 z k  k 2  k 2  k 2  n  x y z 

. Spatial frequence: re-scaling of k

2    n  2 2 2  k z       x  y   

i2 x x   y y   z z E(,,) x y z A e

. Fourier transform to get the plane wave spectrum

 2i x x y  y z  z  E x,,,, y z E   e d  d  d      x y z x y z   26 Propagation of Plane Waves

. Phase of a plane wave x z

2 2  n  2 2 i zncos  i 2z   x y i  i 2zz    e  e  e  e  h x , y ;z

. The spectral component is simply multiplied by a phase factor in during propagation  z z0 z1 2iz   z Ex,,,,,;,,  y z1  E  x  y z 0 e  h  x  y  z  E  x  y z 0  the function h is the phase function . Back-transforming this into the spatial domain:

Propagation corresponds to a convolution Exyz ,,,;,,10  Hxyz  Exyz    2i xx yy  with the impulse response function Hx,y;z    h x , y ;ze d x d y 

2 . Fresnel approximation for propagation: 1 2 2  2   1 2  2  ... x y 2 x y

2  1  i z x2 y 2    2 2 2 1    i x0 y0   i xx0 yy 0  U x, y;z  e   2z  Ux ,y ;0 e z e z dx dy P iz   0 0 0 0  27 Propagation by Plane / Spherical Waves

. Expansion field in simple-to-propagate waves

. 1. Spherical waves 2. Plane waves Huygens principle spectral representation Fourier approach   ik r r '  e  2    ˆ 1 ikz z ˆ E(r')     E(r) d r E(r')  Fxy e  Fxy E(r)  r  r' x' x'

x x

eikr r eik z z

E(x) E(x) z z 28 Diffraction Ranges

Different ranges of edge or slit diffraction as a function of the Fresnel number / distance: z [mm] far field 1. Far field Fraunhofer intensity near left edge

NF = 1 Fraunhofer weak structure

small NF ≤ 1

NF = 5

2. Fresnel

Quadratic NF = 20

approximation Fresnel

of phase regime

ripple structure NF = 100

medium N > 1 F

3. Near field NF = 1000

Behind slit large N >> 1 F NF = 10000

near field slit coherent plane wave 29 Fresnel Diffraction

. Typical change of the intensity profile far zone stop geometrical focus intensity

a z

f z geometrical phase

2 . Normalized coordinates 2  a  2  a  r u     z ; v   r ;     f    f  a

2 1 i 2 . Diffraction integral 2 ia E 2  u E(u,v)   0 ei f / a u  J ( v)e 2  d 2  0  f 0 30 Geometrical vs Diffraction Ranges

. Focussed beam with various wavelengths 2 . Focal region given by Rayleigh length Ru = /NA . Geometrical cone outside focal region . Diffraction structure in focal range

A1/2  = 1 mm  = 2.5 mm  = 5 mm  = 10 mm

geometrical range

diffraction focal range 31 Sampling of the Diffraction Integral

. Fresnel integral: ikz i 2 2 ie  (xx') ( y y')  phase quadratic approximated E(x, y, z)  E(x', y',0)e z dx'dy' z  . Sampling of the initial field: curvature, W(x,y)

. Sampling of the propagator: different distances of points

r1-r2

sag x‘ W(x) . Nearly ideal: P1(x) compensation of wave sag and x distance differences: point quasi flattening max r12 spread function

min r12

light

cone P2(x‘)

wave calculation channel front final plane initial plane 32 Sampling of the Diffraction Integral

. Oscillating exponent :

Fourier transform reduces on 2- phase period 50 . Most critical sampling usually quadratic phase at boundary defines number 40 of sampling points . Steep phase gradients define the 30 sampling . High order aberrations are a problem 20

wrapped 10 phase 2

0 x -6 -4 -2 0 2 4 smallest sampling intervall 33 Fresnel Edge Diffraction

. Diffraction at an edge in Fresnel I(t) approximation 1.5

. Intensity distribution, Fresnel integrals C(x) and S(x) 1 2 2 1  1   1   I(t)     C(t)   S(t)  2  2   2   0.5 scaled argument k 2 t   x   x  2N 0 t z    z F -4 -2 0 2 4 6 . Intensity: - at the geometrical shadow edge: 0.25 - shadow region: smooth profile - bright region: oscillations

27 34 Fresnel Diffraction

. Phase behaviour at edge diffraction . Oscillation of phase in the range of the intensity ripples 35 Fresnel Number

. Fresnel number : critical number for separation of approximation ranges

. Setup with two stops in distance L

final a1a2 start plane sagittal plane N F  stop height p L stop . Fraunhofer diffraction, farfield,

Dominant effect of diffraction: a2 a1 NF < 1 L

. Fresnel diffraction with P considerable influence of spherical wave from axis point P diffraction: NF = 1

. Geometrical-optical range witj neglectable diffraction, near field :

NF >> 1 36 Diffraction at the System Aperture

. Self luminous points: emission of spherical waves . Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave results in a finite angle light cone . In the image space: uncomplete constructive interference of partial waves, the image point is spreaded . The optical systems works as a low pass filter

spherical truncated wave spherical wave point spread function object point

 x = 1.22  / NA

image object plane plane 37 Hybrid model

y yp optical y'p y' system stop image image point

rays

z

real wave front object ideal point reference sphere

Object plane Entrance pupil Exit plane Image plane Op EnP ExP Ip 38 Diffraction inside a system

. Usually neglected: - diffraction inside system with impact on phase and amplitude in exit pupil - mostly extremly good approximation . Special cases: - beam cleanup telescope - very large propagation distances in laser systems

exit stop diffraction pupil

a spreading and intensity redistribution

diffraction angle q =  / a 39

PSF by Huygens Principle

. Huygens wavelets correspond to vectorial field components: - represented by a small arrow - the phase is represented by the direction - the amplitude is represented by the length . Zeros in the diffraction pattern: destructive interference . Ideal point spread function:

point spread function

zero intensity closed loop

single wavelets

sum central peak maximum constructive interference

wave front side lobe peak 1 ½ round trips pupil stop 40

PSF by Huygens Principle

r . Apodization: p point spread variable lengths function of arrows homogeneous pupil: same length of all arrows

I(xp)

apodization: decreasing length of arrows

wave front pupil stop

. Aberrations: point spread variable orientation function of arrows real wave real wavefront front with aberrations central peak reduced

ideal spherical wavefront central peak maximum

ideal wave front pupil stop 41 Perfect Point Spread Function

Circular homogeneous illuminated aperture: optical . Transverse intensity: r 2 axis Airy distribution 2J v  I 0,v 1 I      0 Dimension: DAiry  v  normalized lateral 1.22 coordinate: D  z Airy NA v = 2  x /  NA

2 sinu / 4 . Axial intensity: Iu,0  I  u / 4  0 sinc-function   Airy Dimension: Rayleigh unit R E n lateral normalized axial coordinate RE  2 u = 2  z n /  NA2 NA image plane

1,0

vertical 0,8 lateral Rayleigh axial 0,6

0,4 aperture cone 0,2

0,0 u / v -25 -20 -15 -10 -5 0 5 10 15 20 25 42 Perfect Lateral Point Spread Function: Airy

log I(r) Airy distribution: 10 0

10 -1 . Gray scale picture . Zeros non-equidistant 10 -2

. Logarithmic scale -3 10 . Encircled energy 10 -4

10 -5

10 -6 r 0 5 10 15 20 25 30

Ecirc(r) 1

0.9 3. ring 1.48% 0.8 2. ring 2.79% 0.7 1. ring 7.26% 0.6 peak 83.8% 0.5 DAiry 0.4

0.3

0.2

0.1

0 r / r 0 2 3 4 5 Airy 1 1.831 2.655 3.477 43 Axial and Lateral Ideal Point Spread Function

. Comparison of both cross sections

Ref: R. Hambach 44 Annular Ring Pupil

. Generation of Bessel beams

r 푟푝

z

Ref: R. Hambach 45 Spherical Aberration

. Axial asymmetrical distribution off axis . Peak moves

Ref: R. Hambach 46 Gaussian Illumination

. Known profile of gaussian beams

Ref: R. Hambach 47 Defocussed Perfect Psf

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

z = -2RE z = -1RE focus z = +1RE z = +2RE

normalized intensity

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

constant energy 48

Perfect Axial Point Spread Function

. Axial distribution of intensity 2 2 Corresponds to defocus  sinz  sin u / 4  I(z)  I0    Io    z   u / 4  . Normalized axial coordinate I(z) 1  NA2 u z   z  0.9 2 4 0.8

. Scale for depth of focus : 0.7 Rayleigh length 0.6

0.5  n' 0.4 RE  2  2 n'sin u' NA 0.3

. Zero crossing points: 0.2

equidistant and symmetric, 0.1 Distance zeros around image plane 4R E0 -4 -3 -2 -1 0 1 2 3 4 z/

RE z = 2RE

4RE 49 Deblurring of PSF and Strehl Ratio

pv-value . reference sphere corresponds of wave aberration to perfect imaging image . RMS deviation as an integral plane wave measure of performance aberration 1 W (x, y)  W (x, y) dxdy 0 FExP phase front I( x ) (real )  Strehl ratio I PSF 0,0 SR  reference 1 (ideal ) sphere I 0,0 exit PSF aperture 2  A(x, y)e2iW ( x,y)dxdy SR peak reduced 2 Strehl ratio A(x, y)dxdy  ideal , without aberrations  Approximations . Marechal SR 1 4 2W 2 M rms real with useful SR >0.5 2 2 2 distribution aberrations SRB 1 2 Wrms  broadened . Biquadratic 2 2 4 Wrms r . Exponential SRE e 50 Zernike Polynomials: Measure of Wave Aberrations

. describe deviation from reference sphere with m = + 8 orthogonal basis functions cos + 7 . circular shape

 Zernike Polynomials: conventions (amount/sign) + 5

due to Gram-Smith-Orthogonalization + 4

n + 3 W (,) c Z m (,)   nm n + 2 n mn  sin (m) for m  0 Z 2,7,14 + 1 m m  Z 4,9,16 0 Zn (,) ~ Rn ()  cos (m) for m  0  Z 3,8,15 - 1 1 for m  0 - 2

. Peak-Valey (PV)-values at pupil rim = +1  Fringe - 3

. Constant RMS-value (Orthonormal)  Standard - 4

- 5 k n 2 2 - 6 SRM  1 4   cnm n1 mn - 7 sin Circular obscuration  Zernike-Tatian polynomials - 8 Rectangular pupil  Legendre polynomials n = 0 1 2 3 4 5 6 7 8 51 Zernike Fringe vs Zernike Standard Polynomials

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

Fringe coefficients: . Z 4,9,16 = n² spherical . azimutal order f grows if number increases Standard coefficients - different term numbers

. ZN = 0.05  RMS = 0.05

 SR ~ 1 – 40 ZN² = 0.9 Elimination of tilt No Elimination of defocus @ Zemax 49 Spherical Aberration

. Single positive lens  shorter intersection length for marginal rays  “undercorrected”

c9 = 0 . on axis, no field dependence, circular symmetry . Increase of spherical aberration  growing axial asymmetry around the nominal image plane . paraxial focus  perfect symmetry c9 = 0.3 . best image plane  circle of least RMS, best contrast at special frequency, Z4 = 0,...

 . Example: i c9 = 0.7 Focus a collimated beam with

Paraxial plano-convex lens n = 1.5 focus  worse (red) vs. best (green)

orientation c9 = 1

 spherical aberration i2  i1 differs by a factor of 4: 3 i3 i 2  i  i3 sin i i  2sin i    i  6 2 6  2  24 52 50 Coma: Spot Construction

. Term B2 (2+cos2f) 2y angle 60° . asymmetric ray path for non-axial object point CS ~ CT / 3 circle . construction of spot circles from constant pupil radius ~ ² Tangential coma CT zones: circle radius ~ radius at pupil² Saggital 360°@ pupil  2x360°at image coma Cs tangential . comet shape spot marginal rays central . ray @ interval [1 3]  aspect ratio 2:3 ray

. sagittal coma ~ 1/3 of tangential coma saggital marginal rays

. Coma PSF correspods to spot . 55% of energy in the triangle between tip of spot and saggital coma centroid  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

c7 = 0.3 c7 = 0.5 c7 = 1

53 51 Astigmatism & Petzval

. Term (3B3+B4) y² cosf par S . For a single positive lens: B  chief ray passes surface under oblique angle Tangential T ray fan  projection of surface curvatures  different powers in tangential and sagittal . tangential (blue) and sagittal (red) focal lines Sagittal . Sequence: tangential - circle of least confusion ray fan with smallest spots (best) – sagittal – paraxial Tangential Circle of least focus . Imaging of a circular grid in different planes confusion Sagittal focus tangential sagittal focus best focus focus

. 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 54 55 Psf with Aberrations

. Psf for some low oder Zernike coefficients

. The coefficients are changed between cj = 0...0.7  . The peak intensities are renormalized

coma 5. order astigmatism 5. order spherical 5. order trefoil c = 0.0 coma c = 0.1 c = 0.2 astigmatism c = 0.3 c = 0.4 spherical c = 0.5 defocus c = 0.7