Imaging and Aberration Theory

Lecture 13: Point spread function 2019-01-25 Herbert Gross

Winter term 2018 www.iap.uni-jena.de 2 Schedule - Imaging and aberration theory 2018

1 19.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and 2 26.10. Hamiltonian coordinates mechanics, Hamiltonian coordinates Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical 3 02.11. Eikonal approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 4 09.11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 5 16.11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 6 23.11. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 30.11. 7 Distortion and coma position, various topics, correction options 8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 9 14.12. Chromatical aberrations spherochromatism, secondary spoectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 10 21.12. isoplanatism aberrations, Herschel condition, relation to Fourier optics 11 11.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations special expansion for circular symmetry, problems, calculation, optimal balancing, 12 18.01. Zernike polynomials influence of normalization, measurement 13 25.01. Point spread function ideal psf, psf with aberrations, Strehl ratio 14 01.02. Transfer function transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic 15 08.02. Additional topics and induced aberrations, revertability 3

Contents

1. Introduction 2. Ideal PSF 3. PSF with aberrations 4. High-NA PSF 5. Two-point resolution 4 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 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 Fraunhofer Point Spread Function

. Rayleigh-Sommerfeld diffraction integral,     i  eik r r ' Mathematical formulation of the Huygens-principle E (r)   E(r')   cos 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 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 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 Perfect Point Spread Function Circular homogeneous illuminated aperture: intensity distribution . transversal: Airy 1.22 scale: DAiry  NA 1,0

. axial: sinc n vertical RE  scale NA2 0,8 lateral . Resolution transversal better than axial: x < z 0,6

r optical axis intensity 0,4

0,2 z

0,0 -25 -20 -15 -10 -5 0 5 10 15 20 25 u / v lateral 2 2 image plane 2J1v  sinu / 4 I0,v   I0 Iu,0  I0  v   u / 4  Scaled coordinates according to Wolf : axial : u = 2  z n /  NA2 aperture cone transversal : v = 2  x /  NA

axial Ref: M. Kempe Perfect Point Spread Function

Circular homogeneous illuminated aperture: optical . Transverse intensity: r 2 axis Airy distribution 2J v  I v 1 I 0,     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 11

Ideal Psf

I(r,z) focal point spread spot

axial sinc2 z

optical axis

aperture cone image plane r lateral Airy 12 Abbe Resolution and Assumptions

. 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

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

13 Perfect Lateral Point Spread Function: Airy

. Airy function : I(r) Perfect point spread function for 1 several assumptions 0.9 0.8 . Distribution of intensity: 0.7 0.6 2   2r  0.5 2 J1 NA 0.4    I(r)   0.3  2r  NA 0.2      0.1 0 r . Normalized transverse coordinate 2 4 6 8 10 12 14 16 18 20 DAiry / 2 2ar akr x  krsinu' ak sin' R R . Airy diameter: distance between the two zero points, 1.21976 diameter of first dark ring D  Airy n'sinu' 14 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 15 Perfect Axial Point Spread Function

. Axial distribution of intensity 2 2 Corresponds to defocus  sinz  sinu / 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 16 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 17 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 18 Caustic with Spherical aberration

. Growing spherical aberration shows an asymmetric behavior around the nominal image plane for defocussing

c9 = 0 c9 = 0.3 c9 = 0.7 c9 = 1 Psf for Coma Aberration

. PSF with coma . The 1st diffraction ring is influenced very sensitive

W31 = 0.03  W31 = 0.06  W31 = 0.09  W31 = 0.15  Psf with Coma

. Separation of the peak and the centroid position in a point spread function with coma . From the energetic point of view coma induces distortion in the image

c7 = 0.3 c7 = 0.5 c7 = 1

centroid Variation of Performance with Field Position

. PSF as a function x = 0 x = 20% x = 40% x = 60 % x = 80 % x = 100 % of the field height

. Interpolation is critical

. Orientation of coma in the field y

. Small field area with approximately shift- invariant PSF: isoplanatic patch

x 22 Comparison Geometrical Spot – Wave-Optical Psf

. Large aberrations: Waveoptical calculation shows bad conditioning . Wave aberrations small: diffraction limited, geometrical spot too small and spot wrong diameter . Approximation for the intermediate range:

2 2 DSpot DAiry DGeo

exact wave-optic

DAiry geometric-optic approximated aberrations

diffraction limited, Fourier transform failure of the ill conditioned geometrical model 23 Focal Diameter Determination

1 2 3 4 5 6 Focal diameter: Pupil shape Gauss Super Gauss Gauss Circle Linear / radius w gauss 2a=3w a=w homoge- homog. . Basic scaling factor definiton of no w,m=6 with with neous slit width focus trunc.. no trunc. trunc. trunc. D=2a 2a 

  f with truncation 1. Zero point of # # # 1.426 1.220 1.00  intensity ( Airy )   a at radius a     f no truncation NA  I=0.5 ( FWHM ) 0.375 0.513 0.644 0.564 0.519 0.443  w beamradius w  I=0.1353 (1/e2 ) 0.637 0.831 1.059 0.914 0.822 0.697 =2/ . Additional factor b I=0.01 0.966 1.122 1.491 1.238 1.092 0.908  ( Peak ) D foc  b  NA I=0.001 1.183 2.109 1.695 2.925 1.174 0.969 . Factor differs in size by ( Peak ) one magnitude E=0.86466 0.637 0.818 1.008 0.890 1.378 0.658 =2/

E=0.95 0.779 1.040 1.201 1.104 3.915 1.989

24 Focus Spot Size of a Lens

. Changing the NA/aperture D of a focussing lens: - small values of D: diffraction dominates, Airy formula - large D: geometrical aberrations dominate . Total aberrations: superposition of both effects

Dspot Log Dspot

total diffraction Airy geometrical aberration

total geometrical diffraction aberration Airy

NA / D NA / D Focussing by a Lens: Diffraction and Aberration

Log D . Small aperture: foc f=1000 , 500 , 200 , 100 , 50 , 20 , 10 mm Diffraction limited 1 Spot size corresponds to Airy diameter = 10 m Spot size depends on wavelength

. Large aperture: 0 Diffraction neglectible = 1 m Aberration limited Geometrical effects not  wavelength dependent 550 nm But: small influence of -1 dispersion

-2 -3 -2 -1 Log sinu 0 26 Strehl Ratio

. Important citerion for diffraction limited systems: Strehl ratio (Strehl definition) Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity 2 2iW (x, y) I (real) 0,0 A(x, y)e dxdy D  PSF D   S I (ideal) 0,0 S 2 PSF  A(x, y)dxdy I( x )

. DS takes values between 0...1 1 DS = 1 is perfect

. Critical in use: the complete peak reduced information is reduced to only one Strehl ratio number ideal , without aberrations . The criterion is useful for 'good'

systems with values Ds > 0.5 real with distribution aberrations broadened

r 27

Approximations for the Strehl Ratio

DS . Approximation of 2 defocus Marechal: 2 Wrms  1 Ds 1 4   ( useful for D > 0.5 )  0.9 s   0.8 but negative values possible 0.7 2 2  W   0.6 Bi-quadratic approximation 2  rms  0.5 Ds 1 2    exponential      0.4   0.3 biquadratic 2 0.2  W  exac t 4 2  rms  0.1    0 Marechal Exponential approach   c Ds e 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20

. Computation of the Marechal 2 N 2 N n 2  2   cn0 1 cnm  approximation with the Ds 1        coefficients of Zernike     n1 n 1 2 n1 m0 n 1 28

Strehl Ratio Criterion

. In the case of defocus, the Rayleigh and the Marechal criterion delivers a Strehl ratio of 8 D   0.8106  0.8 S  2

. The criterion DS > 80 % therefore also corresponds to a diffraction limit . This value is generalized for all aberration types

Marechal aberration type coefficient approximated exact Strehl Strehl 8 defocus Seidel a  0.25 0.7944  0.8106 20  2

defocus Zernike c20  0.125 0.7944 0.8106 spherical Seidel a40  0.25 0.7807 0.8003 aberration spherical Zernike c40  0.167 0.7807 0.8003 aberration

astigmatism Seidel a22  0.25 0.8458 0.8572

astigmatism Zernike c22  0.125 0.8972 0.9021

coma Seidel a31  0.125 0.9229 0.9260

coma Zernike c31  0.125 0.9229 0.9260

29 Quality Criteria for Point Spread Function

. Criteria for measuring the degradation of the point spread function: 1. Strehl ratio 2. width/threshold diameter 3. second moment of intensity distribution 4. area equivalent width 5. correlation with perfect PSF 6.power in the bucket

a) Strehl ratio b) Standard deviation c) Light in the bucket d) Equivalent width

STDEV SR / Ds EW LIB

e) Second moment f) Threshold width g) Correlation width h) Width enclosed area

P=50% SM FWHM Ref WEA CW High-NA Focusing

. Transfer from entrance to exit pupil in high-NA:

1. Geometrical effect due to projection y (photometry): apodization y' dy dy/cosu 1 A A  0 4 2 2 with 1 s r u NA R s sinu  n y y y' E e . Tilt of field vector components Ey ey sE s x x' r xE r  2 2  E A0 1 1 s r A     u 2 2 2 E  1 1 s r cos2  

entrance exit image pupil pupil plane R High-NA Focusing

. Total apodization 2 2 2 2 corresponds to astigmatism 1 1 s r  1 1 s r cos2 Axlin (r, )  A0 (r, )  2  4 1 s2r2 . Example calculations

NA = 0.5 NA = 0.8 NA = 0.9 NA = 0.97

A(r) 2 2 2 2

1.5 1.5 1.5 1.5 y

1 1 1 1 x 0.5 0.5 0.5 0.5

0 0 0 0 r -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Vectorial Diffraction for high-NA

. Vectorial representation of the diffraction integral according to Richards/Wolf

 E  iu   i  I  I  cos2 x  0 2     2   4sin 0 / 2 E(r', z')   Ey   E0  e   i  I2 sin 2       Ez    2 I1  cos 

o . Auxiliary integrals I (r', z)  cos sin  (1 cos) J kr'sin  eikz'cos d 0  0   0

o I (r', z')  cos sin 2   J kr'sin  eikz'cos d 1  1  0

o I (r', z')  cos sin  (1 cos) J kr'sin  eikz'cos d 2  2   0

. General: axial and cross components of polarization Vectorial Diffraction at high NA

Linear Polarization

Pupil

 High NA and Vectorial Diffraction

. Relative size of vectorial effects as a function of the numerical aperture

. Characteristic size of errors: I / Io 0 10 error axial lateral

-1 0.01 0.52 0.98 10 axial 0.001 0.18 0.68 -2 10 lateral

-3 10

-4 10

-5 10

-6 10 NA 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 35 Low Fresnel Number Focussing

. Small Fresnel number NF: geometrical focal point (center point of spherical wave) not identical with best focus (peak of intensity)

. Optimal intensity is located towards optical system (pupil)

. Focal caustic asymmetrical around the peak location

wavefront physical geometrical focal focal point a point z

focal shift f focal length f Low NA and Focal Shift

. Systems with small Fresnel number . Same size of: 1. divergence angle and numerical aperture angle 2. focal length and depth of focus 3. influence of geometrical beam shaping and diffraction 1 . Effect of photometric distance law inside depth of focus I(z)  z 2 . Focal shift depends on Fresnel number, apodization and coherence

. For definition of best focus by peak intensity on axis: 2  u  2 sin  I(0, z foc )  u     0 I(z) I 1   4 0  u   z  2 N F     4  . Relative focal shift for circular homogeneous illuminated pupil:  f 1   f  N 2 1 F 12 37 Low Fresnel Number Focussing

I(z) . System with small Fresnel number: Axial intensity distribution I(z) is 1 asymmetric 0.9 u 2 2   N = 2    sin  0.8 F u 4 NF = 3.5 I(z)  I0 1    N = 5   u 0.7 F  2  N F    N = 7.5   F 0.6  4  NF = 10 NF = 20 0.5 N = 100 . Explanation of this fact: F The photometric distance law 0.4 shows effects inside the depth of focus 0.3

0.2 . Example microscopic 100x0.9 system: a = 1mm , z = 100 mm: 0.1 N = 18 0 z / f - 1 F -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Low NA and Focal Shift

. Focal shift as a function of 1. Fresnel number 2. Apodization in linear and logarithmic representation Point Spread Function with Apodization

. Apodization: Non-uniform illumination of the pupil amplitude

. Modified point spread function:

Weighting of the elementary wavelets by amplitude distribution A(xp,yp), Redistribution of interference

. System with wave aberrations: Complicated weighting of field superposition For edge decrease of apodization: residual aberrations at the edge have decreased weighting and have reduced perturbation

. Definition of numerical aperture angle no longer exact possible

. For continuous decreased intensity towards the edge: No zeros of the PSF intensity (example: gaussian profile)

. Modified definition of Strehl ratio with weighting function necessary Point Spread Function with Apodization

. Apodisation of the pupil: I(w) 1 1. Homogeneous - Airy Airy Bessel 2. Gaussian - Gaussian 0.8 Gauss 3. Ring illumination  Bessel 0.6 . Psf in focus: FWHM different convergence to zero for 0.4 larger radii 0.2 . Encircled energy: 0 w same behavior -2 -1 0 1 2 3 E(w) . Complicated: E95% Definition of compactness of the 1 central peak: 0.8 1. FWHM: Airy more compact as Gauss 0.6 Bessel more compact as Airy 2. Energy 95%: 0.4 Gauss more compact as Airy Airy Bessel 0.2 Bessel extremly worse Gauss

0 w 1 2 3 4 Psf of an Annular Pupil

. Farfield of a ring pupil: I(r)

outer radius aa 1 innen radius ai 0.9 ai parameter   1 0.8  = 0.01 aa 0.7  = 0.25 . Ring structure increases with  0.6  = 0.35 0.5 . Depth of focus increases  = 0.50 0.4  = 0.70 0.3 2 z  0.2 nsin 2 u 1  2  0.1

0 r . Application: -15 -10 -5 0 5 10 15 Telescope with central obscuration

. Intensity at focus 2 1 2J1(x) 2 2J1( x) I(x) 2     1 2   x  x  PSF for ring-Shaped Pupil

. Encircled energy curva: - steps due to side lobes - strong spreading, large energy content in the rings

E(r) 1

0.9

0.8

0.7  = 0.01 0.6  = 0.25 0.5  = 0.35 0.4  = 0.50 0.3  = 0.70 0.2

0.1

0 r 0 5 10 15 Psf of a Ring Pupil

. Ring shaped pupil illumination . Depth of focus enlarged . Central peak shrinks lateral 44 Obscuration / PSF

• Central obscuration

Circular aperture Circular aperture Circular aperture Circular aperture 28% central obscuration 28% central obscuration no obscuration 28% central obscuration three spider vanes four spider vanes • Spider obscuration

Ref: K. Uhlendorf 45 PSF Calculation

. System with large grid distortion . Perfect phase on axis due to 4 confocal paraboloids

intensity at exit P2

F1 = F2

P1

P4

F3 = F4 spot at exit

P3 46 PSF Calculation

. System with large grid distortion . Calculation of the PSF with direct integration and with FFT-based algorithm . Large deviation from Airy pattern . Differences in algorithms due to grid distortion, but Strehl nearly identical . Zemax-error due to y-orientation ? 47 Point Resolution According to Abbe

. Transverse resolution of an image: - Detection of object details / fine structures  - basic formula of Abbe x  k  nsin . Fundamental dependence of the resolution from: 1. wavelength 2. numerical aperture angle 3. refractive index 4. prefactor, depends on geometry, coherence, polarization, illumination,...

. Basic possibilities to increase resolution: 1. shorter wavelength (DUV lithography) 2. higher aperture angle (expensive, 75° in microscopy) 3. higher index (immersion) 4. special polarization, optimal partial coherence,...

. Assumptions for the validity of the formula: 1. no evanescent waves (no near field effects) 2. no non-linear effects (2-photon) 48

Incoherent 2-Point Resolution : Rayleigh Criterion

. Rayleigh criterion for 2-point resolution 1 0.61  Maximum of psf coincides with zeros of x  D  neighbouring psf 2 Airy nsinu

I(x) . Contrast: V = 0.15 sum of PSF 1 . Decrease of intensity between peaks 0.8 I = 0.735 I0

0.6

0.4

PSF1 PSF2 0.2

0 x / rairy -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 49

Incoherent 2-Point-Resolution: Sparrow Criterion

. Criterion of Sparrow:  d 2 I(x)  vanishing derivative in the center between two   0  2  point intensity distribution,  d x x0 corresponds to vanishing contrast

. Modified formula I(x) 0.474  xSparrow   0.385 DAiry nsin u 1  0.770 xRayleigh

0.8 . Usually needs a priory information

. Applicable also for non-Airy 0.6 distributions 0.4 . Used in astronomy 0.2

0 x / rairy -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 50

Incoherent 2-point Resolution Criterions

. Visual resolution limit: Good contrast visibility V = 26 % : 0.83  x   0.680 D nsinu Airy

. Total resolution: Coincidence of neighbouring zero points 1.22  x  D  of the Airy distributions: V = 1 Airy nsinu Extremly conservative criterion

0.51  . Contrast limit: V = 0 : x   0.418 D Intensity I = 1 between peaks nsinu Airy 51 2-Point Resolution

. Distance of two neighboring object points . Distance x scales with  / sinu . Different resolution criteria for visibility / contrast V

x = 1.22/ sinu total V = 1 x = 0.68/ sinu visual V = 0.26 x = 0.61/ sinu Rayleigh V = 0.15 x = 0.474/ sinu Sparrow V = 0 52

2-Point Resolution

. Intensity distributions below 10 % for 2 points with different x (scaled on Airy)

x = 2.0 x = 1.22 x = 1.0 x = 0.83

x = 0.61 x = 0.474 x = 0.388 x = 0.25 53 Incoherent Resolution: Dependence on NA

. Microscopical resolution as a function of the numerical aperture

NA = 0.2 NA = 0.3 NA = 0.45 NA = 0.9 Depth of Focus: Diffraction Consideration

. Normalized axial intensity for uniform pupil amplitude 2 sin u  I(u)  I0     u  r . Decrease of intensity onto 80%: depth of focus 1  1 focal z 0.493   R plane 2 diff nsin2 u 2 u

. Scaling measure: Rayleigh length z

- geometrical optical definition beam depth of focus: 1RE caustic I(z)  n' 1 Ru  2  2 n'sin u' NA 0.8

- Gaussian beams: similar formula intensity at r = 0  R  u 2 z n' o -Ru/2 0 +Ru/2 Depth of Focus

. Schematic drawing of the principal ray path in case of extended depth of focus . Where is the energy going ? . What are the constraints and limitations ?

conventional ray path

beam with extended depth of focus

z0

z Depth of Focus

. Depth of focus depends on numerical aperture

1. Large aperture: 2. Small aperture: small depth of focus large depth of focus

Ref: O. Bimber EDF with Complex Toraldo Mask

. I(r,z)

. I(x) I(z) I(z) depth for 80%: 13 RE 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 -3 -2 -1 0 1 2 3 -20 -15 -10 -5 0 5 10 15 20