Extended Nijboer-Zernike Diffraction Theory

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Extended Nijboer-Zernike diffraction theory

Lecturer: Joseph Braat / TU Delft

13-01-2011 9.30 – 11.00 hrs

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 1

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 2

1 What is a diffraction integral? Mathematical operation to obtain light amplitude and intensity in a general point P in image region.

Physical interpretation: take the Huygens secondary waves in the exit pupil and sum their amplitudes in each image point P, with correct amplitude and phase.

Improvement with respect to OPTICS ray optics. Ray optics yields an infinitely high instensity in P if the beam is free of aberration.

Costly operation. Numerical approach: Fast Fourier Transform (multiplex advantage, periodicity and aliasing disadvantage)

Slightest change of exit pupil field, new FFT is needed. SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 3

1) History of diffraction integral  Debye diffraction integral

1672 Wave propagation via secondary Huygens ' wavelets Light impulses; Huygens was not aware of 'modern' diffraction Qualitative model; no explanation for 'forward-only' propagation 1823 Fresnel : harmonic theory of light propagation (wavelength appears!) Strength and phase of secondary waves explained Quantitative description of light amplitude / intensity Very accurate description of optics at interfaces Very satisfactory description of propagation in anisotropic media 1835 Intensity in focus of perfect optical system (George Airy, royal astronomer) 1864 Maxwell 's electromagnetic theory 1885 Lommel , through-focus intensity of perfect optical system 1883 Kirchhoff 's diffraction theory/integral for arbitrary light distribution 1897 Rayleigh diffraction integrals (generalisation of Kirchhoff) 1909 Debye diffraction integral (simplification of one of Rayleigh's formulae)

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 4

2 1) Rayleigh diffraction integrals

Rayleigh diffraction integrals Rayleigh - I 1    expikR EE(,xy ,;zz ') ( x ',',';')y zzP dxd 'y ' f   2π   zRP Rayleigh - II 1    expikR EE(,xy ,;zz ') ( x ',',';')y zzP dxd 'y ' d   2π   zRP y y A

E(x’,y’,z’) ~ P E(z’;k xy ,k ) P y’ r y’ r x x x’ x’

z=z’ z z=z’ z Q

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 5

1) Debye diffraction integral for a focused field

Focused field in the aperture A:  exp ikRQF EEA (',',')xyz 0 (',') xy RQF

Angular spectrum of EA (xyz ', ', ') in aperture plane A :

1 exp ikRQF EE (k , k ; z ') ( x ', y ') exp  i k x '  k y ' dx ' dy ' xy  0 xy 2π A RQF Debye approximation for E (kkz , ; ') y xy A  E(,;')kkzxy  P r y’  2π E ()expQikxky x ik 0   xf yf F  z x’ Ω    kzzf ( z ') Q z=z’ z   (inside cone   0 (outside cone  SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 6

3 1) Debye diffraction integral for a focused field

With the Debye-approximated angular spectrum the Rayleigh integral becomes

Angular spectrum of EA (xyz ', ', ') in aperture plane A :

i E0 xkkzzykkzzf(/)( xzf '),(/)( f yzf ') E(,xyzz ,; ')   2π  kz  x expikxfyfzf ( x x ) k ( y  y ) k ( z  z ) dkdk xy

Angular spectrum has been truncated to the domain of frequencies (plane wave spectrum) subtended by the aperture A. Applicable for large values of the Fresnel number N of the focusing beam:

2 RRQF  QF a NNmax min  ; circular aperture:  . λλ/2  /2 Rλ Practical limit for application of Debye approximation: N  1000

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 7

1) Rayleigh and Debye diffraction integral for field on exit pupil sphere

Coordinate transformation from plane wave spectrum (kkxy , ) to field on exit ' pupil sphere with coordinates (xyzQQQ''' ', ', ).

Rayleigh - I izz(') EE(,,;')xyzz (',')exp(',') x y iΦ xy f  2 0  λ  RRQP'' QF

expik()'' RQP'' R QF dx dy .

Debye approximation i (')zz EE(,,;')xyzz f (',')exp(',') x y iΦ xy  3 0  λ  RQF'

expidxdykr (QP'' r QF ) ' ' .

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 8

4 Transition from Rayleigh to Debye regime

Left column: NA=0.25 Right column: NA=0.50

Upper row: pupil diameter=10

Central row: pupil diameter=100

Lower row: pupil diameter=105 

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 10

5 Properties of the Zernike polynomials

• Complete and orthogonal representation of the wavefront aberration in the exit pupil of an optical system using circle polynomials • Limited to a perfectly spherical support • Accommodate a hard limit (pupil rim) • First proposed on the unit circle by Szegő in 1919 • Introduced in optics by Zernike (1935) and Nijboer (thesis, 1942)

  m m m W(,)  Rn ()an cos(m) bn sin(m) nm00   m m  cn Rn ()exp(im) nm0 

Zernike polynomials

The coordinates on the pupil sphere are given by (x,y,z)  ( , ,z ) with (  , ) the polar coordinates of a general point Q' on the pupil sphere. The axial coordinate follows from the equation of the exit pupil sphere.

Advantages of Zernike polynomial representation of wavefront function  ( , ) : - automatic balancing of aberrations of various order - best focus and lateral position are automatically found - simple expression for the rms wavefront aberration  ,

mm22 2   22 mns,, nc 2 for m  0    with m   2(n  1) 1 for m  0  12

6 Zernike polynomials

Extension to fitting amplitude and phase via complex m Zernike coefficients n ,  mm|| Ai( , ) exp  (  , )  nn Rimexp(  ) nm0

Forward operation is stable. Use inner product method or, m more efficiently, a least squares fitting method to find n . Backward operation ( ) is not unique due to phase excursion larger than 2 . Phase unwrapping is needed, based on continuity property of the phase function (,).

Reference: http://wyant.optics.arizona.edu/zernikes/zernikes.htm

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 14

7 Solving the diffraction integral

Aberration free • Airy (1835) : best focus • Lommel (1885) : out of focus • Zernike (1934) : Circle Polynomials for aberration representation and aberration balancing • Nijboer-Zernike (1942) : Close to best focus, small aberrations • ENZ (2002) : Focal region, general aberrations (old theory is a special case)

The analytic solution of the through-focus diffraction integral

Scalar diffraction integral General pupil-function

Scalar-field

V- functions

8 Nijboer-Zernike theory of aberrations (1942) Airy pattern* (1835)

J1(r) n1 m Jn1(r) U(r)  2  2i n cosm r n,m r

 Best focus, small aberrations  Defocus included for a few low order terms only

* Airy, G. B., "On the diffraction of an object-glass with circular aperture," Transactions of the Cambridge Philosophical Society, Vol. 5, 1835, p. 283-291. 17

Usefulness of the Zernike-polynomials and the Nijboer- Zernike analysis

• Optical design: The polynomials solve the aberration balancing problem in optical design (Maréchal, Hopkins, Welford: 1940’s)  Strehl intensity ! • Diffraction theory: Calculation of diffraction patterns for small aberration values and modest defocusing (Nijboer, 1942, Nienhuis, 1943) •Interferometry: Stable reconstruction of measured data; ‘fringe index’ of polynomials introduced by Wyant c.s. in 1980 (37 ‘first’ Zernike polynomials) • Analytic extension of (aberrated) diffraction integrals to focal volume and arbitrary axial position using Zernike polynomials (A.J.E.M. Janssen, 2002)

9 Description of the defocus term

Cosine rule in  AA'P : 2 W() 2 2 RRRRRRR''2(')cos       n1 with  =sin and zAA ' (positive in the Figure). R Wnznz() 1cos     1cos.   R' 11 For small values of sin , sin246 sin sin 5sin8  Wnz(sin ) 1     . 2816128

With normalised radial coordinate nm sin / sin and dividing by the vacuum wavelength, nzsin2 465  8 W ()1 m 22 nn sin  sin 4  n sin 6  . 0 nnmmm 228640  19

Description of the defocus term

nzsin2 4 W ()1 m 22 n sin  . 0 nnm 220 

For small values of sinm , 2sinnz 2  2,Wf1 m 22   0 nn 20 with the phase defocusing factor nzsin2  ff1 m . | |  / 2, one focal depth ! 0

For larger values of sinm , 21cosnz  ff1 m . | |  / 2, one focal depth ! 0

10 Extension for defocus of the diffraction integral (scalar diffraction, polar pupil coordinates)

2  1 1  A(,)expi(,)  U(r,; f )   exp(if 2 )  d   0   0 expi2r cos( ) d 

Introduction of Zernike polynomials (mapping of amplitude and phase):

  m |m| A(,)expi(,)  n Rn exp(im) nm0 

Diffraction integral (scalar)

In-focus Nijboer-Zernike result (1942) :

Jv1() nm1 Jvn1() Ur(, ;0) 2 in cos m ,  2 r . vvnm, m Extended (through-focus) result for general coefficients  n :

0 nmm Ur(, ; f ) 2 V0 (, vf ) inn V (, vf )exp im nm,

11 Diffraction integral (scalar)

In-focus result (Ph.D. thesis Nijboer, 1942):

J1() nm1 Jn1() Ur(, ;0 ) 2 in cos m nm, Pupil function: mm giiiRm( , ) exp( ) 1 1 nn ( ) cos nm Basic result Zernike diffraction theory (1935):

1 nm m Jr() R ()()Jr d 12 n1 , with Airy disc 0 n m r as limiting case for the aberration-free focusing beam!

Semi-analytic solution of the diffraction integral (scalar) (through focus)

mmm Ur(, ;f ) 2 inn V (, rf )exp( im ) nm, 1 Vrmm( ;ff )  exp( i2 ) R ( ) J (2 rd ) nnm 0

Analytic solution for the V – functions (A.J.E.M. Janssen, 2002):

 p ml1 Jvml2 j() Vrnlj( ;fff ) exp( i ) ( 2 i ) v l ; lj10lv

p mjl111 jl  l  qlj  vmljlj (1)(  2) ; llpjl11 pnm()/2,2,()/2. v  rqnm  24

12 Some calculated patterns

Re{V4,0}, Im{V4,0} and High-frequent pupil ‘noise’; 2 |V4,0| intensity pattern image plane

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 26

13 Extension V-functions to high-NA case

1) Correct treatment of defocus- if 22 exp 1 s0  parameter( s 0  sin  m ), u0 2) Inclusion of ‘radiometric’ effect 1 (aplanatic system with magnification M=0): 2 2 1/4 1s0 

3) Finite-size point-source ~  2  (accounted for by gaussian pupil ‘transmission’): exp f   ~   a complex defocus parameter f C  f  i f is used!

4) Higher than quadratic effects are ‘stored’ in kmax symmetrical polynomials of order 2n leading 2 0 expg  fC  hk R2k () to the formal expression at high NA: k0

Scalar vs vectorial ENZ-theory

Scalar approximation correct for sinm≤ 0.60 For sinm > 0.60 • Cautious use of scalar theory • Fully vectorial diffraction should be considered • Polarization must be included • Some geometric effects become important • Magnification • Mapping from entrance to exit pupil (for instance, Abbe sine condition)

14 Low-NA versus high-NA (|E|-field)

NA  0 NA = 0.95

Focal field description at high value of sinam

V.S. Ignatovski (1919) “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, pp. 1–36

E. Wolf (1959) ‘‘Electromagnetic diffraction in optical systems. I. An integral representation of the image field,’’ Proc.R. Soc. London, Ser. A 253, 349–357. B. Richards and E. Wolf ‘‘Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,’’ Proc. R. Soc. London, Ser. A 253, 358–379.

15 ENZ-formalism for large s0=sinm

For large angles sin(am ≥ 0.60, vector version of the Debye diffraction integral should be evaluated

Refs:

ENZ-formalism for large value of s0

Field in the focal region

with basic functions

for which a series expansion is available.

Ref:

16 Point-spread function computation

Aberration-free, through-focus intensity pattern, s0= 0.95

Point-spread function computation

Field components in the focal region

|Ex| |Ey| |Ez|

17 Point-spread function computation Through-focus behaviour Intensity In focus Defocused

Full aberration-free pupil

Reduced pupil

Annular pupil

Point-spread function computation Through-focus behaviour coma

18 Point-spread function computation o Aberration: astigmatism at 30 , s0 = 0.95

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 38

19 ENZ-based imaging scheme (extended coherent object area) (sheets made by Sven van Haver)

near-field

P2 P 1 imaging system source with illumination system assembly of general source points P object aperture (entrance pupil) image plane

• Illumination part • Near-field calculation • Near- to Far-field (entrance pupil) propagation • ENZ-imaging (entrance pupil to focal region)

ENZ – test object

• Construct an image from dipoles in the object plane and calculate the resulting field on the entrance pupil sphere!

|Ex||Ey||Ez|

Phase Ex Phase Ey Phase Ez

16λ x

20 Example of litho mask imaging

• Construct the resulting image in the focal region with the

ENZ-formalism (s0=0.95)

in front of behind image plane image plane image plane

Near-field calculation

near-field

P2 P 1 imaging system source with illumination system assembly of general source points P object aperture (entrance pupil) image plane

• Near-field calculation

21 Near-field calculation Use FDTD to calculate the near-field produced by the object.

Near- to far- field propagation Stratton-Chu method

near-field

P2 P 1 imaging system source with illumination system assembly of general source points P object aperture (entrance pupil) image plane

• Near- to Far-field (entrance pupil) propagation

22 Near- to far- field propagation

ENZ-based imaging entrance pupil to focal region

near-field

P2 P 1 imaging system source with illumination system assembly of general source points P object aperture (entrance pupil) image plane

• Basic ENZ forward calculation at high NA-value

23 ENZ-based imaging

• Through-focus image

Efficient interfacing

near-field

P2 P 1 imaging system source with illumination system assembly of general source points P object aperture (entrance pupil) image plane interface • Near-field calculation + Stratton-Chu (module) • ENZ forward calculation (module)

24 Efficient interfacing

Task at the interface: • Represent the field at the entrance pupil in terms of a Zernike expansion

Possible approaches: • Coefficient calculation by taking inner products • Least squares approach

Efficient interfacing Test of various sampling schemes equidistantsqrt-sampling Cos-sampling

Sampling scheme

Error of found Zernike coefficients

5 10

0 10 Resulting fit -5 10

-10 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

25 Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 51

Multilayer ENZ-imaging

26 Multilayer ENZ-imaging

• Effective pupil: Homogeneous imaging

pupil distribution

Multilayer ENZ-imaging

Effective pupils coming from one interface

27 Multilayer ENZ-imaging

In each layer, two counter propagating wave patterns are present. Wave amplitudes are derived from standard theory for stratified media and fit to Zernike expansions.

Multilayer ENZ-imaging

• Insert effective backward and forward propagating distributions into Debye Diffraction integral • Apply ENZ semi-analytic solution!

Forward propagating

Backward propagating

28 Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Small numerical aperture imaging

9) Pupil function reconstruction, aberration retrieval

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 57

Special G-functions for quadratic field expressions (energy flow, momentum density, momentum flow)

In the expressions that are quadratic in the field components, we use the general interference term mm'' m m GimmimmklVrfVrfkl, () exp( ') /2 exp ( ' )(nn', nk, )( nl ',, ) nmn,,',' m In the aberration-free case, we typically encounter 0 G0,0(1), 0  0000 GG0,20(1), 0,20(1), GG 2,20(1), 2,20 (1), 0 G2, 2(1), 0  0000 GG0,10(1), 0,10(1), GG 1,10 (1), 1,10 (1), 0 G1, 1(1). 0 

These functions play the role of the II01 , and I 2  integrals in the Richards-Wolf paper on high-NA aberration-free imaging.

29 Through-focus Poynting vector, linearly x-polarized light

z z

Through-focus Poynting vector, left-circularly polarized light

30 Orbital angular momentum in a focused beam

Contents

1) History of diffraction integral  Debye diffraction integral

2) Pupil function, Zernike polynomials

3) Classical and Extended Nijboer-Zernike diffraction theory

4) High-NA ENZ diffraction theory (point source)

5) Imaging of extended objects

6) Imaging in a stratified medium

7) Intensity, energy and momentum flow

8) Pupil function reconstruction, aberration retrieval

9) Small numerical aperture imaging

SPAM Winterschool 2011 Joseph Braat Extended Nijboer-Zernike diffraction theory 62

31 Phase retrieval from through-focus intensity patterns propagation direction

through focus Exit pupil (wavefront pupil Exit + ‘obstruction’)

Linearized retrieval method for Zernike coefficients

Basic principle Observed intensity = analytic expression ≈ linearized analytic expression m = ∑βn x basic-functions

Match experiment to theory by solving a linear system m of equations in n with the aid of recorded images

Through-focus images

Experiment Match Theory

32 Characterisation of optical systems Retrieving a general aberrated pupil function f 

Simulated Intensity

Input Intensity (noise added)

Basic ENZ-retrieval

Predictor-corrector procedure Ref: S. van Haver, J. Eur. Opt. Soc. -RP 1 (2006).

Retrieval at high-NA: basic equation

s0 = sinm

(a,b)  state of polarization

Ψ’s are products of two V-functions

Left- and righthand side are multiplied with Ψ’s and integrated over focal volume  extract Fourier-components

Solved with respect to β’s

Predictor-corrector method applied

33 Conclusion

1) Efficient and accurate scheme for diffraction calculations

2) For a fixed s0-value, V-functions are calculated in advance

3) Robust retrieval of aberration function possible

4) Higher order quantities in aberrated focus (electric and magnetic energy density, Poynting vector components, density of linear and angular momentum, Maxwell pressure tensor and angular momentum tensor) are split out in components related to the specific aberration of the focused beam.

5) ENZ to-do-list: • Extend aberration range for robust retrieval (now limited at 0.25 wave rms) • Include surface waves and propagation in absorbing media • Write down ENZ-intensity for partially polarised beams and birefringent systems • Benchmarking with FFT-based programs (how much faster?).

ENZ-team through the years

1) 2000-2009 Philips Research, target was lens characterisation 2001: analytic result ENZ obtained by Dr. Janssen Braat – Dirksen – Janssen 2009: Dr. Janssen moved to TU/e

2) TUD 2003-2006: Arthur van de Nes (20% Ph.D, postdoc) high-NA focusing and thin layer analysis

3) TUD2003-2006: Cas van de Avoort (20% Ph.D) retrieval

4) TUD 2004-2010: Sven van Haver (master, Ph.D and postdoc 100%) predictor-corrector method, high-NA retrieval, extended imaging, multilayer imaging, ENZ-toolbox

Website: http://www.nijboerzernike.nl