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Medical Photonics Lecture

Optical Engineering

Lecture 2: Geometrical Optics

2019-04-17

Herbert Gross

Speaker: Yi Zhong

Summer term 2019

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Zhong 10.04.Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Zhong 17.04.Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Zhong 24.04.Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 08.05. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Zhong 15.05.Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous

7 Image quality Zhong 29.05. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 05.06.Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 12.06.Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 19.06.Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Photometry Zhong 26.06.Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

12 Illumination systems Gross 03.07.Light sources, basic systems, quality criteria, nonsequential raytrace

13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements

Scheme of Raytrace

zoptical

axis

y j

u'j-1

ij

dj-1

ds j-1

ds j

i'j

u'j

n j

nj-1

mediummedium

surface j-1

surface j

ray

dj

vertex distance

oblique thickness

rr

▪ Ray: straight line between two intersection points

▪ System: sequence of spherical surfaces

▪ Data: - radii, curvature c=1/r

- vertex distances

- refractive indices

- transverse diameter

▪ Surfaces of 2nd order:

Calculation of intersection points

analytically possible: fast

computation

3

Vectorial Raytrace

yj

z

Pj+1

sj

xj

yj+1

xj+1

Pj

surface

No j

surface

No j+1

dj sj+1

intersection

point

ej

normal

vector

ej+1

ray

distance

intersection

point

normal

vector

▪ General 3D geometry

▪ Tilt and decenter of surfaces

▪ General shaped free form surfaces

▪ Full description with 3 components

▪ Global and local coordinate systems

4

▪ Vignetting/truncation of ray at finite sized diameter:

can or can not considered (optional)

▪ No physical intersection point of ray with surface

▪ Total internal reflection

▪ Negative edge thickness of lenses

▪ Negative thickness without mirror-reflection

▪ Diffraction at boundaries

index

j

index

j+1

axis

negative

un-physical

regular

irregular

axis

no intersection

point

axis

intersection:

- mathematical possible

- physical not realized

axis

total

internal

reflection

Raytrace errors

5

Intersection with Surfaces of Second Order

z

y

ray

sign +radius r > 0

sign -radius r < 0

solution 2

solution 1

spherical

surface

▪ Surfaces of second order (conic sections):

analytical computation of intersection points, fast and accurate

▪ Case selection:

- two possible solutions, sign of radius selects the choice

- special case: surfaces with over-hemisphere

- nonsequential raytrace, special logic

necessary

- sign inversion of the z-component

the ray vector for reflection

6

Fresnel Surfaces

▪ Special description of Fresnel surfaces

with circular symmetry

▪ Bezier spline desciption with corresponding

choice of the control points:

modelling of edges

▪ Mathematically:

- surface sag continuous

- derivative with steps

7

Free Shaped Surfaces

( ) ( )nk

m

j

m n

jkmnkj yyxxayxA −−== =

3

0

3

0

, ),(

xy

▪ Free-shaped surfaces: not necessary symmetric

▪ Representations:

- polynomial expansions in x, y

- polynomial expansions with Zernike functions

- spline descriptions, defined on local patches

Bezier, cubic, bi-quadratic , NURBS

▪ Application of free-shaped surfaces:

- compact systems

- systems with special dependencies

on field or aperture

- simulation of measured surfaces

- tolerancing of real surfaces

- array structures, e.g. Fresnel lenses

▪ Cubic spline representation

8

Gradient Lenses

▪ Refocusing in parabolic profile

▪ Helical ray path in 3 dimensions

axis ray bundle

off axis ray bundle

waist

points

view

along z

perspectivic viewy

x

y

x

y'

x'

z

9

Nonsequential Raytrace: Examples

Signal

1 2 3 4

Reflex 1 - 2

Reflex 3 - 2

1

2

3

1. Prism with total internal

reflection

2. Ghost images in optical systems

with imperfect coatings

10

▪ Pupil sampling in 3D for spot diagram:

all rays from one object point through all pupil points in 2D

▪ Light cone completly filled with rays

Pupil Sampling

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

11

Pupil Sampling

▪ Ray plots

▪ Spot

diagrams

sagittal ray fan

tangential ray fan

yp

Dy

tangential aberration

xp

Dy

xp

Dx

sagittal aberration

whole pupil area

12

Raytrace Through a Lens

13

14

Optical Imaging

object

imageoptical

system

▪ Optical Image formation:

All ray emerging from one object point meet in the perfect image point

▪ Region near axis:

gaussian imaging

ideal, paraxial

▪ Image field size:

Chief ray

▪ Aperture/size of

light cone:

marginal ray

defined by pupil

stopimage

object

optical

system

O2field

point

axis

pupil

stop

marginal

ray

O1 O'1

O'2

chief

ray

Optical imaging

15

▪ Single surface

imaging equation

▪ Thin lens in air

focal length

▪ Thin lens in air with one plane

surface, focal length

▪ Thin symmetrical bi-lens

▪ Thick lens in air

focal length

'

1'

'

'

fr

nn

s

n

s

n=

−=−

( )

−−=

21

111

'

1

rrn

f

1'

−=

n

rf

( )12'

−=

n

rf

( )( )

21

2

21

1111

'

1

rrn

dn

rrn

f

−+

−−=

Formulas for surface and lens imaging

16

▪ Imaging by a lens in air:

lens makers formula

▪ Magnification

▪ Real imaging:

s < 0 , s' > 0

▪ Intersection lengths s, s'

measured with respective to the

principal planes P, P'

fss

11

'

1=−

s'

2f'

4f'

2f' 4f'

s-2f'- 4f'

-2f'

- 4f'

real object

real image

real object

virtual object

virtual image

virtual image

real image

virtual image

Imaging equation

s

sm

'=

17

▪ Single surface between two media

Radius r, refractive indices n, n‘

▪ Imaging condition, paraxial

▪ Abbe invariant

alternative representation of the

imaging equation

'

1'

'

'

fr

nn

s

n

s

n=

−=−

y

n'ny'

r

C

ray through center of curvature C principal

plane

vertex S

s

s'

object

surface

image

arbitrary ray

−=

−=

'

11'

11

srn

srnQs

Single Surface

18

19

Cardinal Elements of a Lens

▪ Focal points:

1. incoming ray parallel to the axis

intersects the axis in F‘

2. ray through F is leaves the lens

parallel to the axis

The focal lengths are referenced

on the principal planes

▪ Nodal points:

Ray through N goes through N‘

and preserves the direction

nodal planes

N N'

u

u'

f '

P' F'

sBFLprincipal

planes

backfocalplane

PF

frontfocalplane

f

P principal point

S vertex of the surface

F focal point

s intersection point

of a ray with axis

f focal length PF

r radius of surface

curvature

d thickness SS‘

n refrative index

O

O'

y'

y

F F'

S

S'

P P'

N N'

n n n1 2

f'

a'

f'BFL

fBFL

a

f

s's

d

sP

s'P'

u'u

Notations of a lens

20

▪ Lateral magnification for finite imaging

▪ Scaling of image size 'tan'

tan'

uf

uf

y

ym

−==

z f f' z'

y

P P'

principal planes

object

imagefocal pointfocal point

s

s'

y'

F F'

Magnification

21

▪ Imaging with a lens

▪ Location of the image:

lens equation

▪ Size of the image:

Magnification

Imaging by a Lens

object image

-sf

+s'

system

lens

y

y'

fss

11

'

1=−

s

s

y

ym

''==

22

▪ Ranges of imaging

Location of the image for a single

lens system

▪ Change of object loaction

▪ Image could be:

1. real / virtual

2. enlarged/reduced

3. in finite/infinite distance

Imaging by a Lens

|s| < f'

image virtual

magnified FObjekt

s

F object

s

Fobject

s

F

object

s

F'

F

object

image

s

|s| = f'

2f' > |s| > f'

|s| = 2f'

|s| > 2f'

F'

F'

F'

F'

image

image

image

image

image at

infinity

image real

magnified

image real

1 : 1

image real

reduced

23

▪ Graphical image construction

according to Listing by

3 special rays:

1. First parallel through axis,

through focal point in image

space F‘

2. First through focal point F,

then parallel to optical axis

3. Through nodal points,

leaves the lens with the same

angle

▪ Procedure work for positive

and negative lenses

For negative lenses the F / F‘ sequence is

reversed

Graphical Image Construction after Listing

F

F'

y

y'

P'P

1

2

3

F'

F

y

y'

P'P

1

2

3

24

▪ Two lenses with distance d

▪ Focal length

distance of inner focal points e

▪ Sequence of thin lenses close

together

▪ Sequence of surfaces with relative

ray heights hj, paraxial

▪ Magnification

n

FFdFFF 21

21

−+=

e

ff

dff

fff 21

21

21 =

−+

=

=k

kFF

( ) −=k k

kkk

rnn

h

hF

1'

1

kk

k

n

n

s

s

s

s

s

sm

'

''' 1

2

2

1

1 =

Multi-Surface Systems

25

▪ Apertur stop :

- Transverse limiting shape of the light cone

- at stop or lens mounting

▪ Field stop: :

Limits the size of the

field of view

Field and Aperture Stops

lens with mount

mount acts as

aperture stop

lens with mountingrear stop acts as

limiting aperture

26

Matrix Calculus

▪ Paraxial raytrace transfer

▪ Matrix formulation

▪ Matrix formalism for finite angles

▪ Paraxial raytrace refraction

▪ Inserted

▪ Matrix formulation

111 −−− += jjjj Udyy

1−+= jjjj Uyi in

nij

j

j

j''

=

1' −= jj UU

1−= jj yy

( )1

'

'' −+

−−= j

j

j

j

j

jjj

j Un

ny

n

nnU

'' 1 jjjj iiUU +−= −

=

−

j

jj

j

j

U

yd

U

y

10

1

'

'1

( )

−

−=

j

j

j

j

j

jjj

j

j

U

y

n

n

n

nnU

y

'

'01

'

'

=

j

j

j

j

u

y

DC

BA

u

y

tan'tan

'

27

▪ Linear relation of ray transport

▪ Simple case: free space

propagation

▪ Advantages of matrix calculus:

1. simple calculation of component

combinations

2. Automatic correct signs of

properties

3. Easy to implement

▪ General case:

paraxial segment with matrix

ABCD-matrix :

=

=

u

xM

u

x

DC

BA

u

x

'

'

z

x x'

ray

x'

u'

u

x

B

Matrix Formulation of Paraxial Optics

A B

C D

z

x x'

ray x'

u'u

x

28

▪ Linear transfer of spation coordinate x

and angle u

▪ Matrix representation

▪ Lateral magnification for u=0

▪ Angle magnification of conjugated planes

▪ Refractive power for u=0

▪ Composition of systems

▪ Determinant, only 3 variables

uDxCu

uBxAx

+=

+=

'

'

=

=

u

xM

u

x

DC

BA

u

x

'

'

mxxA == /'

== uuD /'

xuC /'=

121 ... MMMMM kk = −

'det

n

nCBDAM =−=

Matrix Formulation of Paraxial Optics

29

▪ System inversion

▪ Transition over distance L

▪ Thin lens with focal length f

▪ Dielectric plane interface

▪ Afocal telescope

−

−=

−

AC

BDM

1

=

10

1 LM

−=

11

01

f

M

=

'0

01

n

nM

=

0

1L

M

Matrix Formulation of Paraxial Optics

30

▪ Product of field size y and numercial aperture is invariant in a paraxial system

▪ The invariant L describes to the phase space volume (area)

▪ The invariance corresponds to

1. Energy conservation

2. Liouville theorem

3. Constant transfer of information

y

y'

u u'

marginal ray

chief ray

object

image

system

and stop

''' uynuynL ==

Helmholtz-Lagrange Invariant

31

▪ Geometrical optic:

Etendue, light gathering capacity

▪ Paraxial optic: invariant of Lagrange / Helmholtz

▪ General case: 2D

▪ Invariance corresponds to

conservation of energy

▪ Interpretation in phase space:

constant area, only shape is changed

at the transfer through an optical

system

unD

Lfield

Geo sin2

=

''' uynuynL ==

Helmholtz-Lagrange Invariant

space y

angle u

large

aperture

aperture

small

1

23

medium

aperture

y'1

y'2

y'3u1 u2

u3

32

Optical Image Formation

optical

system

object

plane

image

plane

transverse

aberrations

longitudinal

aberrations

wave

aberrations

▪ Perfect optical image:

All rays coming from one object point intersect in one image point

▪ Real system with aberrations:

1. transverse aberrations in the image plane

2. longitudinal aberrations from the image plane

3. wave aberrations in the exit pupil

33

34

Dy’ = - Ds’ tan u’

Longitudinal

Aberration Focus

▪ Spherical aberration: transverse aberration ~ ³

▪ Coma: transverse aberration ~ ²

Ray Aberration Representation

Ref: B. Böhme

34

▪ Longitudinal aberrations Ds

▪ Transverse aberrations Dy

Representation of Geometrical Aberrations

Gaussian image

plane

ray

longitudinal

aberration

D s'

optical axis

system

U'reference

point

reference

plane

reference ray

(real or ideal chief ray)

transverse

aberrationDy'

optical axis

system

ray

U'

Gaussian

image

plane

reference ray

longitudinal aberration

projected on the axis

Dl'

optical axis

system

ray

Dl'o

logitudinal aberration

along the reference ray

35

Representation of Geometrical Aberrations

ideal reference ray angular aberrationDU'

optical axis

system

real ray

x

z

s' < 0D

W > 0

reference sphere

paraxial ray

real ray

wavefront

R

C

y'D

Gaussian

reference

plane

U'

▪ Angle aberrations Du

▪ Wave aberrations DW

36

Transverse Aberrations

▪ Typical low order polynomial contributions for:

defocus, coma, spherical aberration, lateral color

▪ This allows a quick classification of real curves

pprSy cos''' 3=D

)2cos2('''' 2

PpryCy +=D

pprKy cos''' =D

linear:

defocus

quadratic:

coma

cubic:

spherical

offset:

lateral colorDy DyDyDy

ypypypyp

37

Spot Diagram

y'p

x'p

yp

xp x'

y'

z

yo

xo

object plane

point

entrance pupil

equidistant grid

exit pupil

transferred grid

image plane

spot diagramoptical

system

▪ All rays start in one point in the object plane

▪ The entrance pupil is sampled equidistant

▪ In the exit pupil, the transferred grid

may be distorted

▪ In the image plane a spreaded spot

diagram is generated

38

Field Dependence of the Spot Shape

▪ Single plane-convex lens,

BK7, f = 100 mm, l = 500 nm

▪ Spot as a function of

field position

▪ Coma orientation towards the

axis

x

y

39

Wave Aberration

▪ Definition of the peak valley value

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

40

Primary Aberrations

Dy

PryAry

CrySry

pp

pp

+

++

+=D

3

222

23

cos

cos'

▪ Expansion of the transverse aberration Dy on image height y and pupil height r

▪ Lowest order 3 of real aberrations: primary or Seidel aberrations

▪ Spherical aberration: S

- no dependence on field, valid on axis

- depends in 3rd order on apertur

▪ Coma: C

- linear function of field y

- depends in 2rd order on apertur with azimuthal variation

▪ Astigmatism: A

- linear function of apertur with azimuthal variation

- quadratic function of field size

▪ Image curvature (Petzval): P

- linear dependence on apertur

- quadratic function of field size

▪ Distortion: D

- No dependence on apertur

- depends in 3rd order on the field size

41

Polynomial Expansion of Aberrations

▪ Representation of 2-dimensional Taylor series vs field y and aperture r

▪ Selection rules: checkerboard filling of the matrix

▪ Constant sum of exponents according to the order

Field y

Spherical

y0 y 1 y 2 y3 y 4 y 5

Distortion

r

0

y

y3

primary

y5

secondary

r

1

r 1

Defocus

Aper-

ture

r

r

2

r2y Coma primary

r 3

r 3 Spherical

primary

r

4

r

5

r 5 Spherical

secondary

DistortionDistortionTilt

Coma Astigmatism

Image

location

Primary

aberrations /

Seidel

Astig./Curvat.

cos

cos

cos2

cos

Secondary

aberrations

cos

r 3y 2 cos

2

Coma

secondary

r4y cos

r2y

3 cos

3

r2y

3 cos

r1 y

4

r1 y

4 cos

2

r 3y 2

r12

yr 12

y

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

42