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Medical Photonics Lecture

Optical Engineering

Lecture 2: Geometrical Optics

2016-10-26

Herbert Gross

Winter term 2017

2

Contents

No Subject Ref Date Detailed Content

1 Introduction Gross 19.10. Materials, dispersion, ray picture, geometrical approach, paraxial approximation

2 Geometrical optics Gross 26.10. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Components Kempe 02.11. Lenses, micro-optics, mirrors, prisms, gratings

4 Optical systems Gross 09.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

5 Aberrations Gross 16.11. Introduction, primary aberrations, miscellaneous

6 Diffraction Gross 23.11. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

7 Image quality Gross 30.11. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 07.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 14.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 21.12. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Optic design Gross 11.01. Aberration correction, system layouts, optimization, realization aspects

12 Photometry Gross 18.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

13 Illumination systems Gross 25.01. Light sources, basic systems, quality criteria, nonsequential raytrace

14 Metrology Gross 01.02. Measurement of basic parameters, quality measurements 15 Miscellaneous Gross 08.02. Additional materials, repetition, questions, exam preparation

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

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

nkm

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

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

stop

image

object

optical

system

O2field

point

axis

pupil

stop

marginal

ray

O1 O'1

O'2

chief

ray

Optical imaging

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

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

'

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

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

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

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

''

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

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

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 2121

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

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

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

'

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

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

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

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

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

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

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

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

Transverse Aberrations

Typical low order polynomial contributions for:

defocus, coma, spherical aberration, lateral color

This allows a quick classification of real curves

pprSy cos'''3D

)2cos2('''' 2 PpryCy D

pprKy cos''' D

linear:

defocus

quadratic:

coma

cubic:

spherical

offset:

lateral colorDy DyDyDy

yp ypypyp

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

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

Wave Aberration

Definition of the peak valley value

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

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 3y2

cos2

Coma

secondary

r4y cos

r2y

3 cos

3

r2y

3 cos

r1 y

4

r1 y

4 cos

2

r 3y2

r12 y

r 12

y

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

42