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Medical Photonics Lecture 1.2

Optical Engineering

Lecture 4: Components

2017-11-16

Michael Kempe

Winter term 2017

2

Schedule Optical Engineering 2017

No Subject Ref Date Detailed Content

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

2 Geometrical optics Gross 02.11. Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant

3 Diffraction Gross 09.11. Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function

4 Components Kempe 16.11. Lenses, micro-optics, mirrors, prisms, gratings

5 Optical systems Gross 23.11. Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting

6 Aberrations Gross 30.11. Introduction, primary aberrations, miscellaneous 7 Image quality Gross 07.12. Spot, ray aberration curves, PSF and MTF, criteria

8 Instruments I Kempe 14.12. Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes

9 Instruments II Kempe 21.12. Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts

10 Instruments III Kempe 11.01. Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes

11 Optic design Gross 18.01. Aberration correction, system layouts, optimization, realization aspects

12 Photometry Gross 25.01. Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory

13 Illumination systems Gross 01.02. Light sources, basic systems, quality criteria, nonsequential raytrace

14 Metrology Gross 08.02. Measurement of basic parameters, quality measurements

Lenses are key elements in optical systems for

• Optical imaging

• Optical projection

• Light beam shaping, e.g. focusing (energy concentration)

Lenses in Optical Systems

3

from infinity

real image

virtual image

F F'

F'F

F'F

virtual object

location

Virtuell

F F'

virtual image

Imaging by Lenses

4

Lens equation (paraxial approximation, 𝑛 = 𝑛‘): 1

𝑠′−

1

𝑠=

1

𝑓′= −

1

𝑓

Magnification: 𝑚 =𝑦′

𝑦=

𝑠′

𝑠=

𝑓′−𝑠′

𝑓′

F'F

y

f f

y'

s's

F'F

y

f f

y'

s'

s

5

Cardinal Elements of a Refractive 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

N nodal point

S vertex of the surface

F focal point

f focal length PF

r radius of surface

curvature

d thickness SS‘

n refractive 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

Main notations and properties of a lens:

- radii of curvature r1 , r2

curvatures c

sign: r > 0 : center of curvature

is located on the right side

- thickness d along the axis

- diameter D

- index of refraction of lens material n

Focal length (paraxial)

Optical power

Back focal length

intersection length,

measured from the vertex point

2

2

1

1

11

rc

rc

'tan',

tan

'

u

yf

u

yf F

'

'

f

n

f

nF

'' '' PBFLF sffs

Main properties of a lens

Different shapes of singlet lenses:

1. bi-, symmetric

2. plane convex / concave, one surface plane

3. Meniscus, both surface radii with the same sign

Convex: bending outside

Concave: hollow surface

Principal planes P, P‘: outside for meniscus shaped lenses

P'P

bi-convex lens

P'P

plane-convex lens

P'P

positive

meniscus lens

P P'

bi-concave lens

P'P

plane-concave

lens

P P'

negative

meniscus lens

Lens shape

Spherical Lenses Exhibit Aberrations

• Example: spherical aberration

Aspheres - Geometry

z

y

aspherical

contour

spherical

surface

z(y)

height

y

deviation

z

sphere

z

y

perpendicular

deviation rs

deviation z

along axis

height

y

tangente

z(y)

aspherical

shape

Reference: deviation from sphere

Deviation z along axis

Better conditions: normal deviation rs

Reducing the Number of Lenses with Aspheres

Example photographic zoom lens

Equivalent performance

9 lenses reduced to 6 lenses

Overall length reduced

Ref: H. Zügge

436 nm

588 nm

656 nm

xpyp

xy

axis field 22°

xpyp

xy

xpyp

xy

axis field 22°

xpyp

xy

A1A3

A2

a) all spherical, 9 lenses

b) 3 aspheres, 6 lenses,

shorter, better performance

Photographic lens f = 53 mm , F# = 6.5

∆𝑥

∆𝑥

Fresnel Lenses

• Fresnel lenses are refractive lenses with a surface structure

• They are used to reduce weight and length of optical systems

• Significant

aberrations used

for illumination

active

surfaces

linear

Diffractive Optical Elements

• Diffractive optical elements (DOE‘s) are based on diffraction to redirect light

• Different types:

Fresnel zone plates (transmission or phase zones)

Binary diffractive elements

Computer generated diffractive elements

Blazed diffractive element

refractive :

one direction

n

diffractive blazed :

one direction

g

g

+1. order

+2. order

-2. order

-1. order

0. order

diffractive binary :

all directions

DOE‘s for Chromatic Correction

• Example: blazed diffractive element

• One use of DOE‘s in optical systems is for color correction

P. 676 𝑟𝑘 = 2𝜋 ∙ 𝑓 ∙ 𝜆 ∙ 𝑘

For 1st order

tan𝜓𝑘 =𝑠𝑖𝑛𝜃𝑘

𝑛 − 𝑐𝑜𝑠𝜃𝑘

𝑠𝑖𝑛𝜃𝑘 =𝜆

𝑟𝑘+1 − 𝑟𝑘

GRIN lenses

• Gradient index (GRIN) lenses are using a spatially varying index of refraction

• Example: GRIN lenses with radial parabolic index profile

𝑛 𝑟 = 𝑛0 − 𝑛2 ∙ 𝑟²

r

n n0

Such lenses are used, e.g., as relay lenses

0.25 Pitch

Object at infinity

0.50 Pitch

Object at front surface

0.75 Pitch

Object at infinity

1.0 Pitch

Object at front surface

Pitch 0.25 0.50 0.75 1.0

Micro Lens Array

Pupil shape quadratic

No dead zones

Ref: W. Osten

What is Micro Optics ?

17

GRIN rod lens GRIN lens LC-lens liquid lens

mirror micro lens micro prism tilt mirror

Fresnel lens binary DOE holographic DOE waveguide

Micro components with different functionalities

Small dimension: what feature is small ?

One dimension ? All dimensions ?

Fine structure on surface ?

What is Micro-Optic ?

18

x y z

largefine

structuredsmall

x y z LED, photonic crystals

x y z Bragg grating, endoscope

x y z Fiber

x y z Arrays, MEMs, DMDs, DOEs, stacked planar

x y z Linear grating

x y z Coating

x y z Macro optics

6

5

4

3

2

1

No

0

Dimension

Plastic Components of Small Size

Manufacturing of complex geometries in high volumes

Integration of assembly mechanics

Conventional components in glass of

small size

Lenses, prisms,

lightguides

Tilt MEMs mirrors

Combination with

micro-mechanics

and electronics

Simple or Complex Micro-optical Components

20

Ref: Firma Optikron, Jena

Ref: Firma Optikron, Jena

Mirrors

• Mirrors are based on reflection, typically off coated surfaces (dielectric/metal)

• The reflectivity but not the direction depend on the wavelength and polarization

21

n n'

incidence

reflection transmission

interface

E

B

i

i

E

B

r

r

E

Bt

t

normal to the interface

i

i'i

a) s-polarization

n n'

incidence

reflection

transmission

interface

B

E

i

i

B

E

r

r

B

E t

t

normal to the interface

i'i

i

b) p-polarization

In optical systems mirrors are

used to redirect light and to

control aberrations

Reflectivity of silver

Astigmatism for Curved Spherical Mirrors

22

22

11'

'

131'

'

R

s

R

ys

R

s

R

ys

S

T

Image surfaces for a concave mirror

y‘ : image height

𝑠 : stop position

Special cases of flat image shells as a

function of the stop position

a) stop a center:

zero astigmatism

b) stop at distance

0.42 R:

T=0

c) stop at distance

0.29 R:

B = 0 (best plane)

d) stop at mirror:

S = 0

22

stop

C

RR/2

P B STstop

C

R

0.42 R

P BS T

stop

C

R

P BS T

stop

C

R

P BS T

a) astigmatism A = 0 b) tangential flat

c) best image flat d) sagittal flat

0.29 R

23

Avoiding Mirror Obscuration

Avoiding the central obscuration in mirror systems

Field bias or aperture offset as opportunities

Ref.: K. Fuerschbach

22 yxz

222

22

111 yxc

yxcz

22

22 yxRRRRz xxyy

Conic section

Special case spherical

Cone

Toroidal surface with

radii Rx and Ry in the two

section planes

Generalized conic section without

circular symmetry

Roof surface

2222

22

1111 ycxc

ycxcz

yyxx

yx

z y tan

24

Aspherical Surface Types

222

22

111 yxc

yxcz

1

2

b

a

2a

bc

1

1

cb

1

1

ca

Explicite surface equation, resolved to z

Parameters: curvature c = 1 / R

conic parameter

Influence of on the surface shape

Relations with axis lengths a,b of conic sections

Parameter Surface shape

= - 1 paraboloid

< - 1 hyperboloid

= 0 sphere

> 0 oblate ellipsoid (disc)

0 > > - 1 prolate ellipsoid (cigar )

25

Conic Sections

Simple Asphere – Parabolic Mirror

sR

yz

2

2

axis w = 0° field w = 2° field w = 4°

Equation

Radius of curvature in vertex: Rs

Perfect imaging on axis for object at infinity

Strong coma aberration for finite field angles

Applications:

1. Astronomical telescopes

2. Collector in illumination systems

Simple Asphere – Elliptical Mirror

22

2

)1(11 cy

ycz

F

s

s'

F'

Equation

Radius of curvature r in vertex, curvature c

eccentricity

Two different shapes: oblate / prolate

Perfect imaging on axis for finite object and image loaction

Different magnifications depending on

used part of the mirror

Applications:

Illumination systems

Reflection Prisms

Right angle prism

90°deflection

Penta prism

90°deflection

Rhomboid prism

Beam offset

Bauernfeind prism

Beam deviation

Dove prism

image inversion

Properties of Reflection Prisms

Functions

1. Bending of the beam path, deflection of the axial ray direction

Application in instrumental optics and folded ray paths

2. Parallel off-set, lateral displacement of the axial ray

3. Modification of the image orientation with four options:

a. Invariant image orientation

b. Reverted image ( side reversal )

c. Inverted image ( upside down )

d. Complete image inversion (inverted-reverted image)

The number of mirrors is important

Every mirror generates a complete inversion,

No change for even numbers

l/r and u/d separation by roof-edge prisms

4. Off-set of the image position, shift of image position forwards in the propagation direction.

Aberrations introduced

1. Astigmatism

2. Chromatic aberration

3. Spherical aberration in non-collimated beams

Transformation of Image Orientation

Modification of the image orientation

with four options:

1. Invariant image orientation

2. Reverted image ( side reversal )

3. Inverted image ( upside down )

4. Complete image inversion

(inverted-reverted image)

Image side reversal in the

principal plane of one mirror

Inversion for an odd number

of reflections

Special case roof prims:

Corresponds to one reflection

in the edge plane,

Corresponds to two reflections

perpendicular to the edge plane

y

x

y

x

y

x

mirror 1

mirror 2

y - z- folding

plane

z

z

image reversion in the

folding plane

(upside down)

image

unchanged

image

inversion

original

folding planeimage reversion

perpendicular to the

folding plane

Transform of Image Orientation

Rotatable Dove prism:

Azimutal angle: image rotates by the double angle

Application: periscopes

object

Bild

0° 45° 90°angle of prism

rotation

angle of image

rotation 0° 90° 180°

Types of Reflection Prisms: Porro Prism

Porrro Prism

Incoming ray direction inverted in one section

Version with roof-edge:

Ray direction inverted in 3D (retro reflector, cats eye reflector)

roof-edge

D

90°

a

v

Roof-Edge Prism

Roof edge:

- two reflecting surfaces with 90°

- change of lateral coordinate in one section

Critical in practice:

Precision of 90° angle,

typical tolerance 1‘‘

errors cause image split

Coatings critical due to

polarization effects

sA

D

B

C

roof edge

intersection plane

with angle of 90°

intersection

planes with

angles of 2

Application in Binoculars

Double Porro Prism

Abbe-König Roof Prism

Conical Light Taper

Waveguide with conical boundary

Lagrange invariant: decrease in diameter causes increase in angle:

Aperture transformed

Number of reflections:

- depends on diameter/length ratio

- defines change of aperture angle

'sinsin uDuD outin

u'

u

L

Din

/

2

n

Dout

/

2

Reflexion

No j

r i

n

Dispersion Prism

Dispersion prism spatially separate light in its colors

Blue light is refracted more strongly than red light ( normal dispersion)

Application : spectrometer, dispersion control

2sinarcsin2 n

For symmetric case

𝑑𝜑

𝑑𝜆=

2sin 𝛼/2

1 − 𝑛2𝑠𝑖𝑛² 𝛼/2

𝑑𝑛

𝑑𝜆

rot

grün

blau

weiß

red

green

blue

white

𝛼

Ideal diffraction grating:

monochromatic incident collimated

beam is decomposed into

discrete sharp diffraction orders

Constructive interference of the

contributions of all periodic cells

Only two orders for sinusoidal

Ideal Diffraction Grating

grating

g = 1 / s

incidentcollimated

light

grating constant

-1.

-2.

-3.

0.

-4.

+1.

+2.

+3.

+4.

diffraction orders

Grating Diffraction

Maximum intensity:

constructive interference of the contributions

of all periods

Grating equation

g mo sin sin

grating

g

incident

light

+ 1.

diffraction

order

s =

in-phase

grating

constant

Grating Equation

Intensity of grating diffraction pattern

(scalar approximation g >> )

Product of slit-diffraction and

interference function

Maxima of pattern:

coincidence of peaks of both

functions: grating equation

Angle spread of an order decreases

with growing number od periods N

Oblique phase gradient:

- relative shift of both functions

- selection of peaks/order

- basic principle of blazing

2

22

sin

sinsin

ugN

ugN

ug

ug

gNI

mg osinsin

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u = sin

Spectral Resolution of a Grating

Angle dispersion of a grating

Separation of two spectral lines

Complete setup with all orders:

Overlap of spectra possible at higher orders

m

m

d

dD

cos

sinsin 0

NmLA m

0sinsin

0.+1. +2.

+3.

+4.-4.

-3.-2. -1.

0

0.2

0.4

0.6

0.8

1

I(x)

m /g

sin

m( /g

Blaze grating (echelette):

- facets with finite slope

- additional phase shifts the slit diffraction function

- all orders but oen suppressed

Blaze condition is only valid for

- one wavelength

- one incidence angle

Blaze Grating

suppressed ordersworking order

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

mB mB+1 mB+2mB-1mB-2

slit diffraction