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Medical Photonics Lecture 1.2Optical Engineering

Lecture 4: Components

2019-05-08

Michael Kempe

Winter term 2017

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 function4 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, vignetting6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous7 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 raytrace13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements

Passive Components in Optical Systems 1

3

WavefrontManagement

Beam Shaping

Refraction Lenses

Diffraction DOE

Reflection Mirrors

Beam Redirection / Splitting

Reflection Mirrors

Refraction Prisms

Diffraction Gratings

Beam Guiding

Refraction Graded-Index Fibers

Reflection Step-Index Fibers

F F'

stop

C

RR/2

P B ST

stopP BS T

a) ast g at s 0

c) best image flat

ϕ

g

+1. order

+2. order

-2. order

-1. order

0. order

diffractive binary :all directions

Function Principle Components

Passive Components in Optical Systems 2

4

Wavelength Management

Angular Dispersion

Refraction Prisms

Diffraction Gratings

Filtering

Absorption Absorption Filters

Interference Interference and Dielectric Filters

Change Nonlinear Interaction Nonlinear Crystals

PolarisationManagement

Angular Dispersion Birefringence Birefringent Prisms etc.

Filtering

Absorption Dichroic Filters

Reflection Brewster Angle Polarizer

Modification Birefringence

Wavefront Retarder / Rotator

Waveplates

ϕ

ϕ∆

rotgrün

blau

weiß

grating

g = 1 / s

incidentcollimated

light

grating constant

-1.

-2.

-3.

0.

-4.

+1.

+2.

+3.

+4.

diffraction orders

Function Principle Components

Lenses are key elements in optical systems for• Optical imaging• Optical projection • Light focusing (energy concentration)

Lenses in Optical Systems

5

from infinity

real image

virtual image

F F'

F'F

Virtuell

F F'

virtual image

Imaging by Thin Lenses

6

Lens equation (paraxial approximation, 𝑛𝑛 = 𝑛𝑛𝑛): 1𝑠𝑠𝑠− 1

𝑠𝑠= 1

𝑓𝑓′= − 1

𝑓𝑓

Magnification: 𝑚𝑚 = 𝑦𝑦𝑠𝑦𝑦

= 𝑠𝑠𝑠𝑠𝑠

= 𝑓𝑓′−𝑠𝑠𝑠𝑓𝑓𝑠

F'Fy

f f

y'

s's

F'Fy

f f

y'

s'

s

7

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 axisThe focal lengths are referencedon 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 surfacecurvaturer > 0 : center ofcurvature is locatedon the right side

c curvature (c=1/r)

d thickness SS‘

n refractive index

O

O'

y'

y

F F'

SS'

P P'

N N'

n n n1 2

f'

a'

f' BFLf BFL

a

f

s's

d

sP s'P'

u'u

Notations of a lens

''

fn

fn

=−=Φ F# =𝑓𝑓𝐷𝐷

Optical power F-Number

D: effective diameter

n n‘

Different shapes of singlet lenses:1. bi-convex/concave, symmetric2. plane convex / concave, one surface plane3. Meniscus, both surface radii with the same sign

Convex: bending outsideConcave: hollow surface

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

P'P

bi-convex lens

P'P

plane-convex lens

P'P

positivemeniscus lens

P P'

bi-concave lens

P'P

plane-concavelens

P P'

negativemeniscus 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 nm588 nm656 nm

xpyp

∆x∆yaxis field 22°

xpyp

∆x∆y

xpyp

∆x∆yaxis field 22°

xpyp

∆x∆y

A1 A3A2

a) all spherical, 9 lenses

b) 3 aspheres, 6 lenses, shorter, better performance

Photographic lens f = 53 mm , F# = 6.5

∆𝑥𝑥

∆𝑥𝑥

GRIN lenses

• Gradient index (GRIN) lenses are using a spatially varying index of refraction• Example: GRIN lenses with radial parabolic index profile

𝑛𝑛 𝑟𝑟 = 𝑛𝑛0 − 𝑛𝑛2 � 𝑟𝑟𝑟

r

nn0

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

0.25 PitchObject at infinity

0.50 PitchObject at front surface

0.75 PitchObject at infinity

1.0 PitchObject at front surface

Pitch 0.25 0.50 0.75 1.0

Fresnel Lenses

• Fresnel lenses are refractive lenses with a surface structure• They are used to reduce weight and length of optical systems

• Significantaberrations usedfor illumination

activesurfaceslinear

Diffractive Optical Elements

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

• Different types (amplitude or phase zones):− Fresnel zone plates− Binary diffractive elements− Computer generated diffractive elements− Blazed diffractive element

refractive :one direction

ϕ

n

α

diffractive blazed :one direction

ϕ

g

ϕ

g

+1. order

+2. order

-2. order

-1. order

0. order

diffractive binary :all directions

Grating Diffraction

Maximum intensity:constructive interference of the contributionsof all periods

Grating equation

Holds for amplitude gratings too

( )g mo⋅ − = ⋅sin sinθ θ λ

grating

g

incidentlight

+ 1.diffraction

order

∆s = λ

in-phase

θ

θο

gratingconstant

Diffractive Optical Elements (DOE) as Lenses

Phase: Blazed diffractive lens

𝑟𝑟𝑛𝑛2 = 2𝜋𝜋 � 𝑛𝑛 � 𝑓𝑓 � 𝜆𝜆For 1st order

blaze

𝑠𝑠𝑠𝑠𝑛𝑛𝜃𝜃𝑛𝑛 =𝜆𝜆

𝑟𝑟𝑛𝑛+1 − 𝑟𝑟𝑛𝑛

D. T. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge Univ. Press, Cambridge, 1999).

Amplitude: Fresnel zone plate

𝑠𝑠𝑠𝑠𝑛𝑛𝜓𝜓𝑛𝑛 =𝑠𝑠𝑠𝑠𝑛𝑛𝜃𝜃𝑛𝑛𝑛𝑛

DOE‘s for Chromatic Correction

Use of DOE‘s in optical systems is for color correction

Micro Lens Array

Ref: W. Osten

Used for spot array generation or beam homogenization

Use of Micro Lens Arrays for Illumination

Optics Express 18(20):20968-78 · September 2010

The aperture splitting of the lens array provides a plurality of parallel Köhler illumination systems

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 areused to redirect light and tocontrol aberrations

Reflectivity of silver

E-field perpendicular (TE) E-field parallel (TM)

To plane of incidence

Curved Spherical Mirrors

Radius of curvature R Focal length R/2

On-axis imaging with spherical aberration Strong aberrations off-axis dependent on pupil stop position

22

stop

C

RR/2

P B ST

stopP BS T

a) ast g at s 0

c) best image flat

Image plane

( )( ) ( )222

22

111 yxcyxcz

++−+

+=

κ

Explicite surface equation, resolved to zParameters: curvature c = 1 / R

conic parameter κ

Influence of κ on the surface shape Parameter Surface shapeκ = - 1 paraboloidκ < - 1 hyperboloidκ = 0 sphereκ > 0 oblate ellipsoid (disc)

0 > κ > - 1 prolate ellipsoid (cigar )

23

Aspheric Surface: Conic Sections

x

y

z

Simple Asphere – Parabolic Mirror

sRyxz

2

22 +=

axis w = 0° field w = 2° field w = 4°

Surface 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 telescopes2. Collector in illumination systems

Simple Asphere – Elliptical Mirror

222

22

))(1(11)(

cyxyxcz

++−+

+=

κ

F

ss'

F'

Equation

Radius of curvature r in vertex, curvature ceccentricity κ

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 prism90°deflection

• Penta prism90°deflection

• Rhomboid prismBeam offset

• Bauernfeind prismBeam deviation

• Dove prismimage inversion

Properties of Reflection Prisms

Functions1. 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 (reversion, inversion)

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

Aberrations are introduced if used with non-collimated beams1. Monochromatic aberrations: spherical aberration, coma, astigmatism, field curvature,

distortion2. Chromatic aberrations

This holds for a plan-parallel plate too.

Transformation of Image Orientation

Modification of the image orientation with four options:1. Invariant image orientation2. Reverted image ( side reversal )3. Reverted image ( upside down )4. Image inversion y

x

y

x

y

x

mirror 1

mirror 2

y - z- foldingplane

z

z

image reversion in thefolding plane

(upside down)image

unchanged

imageinversion

original

folding planeimage reversion

perpendicular to thefolding 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 imagerotation 0° 90° 180°

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

β

ReflexionNo j

ϕ

r in

32

Optical Fibers: Step-Index

a

θ

θiz

r

n1

Mantel

Kern

Totalreflexion Brechzahl

n1n2

total reflexion

cladding

core

refractive index

∆=𝑛𝑛1 − 𝑛𝑛2𝑛𝑛1

= 0.001 … 0.02

𝜃𝜃𝑐𝑐 = cos−1(𝑛𝑛2/𝑛𝑛1)

Index step

Critical angle

𝑁𝑁𝑁𝑁 = sin𝜃𝜃𝑎𝑎 = 𝑛𝑛1 sin𝜃𝜃𝑐𝑐 = 𝑛𝑛1𝑟 − 𝑛𝑛2𝑟Numericalaperture

𝜃𝜃𝑎𝑎 : acceptance angle

33

Optical Fibers: Step-Index

• V fiber parameter descibes number of guided modes

• Requirement for single mode :

• Approximation for fundamental mode: Gaussian beam

rule-of-thumb for width:

NAaV ⋅=λπ2

r / a

E(r)

0

1

10 2 3

V = 1.0

V = 1.4V = 2.2

V = 3.00.5

Veaw ⋅−⋅+= 5046.20 6128.27993.0

405.2

34

Scalar solution of mode propagation with step-index boundary condition :

Core : r < a ziimm eekrJAzrE⋅−⋅⋅⋅= βϕϕ )(),,(

Cladding : r > a ϕϕ imm ekrKBzrE ⋅⋅= )(),,(

Fundamental Mode of Step-Index Fibers

r

A(r)

Kern Mantel

J0(kr)

K0(γr)

K3(γr)

J3(kr)

Grundmode

Anregungs-mode

fundamental mode

core cladding

Mode matching is requirementfor efficient in- and out-coupling

Azimutal Index

Rad

ial I

ndex

Scalar solution of mode propagation with step-index boundary

condition

:

Core : r < a

z

i

im

m

e

e

kr

J

A

z

r

E

×

-

×

×

×

=

b

j

j

)

(

)

,

,

(

Cladding : r > a

j

j

im

m

e

kr

K

B

z

r

E

×

×

=

)

(

)

,

,

(

_1044557914.unknown

_1044556127.unknown

35

Optical Fibers: Graded-Index

• Continuous index profile in core

• Rays follow curved trajectories with shorter paths compared tostep-index fibers

a

θ

z

r

n1

Mantel

Kern

kontinuierlicheStrahlbiegung Brechzahl

n1n2

𝑀𝑀 ≈𝑉𝑉𝑟4Number of modes:

cladding

core

continuous raydeflection refractive index

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

ααϕ −

⋅⋅=

2sinarcsin2 n

For symmetric case

𝑑𝑑𝜑𝜑𝑑𝑑𝜆𝜆 =

2sin 𝛼𝛼/2

1 − 𝑛𝑛2𝑠𝑠𝑠𝑠𝑛𝑛𝑟 𝛼𝛼/2

𝑑𝑑𝑛𝑛𝑑𝑑𝜆𝜆ϕ

ϕ∆

rotgrün

blau

weiß

redgreen

blue

white

𝛼𝛼

Ideal diffraction grating:monochromatic incident collimatedbeam is decomposed intodiscrete sharp diffraction orders

Constructive interference of thecontributions of all periodic cells

Only two orders for sinusoidalphase grating

Amplitude gratings have lowefficiency

Example: deep sine grating (𝑎𝑎 ≫ 𝜆𝜆)6% amplitude34% phase

Ideal Diffraction Grating

grating

g

incidentlight

+ 1.diffraction

order

∆s = λ

in-phase

θ

θο

gratingconstant

λθθ ⋅=− gmo /sinsin

red

blue

blue

Grating Equation

Intensity of grating diffraction pattern

Product of slit-diffraction andinterference function

Maxima of interference function: grating equation

Angle spread of an order decreaseswith 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

⋅

⋅

⋅⋅=

λπλπ

λπ

λπ

ugN

ugN

us

us

gNI

( ) λθθ ⋅=−⋅ 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 = π/λ sinθ

slit diffraction interference function

𝑢𝑢 =𝜆𝜆𝑠𝑠

𝑢𝑢 =𝜆𝜆𝑔𝑔

𝑢𝑢𝑠𝑠𝜆𝜆

s

g

Blaze grating (echelette):- facets with finite slope- additional phase shifts the slit diffraction function- all orders but one suppressed

Blaze condition is only valid forone wavelength and 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

𝜓𝜓 =𝜃𝜃 + 𝜃𝜃0

2𝜆𝜆 =

2𝑔𝑔𝑚𝑚𝐵𝐵

sin𝜓𝜓 cos 𝜃𝜃0 − 𝜓𝜓

𝜓𝜓𝜃𝜃

Reflection:

Transmission: tan𝜓𝜓 = sin 𝜃𝜃𝑛𝑛 − cos𝜃𝜃for 𝜃𝜃0=0

𝜆𝜆 = g � tan𝜓𝜓 𝑛𝑛− 1+ 1−𝑛𝑛2 tan2 𝜓𝜓

1+tan2 𝜓𝜓

𝜃𝜃𝜓𝜓

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

ddD

θλθθ

λθ

cossinsin 0

⋅−

==

NmLA m ⋅=⋅−=∆

=λ

θθλ

λ 0sinsin

0.+1. +2.

+3.+4.-4.

-3.-2. -1.

0

0.2

0.4

0.6

0.8

1

I(x)

mλ /gsinθ

∆θm(λ+∆λ) /g

𝐿𝐿 = 𝑔𝑔 � 𝑁𝑁 Length of grating

41

Color Glass Filter

• Wavelength filtering based on selective absorption of light

• Insensitive to angle of incidence

• Heating due to absorption

• Limited efficiency

Neutral density Bandpass

Long Pass Short Pass

Source: Schott AG

42

4/2211 λ== dndn

d1

d2

n0

n1

n2

ns

ER0 ER1 ER2

ET0

Luft

Layer 1

Layer 2

Substrat

Air

Substrate

Dielectric Filters

• Wavelength filtering based on interference in multilayer systems

• Sensitive to angle of incidence

• Low absorption• High efficiency

• Design degrees of freedom increase with number of layers

Example: Anti-reflexion coatingFor normal incidence

s

o

nn

nn

=2

1

Interference Condition

Amplitude Condition

43

T

Φ-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R = 0.900R = 0.950R = 0.980R = 0.990R = 0.999

( )ϕ

ϕcos21)1(

2

2

RRRIT −+

−=

Interference Filters

• Wavelength filtering based on multi-path interference

• Highly sensitive to angle of incidence

• Low absorption• Low efficiency• High wavelength

selection

n

Semi-reflective surface

d

𝜆𝜆 =𝑑𝑑

2𝑛𝑛 cos𝜑𝜑

2𝜑𝜑

Medical Photonics Lecture 1.2�Optical Engineering�ContentsPassive Components in Optical Systems 1Passive Components in Optical Systems 2Lenses in Optical SystemsImaging by Thin Lenses�Cardinal Elements of a Refractive Lens�Notations of a lens�Lens shape�Spherical Lenses Exhibit Aberrations�Aspheres - Geometry�Reducing the Number of Lenses with Aspheres�GRIN lenses�Fresnel Lenses�Diffractive Optical Elements�Grating Diffraction�Diffractive Optical Elements (DOE) as Lenses�DOE‘s for Chromatic CorrectionMicro Lens Array Use of Micro Lens Arrays for Illumination �MirrorsCurved Spherical Mirrors�Aspheric Surface: Conic Sections�Simple Asphere – Parabolic Mirror�Simple Asphere – Elliptical Mirror�Reflection Prisms�Properties of Reflection Prisms�Transformation of Image Orientation�Transform of Image Orientation �Application in Binoculars�Conical Light TaperFoliennummer 32Foliennummer 33Foliennummer 34Foliennummer 35�Dispersion Prism�Ideal Diffraction Grating�Grating Equation�Blaze Grating�Spectral Resolution of a GratingFoliennummer 41Foliennummer 42Foliennummer 43