2.3 Aberrations 2.5 Depth of focus Optical Engineering

Universität Karlsruhe (TH)

Optical Engineering

Martina Gerken22.11.2007

4.2

Course outline1. Imaging optics

2.1 Light propagation across interfaces2.2 Photography and optical lenses2.3 Aberrations2.4 Plane parallel plates and reflective prisms 2.5 Depth of focus2.6 Magnifying glass and basic microscope2.7 Modern microscopes2.8 Beam expansion2.9 Telescopes2.10 Optic design

2. Optical sensors3. Optics in data storage4. Introduction to displays5. Fourier optics6. Diffractive optics and holograms7. Integrated optics8. Computerized imaging

4.3

Exercise: Selection of telescope

• Collect the advantages and disadvantages of the two telescopes!

• Decide as a group on one telescope!

4.4

Comparison of telescopes

• Refractor• Terrestrial observation

• Reflector• No spherical/chromatic aberrations• Mirror diameter larger, thus more

light• More compact• More expensive• Prettier• Aberrations due to diffraction on

second mirror

BRASKO 60700 Telescope Bresser Telescope Pluto 114/500

4.5

Important factors for telescope purchase• Two main tasks of telescope:

– Magnify small objects– Brighten faint objects

• Useful magnification limited by diffraction– Max. Magnification ≈ Diameter of objective in mm– Image may be magnified more, but image information is limited– Analog: Not more detail on newspaper image observable with magnifying

glass

• Aberrations and air turbulence ("Seeing") limit magnification further– Max. resolution of terrestrial telescope ≈ 1”– Hubble-Telescope ≈ 0.05” at visible wavelengths

• Larger aperture for brighter images!

• Quality of mount is also critical– Should not move more than 1 sec after touching

4.6



• Analytical solution (optical quadrupole)• Matrix formulation• Sequential raytracing• Non-sequential raytracing• FDTD

4.7

Geometrical optics / Ray optics

ξ << Xλ << X

• Rays describe propagation of light in (most) optical systems sufficiently well, if dimensions X of objects and parts sufficiently larger than:

– Wavelength

– Coherence length

• Wave properties of light neglected

– No interference, diffraction, near field effects ...

• Geometrical optics may be derived from wave optics for wavelength approaching zero

• Direction of ray propagation is direction of wavevectors

4.8

βα

1n 2n1 2sin sinn nα β⋅ = ⋅

1 2n nα β⋅ = ⋅

Paraxial approximation / Gaussian optics• For propagation at small angles relative to optical axis simplified description of

optical parts is possible and analytical solutions exist.

4.9

Spherical surface in paraxial approximation• Approximated

• For rays parallel to optical axis

sn

rnnns−−′′

≈′

2θ1θ

O O’r

s s’

n n’

ϕ

A

S C

rnn

ns−′′

≈′

4.10

Thick lens with spherical surfaces• Calculation of optical power of thick lens• Optical power

– [D]=m-1=Diopter=dpt

fnD −=

d

n=1 nL

B1=G2B2=F

4.11

Thick lens: Focal point

d

n=1 nL

B1=G2B2=F

s1‘

11 1' r

nnsL

L

−≈

11' 1

112 −+−

=−−

=−=L

LL

L

L

nddnrndr

nndss

( )( )

( ) ( ) ( )[ ]11111

11'

12

12

1222

2 −+−−+−

=

+−−+−

=+−

=LLL

LL

LL

LLLLL ndrrnnddnrnr

ddnrnnn

rn

sn

rns

s2‘

s2

• Focal length is distance from principle point to focal point

4.12

Front principle plane• Front principle plane: Locus of intersection of diverging/converging bundle of

rays for parallel outgoing rays– Front principle point H: Intersection of front principle plane with optical axis

Source: Schröder/Treiber, Technische Optik

Positive (converging) system Negative (diverging) system

4.13

Rear principle plane• Rear principle plane: Locus of intersection of parallel ingoing rays with

diverging/converging outgoing rays– Rear principle point H’: Intersection of rear principle plane with optical axis


Positive (converging) system Negative (diverging) system

4.14

Thick lens: Focal length and optical power

f

n=1 nL

B1=G2B2=F

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

21

12

12

21

2

21

1111

''

rrnndrrnnD

ndrrnnrrn

sssf

L

LLL

LLL

L

−+−−−=

−+−−==

H‘

4.15

Possible lens shapes


4.16

Special case: Plano-convex and Plano-concave lenses• For plane surface

• This results in

– Focal length independent of d!


11

−=

Lnrf

∞→2r

4.17

Special case: Spherical lens• For spherical surface

• This results in


21r

nnfL

L

−=

rdrrr 221 =−==

4.18

Special case: Thin lens• For negligible d

• This results in

• For thin lenses distance between principle planes may be neglected

( )( )

( )

L

L

L

L

nndHH

rrn

f

rrnrrf

1'

1111

1

21

12

21

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

−−=

( ) 121 rrnnd LL −<<−

4.19

Notation for imaging surfaces


4.20

Equivalent lens• System with several stages (e.g. objective with several lenses) may be

described by optical quadrupole (“Equivalent lens“)– Normally distance of individual systems measured between H1‘ und H2

– In case of microscopes tube length between focal points used


4.21




4.22

2θ

1θ

z

x

1x 2x

A B 1i

i

xs

θ⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )1 1,x θ ( )2 2,x θOptical system

2 1s s= ⋅12M

Matrix optics / ABCD-matrices

• Propagation of ray at point A completely described by distance x to optical axis and angle θ relative to optical axis

4.23

[ ]5 44' 3'4 12 1...s s= ⋅ ⋅ ⋅ ⋅M M M

2θ1θ

z

x

1x 2x 3x 4x

3θ

4θ

22'M 33'M 44'M12M 2'3M 3'4M

Eingang i Ausgangi

s s⎡ ⎤= ⋅⎢ ⎥⎣ ⎦∏M

Optical system with several components

• Effect of total system Output Input

4.24

2 1 1

2 1 1

10 1

x x Lx

θθ θ

= ⋅ + ⋅= ⋅ + ⋅

2 1 1

10 1 Freiraum

Ls s s⎛ ⎞= ⋅ = ⋅⎜ ⎟⎝ ⎠

M

2θ

1θ

z

x

1x 2x

A B

1. Matrix of translation

L

Translation

4.25

2 1 1

12 1 1

2

1 0

0

x xnxn

θ

θ θ

= ⋅ + ⋅

= ⋅ + ⋅

2 1 11

2

1 0

0 EbeneFläches s snn

⎛ ⎞⎜ ⎟= ⋅ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠

M

2θ

1θ

z

x

1x 2x1x1n 2n

2. Refraction on plane interface

Plane interface

4.26

2 1 11 2 1

2 2

1 0

SphärischeFläches s sn n nn nρ

⎛ ⎞⎜ ⎟= ⋅ = ⋅−⎜ ⎟⎜ ⎟⎝ ⎠

M

2θ1θ

z

x

1x 2x1x ρ

1n 2n

0ρ >

3. Refraction on spherical interface• Snellius at interface• Angle depends on radius

– Convention for curvature and propagation

Spherical interface

4.27

Exercise: Thin lens• All important simple optical elements may be described using the tree

matrices (Translation, plane interface and spherical interface)– Thin lens (convex, concave)– Thick lens (convex, concave)– Mirror (plane, focusing)

• Derive the total matrix for a thin lens! • Derive the dependence of the focal length f and the interface curvatures (Lens

maker's equation for thin lens)!

• Derive the matrix for imaging with a convex lens!• Derive the imaging equation from the resulting matrix!

– Consider that x2 has to be angle-independent

4.28

( ) ( ) ( )2 2 2 1 1 1 2 1, ,SF FR SFs n n L n n sρ ρ= → ⋅ ⋅ → ⋅M M M

z

x 1n2n1n

f f

2h1h

Thick lens• Combination of two spherical interfaces and translation

– Focal length calculated from principal plane

L

SI SITr

4.29

12

A BC D⎛ ⎞

= ⎜ ⎟⎝ ⎠

M

112

2

det nAD BCn

= − =M

Matrix properties in Gaussian optics

• Because of Snellius law:

– Only 3 of the 4 matrix elements may be chosen independently

4.30

2 10A x Bθ= ⇒ =Focusing: 0 B

C D⎛ ⎞⎜ ⎟⎝ ⎠

2 10B x Ax= ⇒ =Optical imaging: 0A

C D⎛ ⎞⎜ ⎟⎝ ⎠

2 1Dθ θ=Deflection of parallel rays:

0A B

D⎛ ⎞⎜ ⎟⎝ ⎠

2 1Cxθ =Generation of parallel rays:

0A BC⎛ ⎞⎜ ⎟⎝ ⎠

Optical elements and their system matrices

4.31

Matrix optics for Gaussian beams and polarization• Matrix optics not limited to Gaussian optics, but may be used for Gaussian

beams as well. – ABCD-Matrices identical– Instead of ray vector s beam parameter q used:

• Polarization may be included as well in matrix algorithm– Jones-vector describes state of polarization– Jones-matrices describe optical elements

( )DCqBAqzq

++

=0

02

11w

iRq π

λ−=

4.32




4.33

Sequential ray-tracing• Rays propagates through optical elements in sequential order• Used for design of imaging optical systems

– Microscope, telescope, camera ...

Source: http://optics.org/dl/2006/OLEJulAugPRODUCTGUIDE.pdf

4.34

Non-sequential ray-tracing• Rays split on surface and may pass optical elements several times• By propagation of Gaussian beams wave phenomena may be included• Used for design including stray light calculation

– Stray light calculation– Back lights, luminaires ...


4.35

FDTD: Finite Difference Time Domain• Solution of Maxwell-equations using space and time grid• Used for micro- and nanosystems

– Integrated optics, photonic crystals, plasmonics ...


4.36

Common optical design software

– At the university additionally COMSOL Multiphysics


4.37

Course outline1. Imaging optics2. Optical sensors

2.1 Spectroscopy2.2 Material characterization2.3 Distance measurement2.4 Angle measurement2.5 Optical mouse

3. Optics in data storage4. Introduction to displays5. Fourier optics6. Diffractive optics and holograms7. Integrated optics8. Computerized imaging

4.38

SpectroscopySpectroscopy:

Frequency or wavelength analysis of light

Applications of spectroscopy– Material characterization– Sensors– Development of light sources

Example: Organic light emitting diode

inte

nsity

/ ar

b. u

nits

4.39

SpectroscopyOptical spectrum characteristic for chemical elements

– Known since appr. 1859 (Kirchhoff, Bunsen), e.g.:

– e.g.: Fraunhofer-lines in solar spectrum (1802, 1814) (Absorption spectra)

In general:

– Glowing solid or liquid objects have continuous spectra (e.g. thermal radiation, black body, Planck's law)

– Glowing gases or vapors have discontinuous spectra(Transition between discrete energy levels in atoms/molecules)

Source: Wikipedia and Gerthsen Physik

4.40

http://www.technoteam.de/

Spectrometer using absorptive filters

• Spectrally broad CCD-chip is illuminated through sequence of filters

4.41

Spectrometers using dispersive elements

• Goal: Spatial separation of light of different wavelengthse.g., on a screen or a CCD-chip

• Use dispersive elementMaterial or structure whose effect on light is wavelength-dependent (e.g. prisms, grating)

• Common types of spectrometers sorted by dispersive element– Prism spectrometer uses dispersion due to refraction – Grating spectrometer uses dispersion due to diffraction

• Common types of spectrometers sorted by functionality– Monochromator selects small wavelength interval– Spectrometer for observation of large wavelength interval– Spectrograph Spectrometer + CCD-camera / film etc.

4.42

Material dispersion• Refractive index depends on type of glass and on wavelength dependent

Source: http://en.wikipedia.org

4.43

Prism spectrometer – Refraction in prism

Minimum deflection angle for symmetric pass:

Dispersion:

Deflection angle:

Angular dispersion

Source: Wikipedia and „Optik Licht und Laser“ von Dieter Meschede

4.44

Prism spectrometer

Advantage: Unambiguous correlation between wavelength and position in image planeDisadvantage: Small dispersion and thus small resolution

Bild 16.3.1 ausNaumannSchröder

Basis of prism

Telescope

Source: „Bauelemente der Optik“ von Naumann und G. Schröder

4.45

Interference – special caseInterference: Superposition of waves with fixed phase relation.Interference is due to wave character of light and cannot be understood using ray

optics.

Example: Two monochromatic waves of identical polarization and identicalamplitude are superposed at position

Principle of superposition for fields.

Irradiance (intensity) at point is:

Spatially modulated interference pattern with minima and maxima

4.46

Interference – general caseGeneral case:

Superposition of fields:

Irradiance (intensity) obtained as temporal average of electrical field, i.e. square of field amplitude A

Interference term results as

Interference depends

on polarization

Only for complex field descriptionDivisor 2 from averaging

4.47

Constructive / destructive interferenceInterference term

Phase difference

Path difference

Constructive interference / Maximum of intensity for

Destructive Interference / Minimum of intensity for

Difference of opt. paths of two partial waves

Constant term

4.48

Grating principle• Interference of spherical waves exiting different slits• Each slit also causes diffraction

• Formula for N slits:

Interference term Diffraction term

Source: Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)

4.49


Grating - intensity with angle• Main maxima for constructive interference of all wave parts

Example:N = 8,d = 2b

Diffraction of slit

1. order interference maximum

4.50

Grating equation• Maximum intensity at:

dm

mλθθ ±=− 0sinsin

Source: Wikipedia and Optics lecture by PD Dr. Seifert, Universität Halle (http://www.physik.uni-halle.de/)

Grating

Effective grating size

Image of light bulb through transmission

grating

4.51

Exercise: Grating period

• Determine the grating period of the given grating!

4.52

Grating ambiguity• Gratings are ambiguous!

Source: http://www.jobinyvon.com

4.53


Grating – width of maximum• For more illuminated slits maximum is narrower

4.54


Grating resolution• Two wavelengths are resolved if

they are at least separated by width of main maximum

Resolution in order m for grating for N illuminated slits

4.55

Types of gratings• Amplitude grating

Slits as origin of fundamental waves.

• Phase gratingModulation of optical path ! Modulation of phase

• Reflection gratingoften blazed

4.56

Blazed gratings• In Littrow configuration groove have such profiles that incident and diffracted

rays are in auto collimation (i.e., α = β). – Efficiency of grating increased

• In this case at the "blaze" wavelength λB we obtain:

Source: http://www.jobinyvon.com

dm Bλα

αβ

=

−=

sin2

4.57

Czerny-Turner monochromator• Light leaving the exit slit (G) contains entire image of entrance slit of selected

color plus parts of entrance slit images of nearby colors. • Rotation of dispersing element causes band of colors to move relative to exit

slit

Source: http://en.wikipedia.org

mdm

dd

θλθ

cos=

4.58

Échelle spectrographs • Prism for dispersion in one direction• Grating for dispersion in orthogonal direction

Source: http://www.andor.com

Spectrograph with standard prism

Spectrograph with Andorpatented prism

4.59

Resolution at 550nm

4 1

4 1

1,79260 , 60

2 104,5 10

12000

nb mmdn d nmd d nm

α

λ

δ λλλ

− −

− −

== = °

= ⋅

= ⋅

=∆

4 116 10

72000

d d nmα λλλ

− −= ⋅

=∆

Comparison grating / prism spectrometer• Reflection grating

– 1200 rules/mm– Width 60mm

• Prism– Glass SF11

4.60

Exercise: Spectroscope

• Build the spectroscope!

• Observe and sketch the spectrum of an incandescent source and a fluorescent lamp!– What influence does the entrance slit have?

4.61

Compilation of questions• When may Gaussian optics be used?• How are the principle planes of a thick lens defined?• What is a thin lens?• What are ABCD-matrices and what are they used for?• What is the difference between sequential and non-sequential raytracing?• How does a spectrometer using absorptive filters work?• Sketch a prism spectrometer!• What is constructive interference?• Sketch the outgoing intensity as a function of angle for a grating!• Why is the wavelength determination with a grating ambiguous?• What limits the resolution of a grating monochromator?• Sketch a grating spectrograph!

Documents

2.3 Aberrations 2.5 Depth of focus Optical Engineering