applied optics 1 - Masaryk University

applied optics 1

mgr. dušan hemzal , ph.d.

web: http://www.physics.muni.cz/~hemzal/vyuka/optometry

applied optics 1:

properties of light

geometrical optics

optical path, Snell’s law

critical angle, Brewster angle

polarisation and intensity of light

Fresnel’s amplitudes

polarisation types of light

birefringence

optical anisotropy

polarisation microscope

interference

coherence, Michelson’s interferometer

diffraction

diffraction of light for basic screens

scattering of light

diffuse scattering, Rayleigh scattering,

theory of Mie, Tyndall effect

applied optics 2:

application of light

HRT, OCT,

GDx, WASCA,

endotellium microscope

bright field/dark field microscopy

phase contrast, Nomarsky contrast

genuscrozonaspis

(400 mil. BC.)

genusdalmanitina

abathochroal eye70 lenses

schizochroal eye700 lenses

holochroal eye15 000 lenses

Descartes

(XVII. century)

Huygens

geometrical optics

LIGHT = electromagnetic waves

sound: waves od pressure (scalar)longitudinal, (can be transverse in solids)

electromagnetic filed: waves of coupled electric and magnetic fieldalways transverseruled by Maxwell’s equations

relation of the quantites allows to treat onlyEkHErrrr

⊥⊥⊥ Er

consider a plane perpendicular to direction of propagation of light: there are two components, Ex, Ey,

x

y

Er

xE

yE

ϕ2

Er

222

yx EEE +=r

ϕcosEEx

r

=

ϕsinEEy

r

=

intensity of light :

plane wave

spherical wave

Intensityof light

types of elmag waves

also: cylindrical wave and other (only approximative solution to Maxwell’s equations)

222

yx EEEI +=≈r

( )xktEEr

rr

−= ωcos0

( )krtr

EE −= ωcos0r

konstEI == 20

2

20

r

EI =

in contrast to intensity of electric field E

propagation of light

Fermat’s principle:between two fixed points, light traverses in shortest possible time

optical path δ is defined through the geometric path length d, weighted by index of refraction on the medium n:

consequences:

- in homogeneous medium , light propagates straight lines

- at interface , angle of reflection = angle of incidenceSnell’s law holds for refraction

nd=δ

the Fermat principle can be reformulated to require shortest optical path

2211 sinsin αα nn =

1α 1α

2α

for subsequent media, the optical path is additive: ...2211 ++= dndnδ

Interesting points of Snell’s law:

critical angle(and total reflection)

propagation of light through inhomogeneous media:numerical analysis only (rays are bent) via series of thin homogeneous slabs

2211 sinsin αα nn =

1n

21 nn <

1

2sinn

nc =α

1

2tann

nc =α

application: optical fibres

Brewster’s angle application: polarization of light

propagation of light

Fresnel’s amplitudes

reflection and refraction linearly polarized light

Fresnel’s formulas

either for electric field: r, tor for intensity of light: : R=r2, T=t2

RI

I

i

R = TI

I

i

T =

conservation of energy requires(disregarding absorption):

iTR III =+

1=+TR

iθtθ

situation at planar interface of two homogeneous media:

general incoming light can de decomposed always decomposed to s- and p- components :

as electric field must be perpendicular to direction of propagation, the vector of electric intensityhas a component in the plane of incidence and perpendicular to it

pE

sE

Ep component within the plane of incidence is called p-polarised (parallel)Es component perpendicular to the plane of incidence is called s-polarised (senkrecht):

2

21

21

coscos

coscos

+−=

ti

tis nn

nnR

θθθθ

2

21

21

coscos

coscos

+−=

it

itp nn

nnR

θθθθ

Fresnel’s formulas: amount of light reflected/refracted through the interface in the two polarisations:

using the Snell’s law, huge number of formulas can be obtained: ti nn θθ sinsin 21 =


Fresnel’s amplitudes:

for light passingfrom optically thinner Into optically densermedium

only Brewster’s angle is present

note that atBrewster’s angle : Rp=0



Fresnel’s amplitudes:

for light passingfrom optically denser Into optically thinnermedium

both Brewster’s angle and critical angleare present

polarisation of light

-

linearly polarised light

electric field is perpendicular to direction of propagation of light

componetns Ex and Ey are in phase(or in anti-phase)

the endpoints of E lie on a straight line(it’s tangent is given by ratio of Ex and Ey)

elliptically polarised light :

electric field is perpendicular to direction of propagation of light

components Ex and Ey oscillate with general phase shift (π/2 in the image)

the endpoints of E lie on an ellipse(it’s tilt and aspect is given by ratio of Ex and Ey)

(for Ex = Ey we get circularly polarised light )(for phase shift π/2 the ellipse is axial)



common emission produces linearly polarized photons, but polarization of individual photonsis not correlated in any way

unpolarised light (daylight):

Each photon is produced independently of others, so within a light beam, all possible orientations of polarization are present in practically equal amounts.We call the light randomly polarized, although each photon itself keeps its polaristion statestrictly fixed (until further interaction with matter).

linearly polarized light: Ex and Ey are in phase

-

( )δω +−= zktEE zy cos0( )zktEE zx −= ωcos0

The trajectory of the electric vector endpoint is obtained upon exclusion of : zkt z−ω

δδ 2

00

2

0

2

0

sincos2 =−

+

E

E

E

E

E

E

E

E yxyx

this curve represents an ellipse (circle) or straight line

– depending on the value of δ

0=δ :00 E

E

E

E yx = πδ = :00 E

E

E

E yx −=

the electric filed components of a light wave oscillate harmonically:(for simplicity we consider components of same magnitude)

2

πδ ±= : 12

0

2

0=

+

E

E

E

E yx

linearly polarized light,

with inclinationx

y

E

E=ϕtan

circularly polarized light

right- or left- , depending on value of δ


linear polariser – only light with single specified direction of polarization is passed through

this direction is designated orientation of polariser

The action of the polarizer can be constructed by projecting Ex a Ey into

direction of polarizer orientation and subsequent addition of the produced vectors

pr

Er

ϑϕ

xE

yE

x

y

During construction the oscillation

of the electric field is not taken

into account –

only the full magnitudes

are projected


pr

Er

ϑϕ

xE

yE

x

y

in the simple case of linear

polarisation, the full vector E

can be projected directly

in case of elliptic polarization

the two components must be

projected separately


linear polariser – only light with single specified direction of polarization is passed through

this direction is designated orientation of polariser

The action of the polarizer can be constructed by projecting Ex a Ey into

direction of polarizer orientation and subsequent addition of the produced vectors

law of Malus – describes passage of daylight light through two subsequent polarisers

centered on a common axis and rotated by angle :ϑ

ϑ20 cosII =

elliptically polarized light cannot be eliminated by a single polariser:

In every instant of time at least one of the components Ex and Ey is nonzero, and, hence,

for any orientation of polarizer some light always passes through

(this fact is used in polarisation microscope for observation of the birefringent samples)

linear polarisation in nature – atomic emission

elliptic polarisation in nature – microscopic structures on beetles elytra

– passage through anisotropic medium


interference of light


δ is the phase shift between the waves (of same polarisation)

Interference of two monochromatic light waves:

visibility of interference fringes:

coefficient describes the level of coherence of the two waves

in nature, interference of daylight appears only when the objects, that interactwith light, have dimensions comparable with wavelength of the light in question

more precisely, for interference to occur between two photons, there mustbe a correlation between their phases. To describe it, we introduce the notionof coherence .apart from using thin media, coherence can be reached using eg. lasers as sources of light

In these cases, the common formula for addition of two sources of light, I=I1+I2 ,ceases to be correct. Instead of addition, we talk about interference of light.

Comparison to birefringence, when also two light waves are created:

the ordinary and extra ordinary rays

have the same frequencypropagate in the same directioncan have comparable magnitudeshave perpendicular polarization

In result, no interference is produced. On the cintrary, elliptically polarized light appears.

conditions of coherence: the two waves

- have the same frequency (wavelength/energy)- propagate in the same direction- have the same direction of polarization- have the same magnitude

The more we fail to comply with the above requirements,the less interference visibility we gain


two plane waves of same intensity I0=E02 and phase shift δ

example: Young’s experiment with two slits

2cos4 2

0

δII =

a

for the waves just add upKπδ 2,0=for the waves cancel each otherKππδ 3,=

ϑ

d∆

ϑ

the phase shift between the slits light wavesdepends on the angle towards the screen:

ϑsinad =∆

Hence, the interference maxima and minima alternate at the screen:

maximum:

minimum:

λϑ mnadn ==∆ sin

( )2

12sinλϑ += mna KK ,2,1,0,1,2 −−=m

example: thickness of a layer (bubble wall, oil film) that produces a colored interference is about half the wavelength of the observed color


in simple cases, the interference can be evaluated directly, consider eg.

Michelson interferometer

used for creation of two coherent light waves by splitting the incoming light.length of one of the arms can be changed to tunethe phase shift between the wavesat the output, interference fringes are seen

thus, sacrificing the possibility to see the image of the object directly, details with size of fraction of the wavelength can be studied

the beam splitter is metalised to achievesplitting with aplitude ratio 1:1

the (optional) phase compensator can be used to balance the light passage in both arms

in applications, one of the mirrors is replaced by sample; the sample morphology is revealed as changeof sample surface distance from beam splitter changesthe phase difference between the arms

(gradual) change of the phase difference results in shifting the interference fringesby counting the maxima (or minima) passed across the image the distance can bemeasured as


wave packets

real light cannot actually be described by monochromatic plane wave. rather, every photon seems to be a mixture of many plane waves with neighbouring wavelengths:

in this way, the so called wave packet is constructed, which propagates through space

It turns out, that including the nearby wavelengths, the wave packet is limited in spaceand its (coherence ) length can be given as

interference in the language of wave packets

description of interference gets extremely simple using the notion of wave packets:

two photons will interfere, if their wave packets s ucceed to meet in space and time.

the interference will be strong if the packets overlap significantly and vice versa. this type of coherence is called the time coherence, as timing of the packets is essential.


(elastic) scattering of light

diffuse scattering- most common type of scattering, depends on surface morphology and

bulk inhomogeneities in the material

Mie scattering – occurs at spherical particles dispersed in otherwise homogeneous medium

Rayleigh scatteringparticles are much smaller than wavelength

Tyndall scatteringparticles of irregular size, usually prolongate,with size comparable to wavelength of imapctin light


- realized by high number of uncorrelated light reflection at surface and bulk inhomogeneities

- usually unwanted, as all-direction light is producedthat hampers the optical image and causes losses of light intensity

application:

- homogenisation of light intensitywithin light sources

- endotellium microscope

examples of good diffuse scatterers includeobjects with sharp edges, such as scratches etc


Rayleigh scattering

40

1

λ≈

I

I

- scattering over small spherical particles

- photons with shorter wavelengths are scattered with higher efficiency:

ukazy.astro.cz

this is the explanation of blue color of the sky (it should, in principle be violet, but spectral profile of sunlight comes into question)

- the scattering takes place at N2 a O2 molecules

- water droplets within clouds are already to bigand scatter all wavelengths equally (clouds are grey)

upon Rayleigh scattering, the light is - scattered to all directions, with preference of the

original direction- partially polarized, especially in the perpendicular direction

Tyndall scattering

scattering over small, but prolongated particles

- also effective with λ4 inverse, butstronger than Rayleigh scattering

- a typical Tyndall cone is produced

occurence: generally colloid solutions,smoke, milk

note that, due to sun distance, the light enters the opening as parallel beam – there is no reason for occurrence of the diverging cone of illumination apart from Tyndall scattering

Tyndall scattering

the iris stroma contains collagen and elasticfibers.

while brown colour of eyes is caused by the presence of melanin pigment,blue colour of eyes is due to Tyndall scatteringat the fibers within iris stroma

- 15. chromozome, genes OCA (P protein) and HR2

- OCA malfunction leads to albinism - mutation of HR2 causes lesser deposition of melanin

within eyes by affecting the function of the P protein

- P protein transports tyrosin to the cells- tyrosin is precursor of melanin

- HR2 mutation can be tracked to a single common ancestorof blue-eyed people, some 10 000 years ago

tyrosin

eumelanin polymer

applied optics 1 - Masaryk University

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