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 =
reflection and refraction linearly polarized light
Fresnel’s amplitudes:
for light passingfrom optically thinner Into optically densermedium
only Brewster’s angle is present
note that atBrewster’s angle : Rp=0
reflection and refraction linearly polarized light
reflection and refraction linearly polarized light
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)
polarisation of light
polarisation of light
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
polarisation of light
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
polarisation of light
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
polarisation of light
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
polarisation of light
interference of light
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
interference of light
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
interference of light
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
interference of light
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.
interference of light
(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
(elastic) scattering of 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
(elastic) scattering of light
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