basic optics - vutbr.czphysics.fme.vutbr.cz/~jirka/fp/l_02_basic_optics.pdf · 2017. 2. 20. ·...
TRANSCRIPT
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Basic optics
• Geometrical optics and images
• Interference
• Diffraction
• Diffraction integral
we use simple models
that say a lot!
more rigorous approach
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Basic optics
• Geometrical optics and images
• Interference
• Diffraction
• Diffraction integral
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Images
Édouard Manet: A Bar at the Folies-Bergère (1882)
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http://epod.usra.edu/blog/2012/10/inferior-mirage-on-a-desert-road.html
Mirage
light rays are bent to
produce a displaced image
of distant objects or the sky
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Plane mirrors
convention:
- light entering from the left
- positive distances: O, I on the left
- real image: i > 0
- virtual image: i < 0
object distance image distance
Point objects
(here, we have
virtual image)
Mirror
ImageObject
p > 0 i < 0
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• virtual image
• the same orientation and
size (height) as object
Extended objects
Plane mirrors
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Head
Eye
Foot
Mirror
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r, f > 0
r, f < 0
Spherical mirrors
convex
concaveReal
focus
Central axis
Virtual
focus
Central axis
Focal lengthRadius
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Axis
Mirror
Image formation
Spherical mirrors
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4 raysRay tracing
Spherical mirrors
- real image: i > 0
- virtual image: i < 0
Concave mirror:
p > f : real image
p = f : image at infinity
p < f : virtual image
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Spherical mirrors
4 raysRay tracing
- real image: i > 0
- virtual image: i < 0
Convex mirror:
image is always
- virtual
- erect
- minified
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Magnification m
erect image: m > 0
inverted image: m < 0
Spherical mirrors
- real image: i > 0
- virtual image: i < 0triangles ABV and DEV are similar
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Spherical refracting surfaces
- real image: i > 0
- virtual image: i < 0
r < 0
r < 0
r < 0
r > 0
r > 0
r > 0
Real
image
Real
image
Virtual
image
Virtual
image
Virtual
image
Virtual
image
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Axis
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Thin lens
0 (thin lens)(thick lens)
for both refracting surfaces
Air
Glass
Axis
p i
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Thin lens
< 0
f > 0
> 0
> 0< 0
f < 0
converging lens
diverging lens
Extensions
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f > 0
3 rays
Ray tracing: converging lens
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Ray tracing: diverging lens
f < 0
3 rays
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diverging lensconverging lens
Thin lens (bottom line)
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Thin lens: magnification m
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Simple magnifier
angular magnification:
To distant virtual image
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Compound microscope
Objective
Eyepiece
Parallel
rays
To distant virtual image
The lateral
magnification produced
by the objective lens
The overall
magnification
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Refracting telescope
Objective
Eyepiece
Parallel
rays
To distant
virtual image
Parallel
rays from
distant
object
(angular magnification
of the telescope)
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Aberrations (image errors)
- aberrations can be balanced
- image fidelity is limited only by diffraction
examples
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Basic optics
• Geometrical optics and images
• Interference
• Diffraction
• Diffraction integral
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Interference
What will happen if we add waves?
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Double-slit experiment (Young’s experiment, 1801)
Incident
wave
An interference
pattern
Superposition
of waves
u – a suitable component
of E- or H- vector
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assume
Incident
wave
Path length difference
(maxima)
(minima)
Different phases due to
different paths
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(maxima)
(minima)
(two coherent sources)
(two incoherent
sources)
(one source)
(m for maxima)
(m for minima)
Inte
nsi
ty
Intensity in double-slit experiment
(missing – sign)
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particles
http://www.feynmanlectures.caltech.edu/III_01.html
waves
Double-slit experiment with ...
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Interference from thin films
Ray reflected at A
Ray reflected at B
Incident wave
For simplicity we assume
1. Normal incidence
2. Double beam interference
Phase difference
phase shift arising
from reflection at Bphase shift arising
from reflection at A
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Ray reflected at A
Ray reflected at B
Incident wave
Example: air - n2 - air
(maxima)
(minima)
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(maxima)
(minima)
Example: air - n2 - air
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(Newton rings)
Example: glass - air - glass
Incident
light
Glass
Glass
Air
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Temporal coherence
coherence length coherence time
Define:
monochromatic wave -
- perfectly coherent
pulse (wave-packet) -
- less coherent
white light -
- incoherent
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Interference and temporal coherence
coherence length coherence time
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Interference and temporal coherence
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Spatial and temporal coherence
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Michelson interferometer
Movable
mirror
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incident wave
...
...
transmitted
wave
reflected
wave
Interference from thin films (again)
Now we consider
1. Arbitrary incident angle
2. Multiple beam interference
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incident wave
...
...
transmitted
wave
reflected
wave
geometrical series
The amplitude of the resultant transmitted wave
Interference from thin films
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incident wave
...
...
transmitted
wave
reflected
wave
geometrical series
The amplitude of the resultant reflected wave
Interference from thin films
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incident wave
reflected wave transmitted wave
For simplicity assume symmetric structure
and pure real numbers
Interference from thin films
(relative transmitted intensity)
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integer
Spectral response (thin film, FP etalon)
(relative transmitted intensity)
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The free spectral range, FSR
Spectral response (thin film, FP etalon)
(relative transmitted intensity)
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with loss
Spectral response (thin film, FP etalon)
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Spectral analyzer
0
const.
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Basic optics
• Geometrical optics and images
• Interference
• Diffraction
• Diffraction integral
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Huygens-Fresnel principle
Every point of a wavefront at a given instant in time,
serves as a source of spherical secondary waves. The
amplitude of the optical field at any point beyond is
the superposition of all these wavelets.
A wavefront
at t = 0
The new wavefront
at t = ∆t
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Huygens-Fresnel principle
Every point of a wavefront at a given instant in time,
serves as a source of spherical secondary waves. The
amplitude of the optical field at any point beyond is
the superposition of all these wavelets.
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Diffraction
Every point of a wavefront at a given instant in time,
serves as a source of spherical secondary waves. The
amplitude of the optical field at any point beyond is
the superposition of all these wavelets.
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Diffraction
Every point of a wavefront at a given instant in time,
serves as a source of spherical secondary waves. The
amplitude of the optical field at any point beyond is
the superposition of all these wavelets.
Incident
wave
Diffracted
wave
Screen
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Diffraction from a single slit
z
x
z
radiates a wavelet
The superposition of all these wavelets:
a source at x
x
?
some “constant”
Incident
wave
Screen
Path length difference
in the Fraunhofer region (far-field region)
Amplitude of diffracted wave
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Fourier transform and diffraction
The amplitude of diffracted wave is proportional
to the Fourier transform of the field distribution
across the aperture ( = the aperture function).
z
x
?Incident
wave
Screen
aperture function
(we will prove it later)
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... back to diffraction from a single slit
z
x
?Incident
wave
Screen
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Diffraction from a single slit (results)
z
x
?Incident
wave
Screen(minima)
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Diffraction from a single slit (results)
(minima)
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Diffraction from a circular aperture
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diameter
Airy rings
first minimum
Diffraction from a circular aperture
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Resolution of imagining systems
Rayleigh’s criterion for the minimum
resolvable angular separation
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Diffraction from a double slit
z
x
substitution
Diffraction factor – due to the
diffraction by a single slit
Interference factor – due to the
interference between two slits
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Diffraction factor – due to the
diffraction by a single slit
Interference factor – due to the
interference between two slits
Diffraction from a double slit
diffraction by
a single slit
interference
between two slits
diffraction from
a double slit
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Diffraction factor – due to the
diffraction by a single slit
Interference factor – due to the
interference between two slits
Diffraction from a double slit
diffraction by a
single slit
interference fringes for
a double slit system
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Diffraction gratings (multiple slits)
(maxima)
Path length difference
(grating orders)
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Diffraction gratings (multiple slits)
Path length difference
Diffraction factor – due to the
diffraction by a single slit
Interference factor – due to the
interference from N slits
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Diffraction gratings (multiple slits)
Diffraction factor – due to the
diffraction by a single slit
Interference factor – due to the
interference from N slits
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X-ray diffraction
(Bragg’s law)
Incident
x rays
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Davisson, C. J., "Are Electrons Waves?,"
Franklin Institute Journal 205, 597 (1928)
Electron diffraction
(Bragg’s law)
Incident
electron beam
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Basic optics
• Geometrical optics and images
• Interference
• Diffraction
• Diffraction integral
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Angular spectrum representation
arbitrary wave = superposition of plane waves
=
in homogeneous medium
+z
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Angular spectrum representation (more details)
Plane wave
Wave function:
real complex
Superposition of plane waves:
For
(IFT)
(FT)
we choose + sign, i.e., we
assume propagation in +z
(possible reflections are
neglected)
for EM waves – scalar
approximation
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Propagation of waves
+z
?
paraxial approximation
known
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Propagation of waves
paraxial approximation
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(calculation of the integral)
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Fresnel-Kirchhoff diffraction formula
Fraunhofer approximation:
only for
+z
?known
Diffraction integral