Upload
leminh
View
217
Download
0
Embed Size (px)
Citation preview
April 03LASERS 51
Diffraction• Interference with more than 2 beams
– 3, 4, 5 beams– Large number of beams
• Diffraction gratings– Equation– Uses
• Diffraction by an aperture– Huygen’s principle again, Fresnel zones, Arago’s spot– Qualitative effects, changes with propagation distance– Fresnel number again– Imaging with an optical system, near and far field– Fraunhofer diffraction of slits and circular apertures– Resolution of optical systems
• Diffraction of a laser beam
April 03LASERS 51
Interference from multiple aperturesBright fringes when OPD=nλ
dnLx λ
=source
two slitsscreen
OPD
d
L
x
Complete destructive interference halfway between
40
position on screen
Inte
nsity
source threeequally spacedslits screen
d
OPD 1
OPD 2 40
position on screen
Inte
nsity
OPD 1=nλ, OPD 2=2nλall three waves interfere constructively
OPD 2=nλ, n oddouter slits constructively interferemiddle slit gives secondary maxima
April 03LASERS 51
Diffraction from multiple apertures• Fringes not sinusoidal for
more than two slits• Main peak gets narrower
– Center location obeys sameequation
• Secondary maxima appearbetween main peaks– The more slits, the more
secondary maxima– The more slits, the weaker the
secondary maxima become• Diffraction grating – many slits, very narrow spacing
– Main peaks become narrow and widely spaced– Secondary peaks are too small to observe
2 slits
3 slits
4 slits
5 slits
April 03LASERS 51
Reflection and transmission gratings
• Transmission grating – many closely spaced slits• Reflection grating – many closely spaced reflecting regions
Inputwave
Huygenswavelets
screen
path length toobservation point
opaque
transmittingopening
Inputwave
wavelets
screen
path length toobservation point
absorbing reflecting
Transmission grating Reflection grating
April 03LASERS 51
Grating equation – transmission grating with normal incidence
• Θd is angle of diffracted ray• λ is wavelength• l is spacing between slits• p is order of diffraction
Θdinput
Diffracted light
lp
dλθ =sin
Except for not making a small angle approximation, this is identical to formula for location of maxima in multiple slit problem earlier
April 03LASERS 51
Diffraction gratings – general incidence angle
• Grating equationlp
idλθθ =− sinsin
l=distance between grooves (grating spacing)
Θi=incidence angle (measured from normal)
Θd=diffraction angle (measured from normal)
p=integer (order of diffraction)
Θi
Θd
• Same formula whether it’s a transmission or reflection grating– n=0 gives straight line propagation (for transmission grating) or
law of reflection (for reflection grating)
April 03LASERS 51
Intensities of orders – allowed orders• Diffraction angle can be found only for
certain values of p– If sin(Θd) is not
between –1 and 1,there is no allowed Θd
• Intensity of other ordersare different dependingon wavelength, incidence angle,and construction of grating
• Grating may be blazed to makea particular order more intense thanothers– angles of orders unaffected by blazing
inputbeam
strong diffractedorder
weak diffractedorder
Blazed grating
April 03LASERS 51
Grating constant (groove density) vs. distance between grooves
• Usually the spacing between grooves for a grating is not given – Density of grooves (lines/mm) is given instead–– Grating equation can be written in terms of grating
constant
lg 1
=
( ) λpgid =Θ−Θ sin)sin(
April 03LASERS 51
Diffraction grating - applications
• Spectroscopy– Separate colors, similar to
prism
• Laser tuning– narrow band mirror– Select a single line of
multiline laser– Select frequency in a
tunable laser
• Pulse stretching and compression– Different colors travel
different path lengths
grating
Θ
Littrow mounting – input and output angles identical
( )dλ
=Θsin2
grating
1storder
2ndorder
negativeorders
two identicalgratings
April 03LASERS 51
Fabry-Perot InterferometerInput
Reflected field
Transmitted
field
Partially reflecting mirrors
transmitted through first mirror Beam is partially reflected and
partially transmitted at each mirror
All transmitted beams interfere with each other
All reflected beams interfere with each other
OPD depends on mirror separation
• Multiple beam interference – division of amplitude– As in the diffraction grating, the lines become narrow as
more beams interfere
April 03LASERS 51
Fabry-Perot Interferometer
• Transmission changes with frequency– Can be very narrow range where transmission is high
• Width characterized by finesse• Finesse is larger for higher reflectivity mirrors
– Transmission peaks are evenly spaced• Spacing called “Free spectral range”• Controlled by distance between mirrors, fsr=c/(2L)
• Applications– Measurement of laser linewidth or other spectra– Narrowing laser line
trans
mis
sion
0
1
frequency or wavelength
free spectral range,
fsrLinewidth=fsr*finesse
April 03LASERS 51
Diffraction at an aperture—observations
Light through aperture on screen downstream
Aperture
• A careful observation of the light transmitted by an aperture reveals a fringe structure not predicted by geometrical optics
• Light is observed in what should be the shadow region
April 03LASERS 51
Pattern on screen at various distances
25 mm from screen, bright fringes just inside edges
250 mmlight penetrates into shadow region
2500 mmpattern doesn’t closely resemble mase
Far field – at a large enough distance shape of pattern no longer changes but it gets bigger with larger distance. Symmetry of original mask still is evident.
Intermediate fieldNear Field2.5mm
Immediately behind screen
April 03LASERS 51
Huygens-Fresnel diffraction
• Each wavelet illuminates the observing screen• The amplitudes produced by the various waves at the
observing screen can add with different phases• Final result obtained by taking square of all amplitudes
added up– Zero in shadow area– Non-zero in illuminated area
Point source
screen with aperture
observing screen
Wavelets generated in hole
April 03LASERS 51
Fresnel zones
• Incident wave propagating to right• What is the field at an observation point a
distance of b away?• Start by drawing a sphere with radius
b+λ/2• Region of wave cut out by this sphere is
the first Fresnel zone• All the Huygens wavelets in this first
Fresnel zone arrive at the observation point approximately in phase
• Call field amplitude at observation point due to wavelets in first Fresnel zone, A1
incidentwavefront
observationpoint
b
λ/2b +First Fresnel zone
April 03LASERS 51
Fresnel’s zones – continued • Divide incident wave into
additional Fresnel zones bydrawing circles with radii,b+2λ/2, b+3λ/2, etc.
• Wavelets from any one zoneare approximately in phaseat observation point– out of phase with wavelets from a
neighboring zone• Each zone has nearly same area• Field at observation point due to second Fresnel zone
is A2, etc.
incidentwavefront
observationpoint
b
b b +λ +λ/2
• All zones must add up to the uniform field that we must have at the observation point
April 03LASERS 51
Adding up contributions from Fresnel zones
• A1, the amplitude due to the first zone and A2, the amplitude from the second zone, are out of phase (destructive interference)– A2 is slightly smaller than A1 due to area and distance
• The total amplitude if found by adding contributions of all Fresnel zones
A=A1-A2+A3-A4+…
minus signs because the amplitudes are out of phaseamplitudes slowly decrease
So far this is a complex way of showing an obvious fact.
April 03LASERS 51
Diffraction from circular apertures• What happens if an aperture the diameter of the
first Fresnel zone is inserted in the beam?• Amplitude is twice as high
as before inserting aperture!!– Intensity four times as large
• This only applies tointensity on axis
incidentwavefront
observationpoint
b
b b +λ +λ/2
Blocking two Fresnel zones gives almost zero intensity on axis!!
April 03LASERS 51
Fresnel diffraction by a circular aperture• Suppose aperture size and observation distance chosen so
that aperture allows just light from first Fresnel zone to pass– Only the term A1 will contribute– Amplitude will be twice as large as case with no aperture!
• If distance or aperture size changed so two Fresnel zones are passed, then there is a dark central spot– alternate dark andlight spots alongaxis– circular fringesoff the axis
April 03LASERS 51
Fresnel diffraction by circular obstacle—Arago’s spot
• Construct Fresnel zones just as before except start with first zone beginning at edge of aperture
• Carrying out the same reasoning as before, we find that the intensity on axis (in the geometrical shadow) is just what it would be in the absence of the obstacle
• Predicted by Poisson from Fresnel’s work, observed by Arago (1818)
incidentwavefront
observationpoint
b
b+λ/2
April 03LASERS 51
Character of diffraction for different locations of observation screen
• Close to diffracting screen (near field)– Intensity pattern closely resembles shape of aperture, just like
you would expect from geometrical optics– Close examination of edges reveals some fringes
• Farther from screen (intermediate)– Fringes more pronounced, extend into center of bright region– General shape of bright region still roughly resembles
geometrical shadow, but edges very fuzzy• Large distance from diffracting screen (far field)
– Fringe pattern gets larger– bears little resemblance to shape of aperture (except symmetries)– Small features in hole lead to larger features in diffraction pattern– Shape of pattern doesn’t change with further increase in distance,
but it continues to get larger
April 03LASERS 51
How far is the far field?
zAF number, Fresnel
wavelengthaperture of areaA
screenobservingtoaperturefrom distancez
λ
λ
=
==
=
Fresnel number characterizes importance of diffraction in any situation
• A reasonable rule: F<0.01, the screen is in the far field– Depends to some extent on the situation
• F>>1 corresponds to geometrical optics• Small features in the aperture can be in the far
field even if the entire aperture is not• Illumination of aperture affects pattern also
April 03LASERS 51
Imaging and diffraction
• Image on screen is image of diffraction pattern at P– Same pattern as diffraction from a real aperture at image location
except:• Distance from image to screen modified due to imaging equation• Magnification of aperture is different from magnification of diffraction
pattern
• Important: for screen exactly at the image plane there is no diffraction (except for effects introduced by lens aperture)
screen with aperture
observing screen at image of plane P
Lens Image of aperture
Diffraction pattern at some plane, P
April 03LASERS 51
Imaging and far-field diffractionscreen with aperture
f
observing screen
Lens
• Looking from the aperture, the observing screen appears to be located at infinity. Therefore, the far-field pattern appears on the screen even though the distance is quite finite.
April 03LASERS 51
Fresnel and Fraunhofer diffraction• Fraunhofer diffraction = infinite observation distance
– In practice often at focal point of a lens– If a lens is not used the observation distance must be large – (Fresnel number small, <0.01)
• Fresnel diffraction must be used in all other cases• The Fresnel and Fraunhofer regions are used as synonyms
for near field and far field, respectively– In Fresnel region, geometric optics can be used for the most part;
wave optics is manifest primarily near edges, see first viewgraph– In Fraunhofer region, light distribution bears no similarity to
geometric optics (except for symmetry!)– Math in Fresnel region slightly more complicated
• mathematical treatment in either region is beyond the scope of this course
April 03LASERS 51
Fraunhofer diffraction at a slit• Traditional (pre laser)
setup– source is nearly
monochromatic
• Condenser lens collects light
• Source slit creates point source– produces spatial coherence at the second slit
• Collimating lens images source back to infinity– laser, a monochromatic, spatially coherent source, replaces all this
• second slit is diffracting aperture whose pattern we want• Focusing lens images Fraunhofer pattern (at infinity) onto
screen
f2
Lightsource
Condenserlens
smallsource slit
Collimatinglens
Focusinglens
Observationscreen
f1
Diffractingslit
April 03LASERS 51
Fraunhofer diffraction by slit—zeros• Wavelets radiate in all
directions– Point D in focal plane is at
angle Θ from slit, D=Θf– Light from each wavelet
radiated in direction Θ arrives at D
• Distance travelled is different for each wavelet
• Interference between the light from all the wavelets gives the diffraction patter
– Zeros can be determined easily
• If Θ=λ/d, each wavelet pairs with one exactly out of phase– Complete destructive interference– additional zeros for other multiples of λ, evenly spaced zeros
field radiated bywavelets at angle
λ/2
f
D
Slit width = d
Θ
Θ
= λ fd
λ
April 03LASERS 51
Fraunhofer diffraction by slit—complete pattern
• Evenly spaced zeros• Central maximum brightest, twice as wide as
others
slit
Diffraction pattern, short exposure time
Diffraction pattern, longer exposure time
April 03LASERS 51
Multiple slit diffraction• In multiple slit patterns discussed earlier, each slit
produces a diffraction pattern• Result: Multiple slit interference pattern is
superimposed over single slit diffraction pattern
position on screen
Inte
nsity
Three-slit interference pattern with single-slit diffraction included
April 03LASERS 51
Fraunhofer diffraction by other apertures• Rectangular aperture
– Diffraction in each direction is just like that of a slit corresponding to width in that direction
– Narrow direction gives widest fringes
• Circular aperture– circular rings– central maximum brightest– zeros are not equally spaced– diameter of first zero=2.44λf2/d
where d= diameter of aperture– Note: this is 2.44λf/#– angle=1.22λ/d
April 03LASERS 51
Resolution of optical systems• Same optical system
as shown previously without diffracting slit – produces image of
source slit on observing screen
– magnification f2/f1
f2
Lightsource
Condenserlens
smallsource slit
Collimatinglens
Focusinglens
Observationscreen
f1
• We’ve assumed before that the source slit is very small, let’s not assume that any more– each point on source slit gives a point of light on screen– if we put the diffracting aperture back in, each point gives rise to
its own diffraction pattern, of the diffracting slit– ideal point image is therefore smeared
April 03LASERS 51
Resolution of optical systems (cont.)• With two source
slits we can ask the question, will we see two images on the observation screen or just a diffraction pattern? f2
Lightsource
Condenserlens
Collimatinglens
Focusinglens
Observationscreen
f1
Diffractingslit
screen withtwo source slits
• Answer: If the spacing between the images is larger than the diffraction pattern, then we see images of two slits, i.e. they are resolved. Otherwise they are not distinguishable and we only see a diffraction pattern
Main lobe of pattern due to one slit
Rayleigh criterion-images are just resolved if minimum of one coincides with peak of neighbor
April 03LASERS 51
Resolution of optical systems (cont.)• Limiting aperture is usually a round aperture stop, so
Rayleigh criterion is found using diffraction pattern of a round aperture
/#22.122.1distance resolvable minimum fDfR λλ
===f= focal lengthD=diameter of aperture stopR= distance spots which are just resolved
Diffraction Limited System: Resolution of an optical system may be worse than this due to aberrations, ie not all rays from source point fall on image point. An optical system for which aberrations are low enough to be negligible compared to diffraction is a diffraction limited system.
If geometrical spot size is 2 times size of diffraction spot, then system is 2x diffraction limited, or 2 XDL
April 03LASERS 51
Resolution of spots and Rayleigh limit
A A A
Well resolved Rayleigh limit Slightly closer, are you sure it’s really two spots?
• At the Rayleigh limit, two spots can be unambiguously identified, but spots only slightly closer merge into a blur
April 03LASERS 51
Diffraction of laser beams• Till now, disscussion has been of uniformly illuminated
apertures– mathematical diffraction theory can treat non-uniform
illumination and even non-plane waves• A TEM00 laser beam has a Gaussian rather than uniform
intensity pattern– no edge to measure from so we use 1/e2 radius, w– wo is radius where beam is smallest (waist size)– relatively simple formulae for diffraction apply both in near field
(Fresnel) and far field (Fraunhofer) zones– only far field result will be presented here
0
,angle half divergence fieldfar wπλθ =
0
,radius beam fieldfar wzw
πλ
=
April 03LASERS 51
Diffraction losses in laser resonators2a
L
• Light bounces back and forth between mirrors• Spreads due to diffraction as it propagates• Some diffracted light misses mirror and is not fed back• Resonator Fresnel Number measures diffraction losses
If index of refraction in laser resonator is not 1, multiply by nL
aFλπ 2
=