P Incident Wave (wavelength l) y L a Lecture 3 Diffraction & Spectroscopy We will discuss N split interference (diffraction) and single split diffraction Multi slit Interference, I = 4I1cos 2(f/2) (Fo[.]
Trang 1Incident Wave (wavelength l ) y
L a
Lecture 3:
Diffraction
&
Spectroscopy
Trang 2We will discuss N-split interference (diffraction) and single-split diffraction
Multi-slit Interference, I = 4I 1 cos 2 (f/2)
(For point sources, I 1 = constant.)
and
Single-slit Diffraction, I 1 (q)
(For finite sources, I 1 = I 1 (q).)
to obtain
Total Interference Pattern, I = 4I 1 (q)cos 2 (f/2)
(2 slits)
(Remember how f is related to q: f/2p = d/l = (dsinq)/l d q/l)
Trang 3 Multiple-slit Interference formula
Diffraction Gratings
Optical Spectroscopy
Spectral Resolution
Single-Slit Diffraction
Interference + Diffraction
Applications: X-ray Crystallography
Content
Trang 4General properties of N-Slit Interference
• The positions of the principal maxima of the intensity patterns always occur at f = 0, 2p, 4p, [f is the phase between adjacent slits]
(i.e., dsinq = ml, m = 0, 1, 2,…).
• The principal maxima become taller and narrower as N increases
• The intensity of a principal maximum is equal to N 2 times the maximum intensity from one slit The width of a principal maximum goes as 1/N.
• The # of zeroes between adjacent principal maxima is equal to N-1
The # of secondary maxima between adjacent principal maxima is N-2.
-2p
I 0
16I1
N=4
-2p
I 0
25I1
N=5
-2p
I
0
9I1
N=3
0 10
20 25
0 h5( ) x
10
10 -l/ d 0 l/ x d
f q
0
5
9
0
g( ) x
10
f q -l/ d 0 l/ d 10 0 10
0
10 16
0 h( ) x
10
10 -l/ d 0 l/ x d
f q
Trang 5exercise 1 Light interfering from 10 equally spaced slits initially illuminates
a screen Now we double the number of slits, keeping the
spacing constant
What happens to the net power on the screen?
a stays the same b doubles c increases by 4
Trang 6exercise 1 Light interfering from 10 equally spaced slits initially illuminates
a screen Now we double the number of slits, keeping the
spacing constant
What happens to the net power on the screen?
a stays the same b doubles c increases by 4
If we double the number of slits, we expect the net
power on the screen to double How does it do this…
The location and number of the principle maxima (which
have most of the power) does not change.
The principle maxima become 4x brighter.
But they also become only half as wide.
Therefore, the net power (integrating over all the peaks) increases two-fold, as we would expect.
We will soon see that we often use such an array of slits (also called a “diffraction grating”) to perform very precise
metrology, e.g, spectroscopy, crystallography, etc
Trang 7N-Slit Interference – Summary
The Intensity for N equally spaced slits is given by:
L
y
and d
sin
=
l
q l
q l
d p
f 2
2 1
) 2 / sin(
) 2 / sin(
=
f
f
N I
*
* Note: we can not be able to use the small angle approximations if d ~ l.
y
L
d
q
As usual, to determine the pattern at the screen
(detector plane), we need to relate f to q or y = Ltan q :
**
Trang 8Example 1
In an N-slit interference pattern, at what angle qmin does the
intensity first go to zero? (In terms of l , d and N.)
Trang 9Example 1
In an N-slit interference pattern, at what angle qmin does the
intensity first go to zero? (In terms of l , d and N.)
But f = 2p(d sinq)/l 2pd q/l = 2p/N Therefore, qmin l /Nd
As the illuminated number of slits increases, the peak widths decrease!
This is a general feature: Wider slit features narrower patterns
in the “far field”
2 1
sin( / 2) sin( / 2)
N
N
f
has a zero when Nfmin/2 = p, or fmin = 2p/N.
Trang 10Optical spectroscopy – how we know
about the world
• Quantum mechanics definite energy levels, e.g.,
of electrons in atoms or molecules
• When an atom transitions between energy levels
emits light of a very particular frequency.
• Every substance has it’s own “signature” of what
colors it can emit.
• By measuring the colors, we can determine the
substance, as well as things about it’s surroundings (e.g., temperature, magnetic fields), whether it’s
moving (via the Doppler effect), etc.
Optical spectroscopy is invaluable in materials
research, engineering, chemistry, biology, medicine…
But how do we precisely measure wavelengths???