1. Thin-Film Interference Summary
Path length difference:
(Assume near-normal incidence.)
destructive
constructive
where
ray-one got a phase change of 180o due to reflection from air to glass.
the phase difference due to path length is:
then total phase difference: f = f’+180.

2. Newton’s Rings
The air between the glass plates acts like a thin film.
Since the thickness of the film changes over the radius of the plates, alternating bright and dark fringes form, when the plates are illuminated. Because of the curvature of the upper piece, the film thickness varies more rapidly at larger radius. Thus the fringe separation is smaller toward the outside.

3. Two (narrow) slit Interference
According to Huygens’s principle,
each slit acts like a wavelet. The
the secondary wave fronts are
cylindrical surfaces.
Young’s double-slit experiment
Upon reaching the screen C, the
two wave interact to produce an
interference pattern consisting of
alternating bright and dark bands
(or fringes), depending on their
phase difference.
Constructive vs. destructive
interference

4. y ~ D*tan(θ)~ D*(m+1/2)λ/d
Interference Fringes
For D >> d, the difference in path lengths between the two waves is
A bright fringe is produced if the path lengths differ by an integer number of wavelengths,
y ~ D*tan(θ)~ D*mλ/d
A dark fringe is produced if the path lengths differ by an odd multiple of half a wavelength,
y ~ D*tan(θ)~ D*(m+1/2)λ/d
m is referred to as the order of the fringes:
m = 0, central maximum or first minimum
m = +/- 1, the first maximum or second minimum ……

5. Which of the following statements is incorrect ?
READING QUIZ 1
Which of the following statements is incorrect ?
A| In the ideal two slit interference experiment the slit size (a)
is: a << λ , where λ is the wavelength of the light.
B| In the single slit diffraction pattern the slit size (a)
is: a ~ λ or larger.
C| In a two slit interference experiment the fringe separation
between dark fringes is independent of the wavelength λ.
D| A diffraction grating has a very high resolution capability
which depends on a large number of slits per centimeter.

6. Dark and Bright Fringes of Single-Slit Diffraction

7. Phasor Diagram f2 f1

8. Phasor Diagram for Single-Slit Diffraction
The superposition of wavelets can be illustrated by a phasor diagram. If the slit is divided into N zones, the phase difference between adjacent wavelets is
total phase difference:

9. Intensity Distribution 1
maxima:
central maximum because or
minima:

10. Intensity Distribution 2
for small q y
y ~ D*θ
Bright fringe: D*(m+1/2)λ/a
Dark fringe: D*mλ/a
Width: D*λ/a except central maximum
Fringe widths are proportional to λ/a.
Width of central maximum is twice any other maximum.
Width = D*λ/a – D*(-1)λ/a = 2D*λ/a

11. Young’s Double-Slit Experiment Revisited
Intensity pattern for an ideal double-slit experiment with narrow slits (a<<λ)
Light leaving each slit has a unique phase. So there is no superimposed single-slit diffraction pattern but only the phase difference between rays leaving the two slits matter. a
slit separation
where I0 is the intensity if one slit were blocked
If each slit has a finite width a (not much smaller than λ), single-slit diffraction effects must be taken into account!

12. a) Decreasing slit separation b) Moving the screen closer to the slits
Physics 241 Warmup Quiz 2
In Young’s two-slit interference experiment, which of the following will increase the spacing between the bright fringes?
a) Decreasing slit separation
b) Moving the screen closer to the slits
c) Reducing the wavelength of light
d) Submerging the entire apparatus in water (assuming that the experiment can still be performed properly in water)
e) Increasing the width of each slit

13. Diffraction from Slit(s) of Finite Width
Single slit of width
or greater
where
Two slits of width
or greater a

14. Intensity Distribution from Realistic Double-Slit Diffraction
replace by
double-slit intensity
single-slit intensity envelope
How many maxima will fit between central max and first envelope min?:

15. DOCCAM 2
7B-17 LASER AND TWO SLIT REALISTIC
DIFFRACTION PATTERN

16. Diffraction by a Circular Aperture
The diffraction pattern consists of a bright circular region and concentric rings of bright and dark fringes.
The first minimum for the diffraction pattern of a circular aperture of diameter d is located by
geometric factor
Resolution of images from a lens is limited by diffraction.
Resolvability requires an angular separation of two point sources to be no less than θR where central maximum of one falls on top of the first minimum of the other:
Rayleigh’s criterion

17. Diffraction Gratings
Devices that have a great number of slits or rulings to produce an interference pattern with narrow fringes.
One of the most useful optical tools. Used to analyze wavelengths.
up to thousands per mm of rulings D
Types of gratings:
transmission gratings
reflection gratings
Maxima are produced when every pair of adjacent wavelets interfere constructively, i.e.,
mth order maximum

18. Spectral Lines and Spectrometer
Due to the large number of rulings,
the bright fringes can be very narrow and are thus called lines.
For a given order, the location of a
line depends on wavelengths, so light waves of different colors are
spread out, forming a spectrum.
Spectrometers are devices that can
be used to obtain a spectrum, e.g.,
prisms, gratings, …

19. 7B-17 LASER AND DIFFRACTION GRATING
DOCCAM 2
7B-17 LASER AND DIFFRACTION GRATING

20. X Ray Diffraction
X rays are EM radiation of the wavelength on the order of 1 Å, comparable to atomic separations in crystals.
X rays are produced, e.g., when core electrons in atoms are inelastically excited. They are also produced when electrons are decelerated or accelerated.
Vacuum tubes, synchrotrons, …
Standard gratings cannot be used as X ray spectrometers.
(Slit separation must be comparable to the wavelength!)
Von Laue discovered the use of crystals as 3-dimensional diffraction gratings.
Nobel 1914

21. a) Increasing slit separation b) Moving the screen closer to the slits
Physics 241 –10:30 Quiz 3, December 6, 2011
In Young’s two-slit interference experiment, which of the following will increase the spacing between the dark fringes?
a) Increasing slit separation
b) Moving the screen closer to the slits
c) Reducing the wavelength of light
d) Submerging the entire apparatus in water (assuming that the experiment can still be performed properly in water)
e) None of the above

22. In Young’s two-slit interference experiment, which
Physics :30 Quiz 3, December 6, 2011
In Young’s two-slit interference experiment, which
of the following will decrease the spacing between
the bright fringes?
a) Decreasing slit separation
b) Moving the screen further from the slits
c) Increasing the wavelength of light
d) Submerging the entire apparatus in water
(assuming that the experiment can still be
performed properly in water)
e) Decreasing the width of each slit

23. a) Increasing slit separation b) Moving the screen closer to the slits
Physics 241–11:30 Quiz 3, April 26, 2011
In Young’s two-slit interference experiment, which of the following will decrease the spacing between the dark fringes?
a) Increasing slit separation
b) Moving the screen closer to the slits
c) Increasing the wavelength of light
d) Submerging the entire apparatus in water (assuming that the experiment can still be performed properly in water)
e) None of the above