Classical Optics

Classical Optics (course no. 203-12181)
Exam: '‫מועד א‬
12.07.2007
You have a choice. Answer only 11 questions out of 13. Do not submit more than 11
solutions for the questions or else you might jeopardize your maximum score. All
questions have equal weight. One page of formulas (both sides) in your handwriting is
permitted. Good luck!
1. Pulses of UV (λ = 325 nm) lasting 2.0 ns each are emitted from a laser that has a
beam of diameter 2.5 mm. Given that each burst carries an energy of 6.0 J, (a)
determine the length in space of each wavetrain, and (b) find the average energy
per unit volume for such a pulse. (c) How many photons are contained in each
pulse?
2. (a) Monochromatic light (λ = 500 nm) having an irradiance of 400 W/m2 is
incident normally on the cornea (nc = 1.376) of the human eye. If the person is
swimming under the water (nw=1.33), determine the transmitted irradiance into
the cornea. (b) Write an expression for the focal length (fw) of a thin lens
immersed in water in terms of its focal length when it is in air (fa).
3. (a) Use the Fresnel Equations to prove that light incident at θp = π/2 –θt results in
a reflected beam that is polarized. (b) Describe the polarization. (c) Show that
tan(θp) = nt/ni (where the subscripts represent t = transmitted and i = incident
media) and calculate the polarization angle for external incidence on a plate of
glass (ng = 1.52) in air.
4. Two identical converging (convex) lenses each have focal lengths of 15 cm and
are separated by a distance of 6 cm. (a) If an object of height 1 cm is placed 10 cm
in front of the first lens, find the position and height of the final image. (b) What
is the total magnification of this lens system? Is the image erect or inverted?
Nq e2
5. Using the dispersion equation, n (ω ) = 1 +
ε o me
2
∑ω
j
fj
2
oj
−ω2
, show that (a) the
c
for high-frequency
1 + Nq / 2ε o meω 2
electromagnetic waves (such as x-rays). (b) What is the meaning of fj in the above
expression? (c) Calculate the phase velocity, vph, for these waves?
group
velocity
is
given
by
vg =
2
e
6. An ionized gas or plasma is a dispersive medium for electro-magnetic waves.
Given that the dispersion equation is ω2 = ωp2 + c2k2, where ωp is the constant
plasma frequency, determine expressions for both the phase and group velocities,
v and vg,and show that vvg=c2.
7. A beam of light is incident normally on a quartz plate (no = 1.5443 and ne =
1.5534) whose optic axis is perpendicular to the beam. If the wavelength in
vacuum is λo = 589.3 nm, compute the wavelengths, velocities and frequencies of
both the ordinary and extraordinary waves.
8. A calcite sheet (no = 1.6584 and ne = 1.4864) is cut so that its faces are parallel
and lie in the xy-plane. The optic axis is along the y-direction. (a) What are the
allowed thicknesses of the sheet if a linear polarized beam (λo = 589 nm) at
normal incidence (propagating in the z-direction) with its incident electric field at
an angle θ = +30º with respect to the y-axis is to be converted into a linearly
polarized beam with its emergent electric field an angle θ = -30º with respect to
the y-axis? (b) If the electric field amplitude is 10 V/m, write an expression
describing the emergent electric field as a function of time.
9. One of the mirrors of a Michelson Interferometer
is moved and 1000 fringe-pairs shift past the
hairline in a viewing telescope during the
process. (a) What are the conditions for
constructive interference for this interferometer?
Describe each variable. (b) If the device is
illuminated with 500-nm light, how far was the
mirror moved? (c) What is the purpose of the
optical element that is labeled C in the figure on
the right?
10. Consider an N-slit Fraunhofer diffraction pattern with slit width b and slit spacing
a, where a = 2b. (a) Write the conditions for the principal maxima and subsidiary
maxima for the phase α where α=(ka/2)sinθ. (b) What is the relative irradiance
(as a fraction of I(0)) of the subsidiary maxima in a three-slit Fraunhofer
diffraction pattern? (c) Draw a graph of the irradiance distribution I versus sinθ
for 0 ≤ sinθ ≤ 2λ/a for systems containing two, three and four slits.
11. Using Lloyd's mirror (as shown below), X-ray fringes were observed, the spacing
of which was found to be 0.0025 cm. The wavelength used was 0.833 nm. (a) If
the source-screen distance was 300 cm, how high above the mirror plane was the
point source of X-rays placed? (b) Write an expression that describes the
irradiance I(y) as a function of the distance y along the detection plane, where Io is
the maximum irradiance. Plot I(y) vs y.
12. Consider Fresnel diffraction. A long narrow slit 1.00 mm wide is illuminated by
light (λ = 500 nm) coming from a point source 0.90 m away emitting with a
power of 100 W. (a) Determine the irradiance at a point 2.0 m beyond the screen
when the slit is centered on, and perpendicular to, the line from the source to the
point of observation. (b) What is the dimensionless arc length (i.e., "string
length") representing the slit width along the Cornu spiral? Is the central
diffraction point a local maximum or minimum? Explain.
13. (a) A Collimated beam of microwaves impinges on a metal screen that contains a
long horizontal slit that is 20 cm wide. A detector moving parallel to the screen in
the far-field region locates the first minimum of irradiance at an angle of 36.87º
above the central axis. Determine the wavelength of the radiation.
(b) Suppose that we have a laser emitting a diffraction-limited beam (wavelength
of 632.8 nm) with a 2-mm diameter. What is the diameter of the central light spot
that would be produced on the surface of the Moon which is 3.76 x 108 m from
the Earth.
εo = 8.854 × 10-12 C2/N⋅m2
µo = 4π × 10-7 N⋅s2/C2