THE UNIVERSITY OF HULL
Department of Physical Sciences (Physics
Level 5 Examination
May 2008
Optical Physics
Thursday 29 May 2008, 09.30 to 11.30
2 hours
Answer THREE questions, ONE from each section.
Do not open or turn over this exam paper, or start to write anything until
told to by the Invigilator. Starting to write before permitted to do so may
be seen as an attempt to use Unfair Means.
Module 04212
CONTINUED
Page 1 of 7
SECTION A: APPLIED OPTICS
1.
(i) State the Huygens-Fresnel Principle and indicate how it explains the
phenomenon of diffraction.
[4 marks]
(ii) A source uniformly illuminates an aperture with a plane wave at normal
incidence. Use the Huygens-Fresnel Principle to show that the electric field Ep
at a point, p, a distance rs beyond the aperture is given by
E p  
Es
exp( ikrs )dS
rs
where Es is the electric field amplitude of the aperture element dS and k is the
wave-number.
[4 marks]
(iii) State the assumptions used to simplify the amplitude and phase terms of
the integral above in the Fraunhofer limit. Discuss the conditions for which the
assumptions are reasonable.
[4 marks]
(iv) For a one-dimensional aperture, use the Fraunhofer assumptions to find
an expression for Ep in terms of the distance from the mid-point of the slit to p.
[4 marks]
(v) The first minimum in the diffraction pattern from a single slit of width b
occurs at an angle  given by the relationship bsin =. A beam of wavelength
5  10-7 m is diffracted by a slit of width 1  10-3 m. What is the beam width at
a distance 10 m away from the slit?
[4 marks]
Module 04212
CONTINUED
Page 2 of 7
2.
(i) Write the general equation of the electric field of a monochromatic wave in
terms of its two orthogonal plane-polarised components. Use the general
equation to give the conditions which describe
(a)
(b)
(c)
unpolarised,
plane-polarised, and
circularly polarised
light. For each case show how , the angle between the electric field vector
and the x axis, varies with time.
[8 marks]
(ii) Discuss the refractive index properties of a uniaxial birefringent material.
Define the
(a)
(b)
ordinary ray and
extraordinary ray.
Indicate how the refractive index of both rays varies with propagation
direction.
[7 marks]
(iii) A plane-polarised beam of unit amplitude is incident on a quarter wave
plate. The polarisation direction of the beam is at 60 to the optic axis of the
plate. Write the equation of the electric field of
(a)
(b)
the incident and
transmitted
beams in terms of the two orthogonal components parallel and perpendicular
to the optic axis of the crystal. What is the polarisation state of the output
wave? How should the quarter-wave plate be configured to obtain a circularly
polarised output?
[5 marks]
Module 04212
CONTINUED
Page 3 of 7
SECTION B: OPTOELECTRONICS
3.
e m
(i) Poisson statistics for a random process gives P  m
for the
a!
probability of a occurrences in an interval where m is the average number of
events in this interval. If, on average, 2 photons per second fall upon a
photodetector calculate the probability of receiving 0, 1, 2, 3 and 4 photons in
1 second. Sketch the resulting probability distribution and comment on its
significance to noise in a photodetector.
[4 marks]
a
(ii) Write down the expression relating signal current to the received
power in an ideal photodetector, defining the terms involved. Prove that for
a background-limited ideal photodetector the minimum detectable optical
power is
2hPb B
Ps min 

where Pb is the received background power, B the bandwidth and  the
quantum efficiency. (You may assume the mean-square current
fluctuations arising from shot (quantum) noise is i n  2eIB ).
2
[6 marks]
(iii) Outline the operating principle of the semiconductor photodiode detector
and sketch the physical construction of a typical device.
[5 marks]
(iv) A GaP photodiode has a quantum efficiency of 0.2 at wavelength of
400nm. Calculate:
(a) Its output current if it receives an optical power of 110-6W at 400nm.
(b) The minimum detectable power Psmin if it is background limited and the
received background power is Pb = 10-10W at 400nm and B = 108Hz.
(c) The bandgap of GaP if the detector has a cut-off wavelength of 550nm.
[5 marks]
[c = 3  108 ms-1, e = 1.6  10-19C, h = 6.6  10-34J s]
Module 04212
CONTINUED
Page 4 of 7
4.
(i) With reference to a simple two-level atomic system explain the terms
spontaneous emission, absorption and stimulated emission. Relate these to
the corresponding Einstein coefficients and state what condition must be met
to achieve optical gain.
[7 marks]
(ii) How is optical feedback implemented in a typical laser system?
[2 marks]
(iii) Give a brief account of how optical gain is produced in a HeNe laser and
provide a sketch of a practical laser system.
[5 marks]
(iv) A HeNe laser has a 300mm long gain length and resonator mirrors with
(power) reflection coefficients of 0.93 and 0.99 spaced by 450mm. Calculate
(a) The optical gain coefficient to reach the threshold for laser action
(b) The frequency spacing of the axial modes of the laser
and
(c) Estimate how many axial mode frequencies will appear in the laser
output if the laser transition has a spectral width of 1.8710-3nm (full
width at half maximum).
[6 marks]
[c = 3  108ms-1]
Module 04212
CONTINUED
Page 5 of 7
SECTION C: E AND M FIELDS
5.
(i) Consider a series circuit consisting of a battery with emf = , a capacitor,
C, resistor, R, and switch. Initially the capacitor is uncharged and the switch
is open.
(a) Show that the magnitude of the charge on the capacitor, Q,
increases in time, t, when the switch is closed according to the
expression
Q = QMAX (1 – e-t / RC)
[5 marks]
(b) Sketch a graph of the charge stored by the capacitor as a function
of time and indicate the magnitude of the upper limiting value, QMAX.
Give a definition of the time constant of the circuit and add construction
lines to the graph to illustrate what it represents.
[3 marks]
(ii) Below is a circuit diagram for a simple sawtooth oscillator. The neon bulb
is a light that is based on a gas discharge (similar to shop signs). Initially, at
low voltage, the neon acts as an insulator (infinite resistance). Once the
voltage across the bulb reaches 90V the gas is said to break-down and
become a conductor (resistance essentially zero).
(a) After the switch is closed, how long does it take for the voltage
across the neon bulb to reach 90V?
[5 marks]
(b) Why does the voltage across the neon bulb decrease rapidly once
it breaks down?
[2 marks]
(c) If the neon bulb stops conducting once the voltage across it
reduces to 65V, draw a graph of the voltage waveform measured
across the output terminals as a function of time. Be sure to label the
axes so that the timescales and voltages are clearly indicated for at
least three cycles.
[5 marks]
Module 04212
CONTINUED
Page 6 of 7
6.
(i) When a dielectric is placed in an electric field, the material may become
polarized. Draw a sketch to illustrate what is meant by this.
[2 marks]
(ii) If a parallel plate capacitor is charged and then disconnected from the
voltage source, describe how the subsequent insertion of a dielectric material
between the plates affects (a) the capacitance of the system, (b) the voltage
across the plates and hence (c) give a definition of the relative permittivity
(dielectric constant) r of the material.
[6 marks]
(iii) Capacitors in series can act in a similar way to a potential divider. When
a DC voltage is applied as shown below, the charge in the system is
distributed so that it is the same on each capacitor. Show that the expression
for V2 given below holds in this case.
V2  V0
C1
C1  C2
[6 marks]
(iv) Using the result in (iii), analyse the following problem. A man, standing
on one foot on a grounded metal floor, places one hand on a television screen
(an older style cathode ray tube, CRT, in this case). The voltage inside the
CRT is 25kV and the glass screen thickness is 6.3mm. His shoes have
insulating soles, 10 mm thick, made from rubber. Assuming that the area of
his hand on the screen is very similar to the area of the sole of his shoe in
contact with the floor and that his body is a perfect conductor, calculate the
voltage across the sole of the shoe.
Assume r = 4.7 for glass and 7 for rubber. [Also 0 = 8.85 × 10-12 Fm-1
although this is not strictly required]
[6 marks]
Module 04212
END
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