The maximum possible score for this assignment is 60. Induced optical effects Question 1: In lectures, we studied the microscopic...

The maximum possible score for this assignment is 60. Induced optical effects Question 1: In lectures, we studied the microscopic origin of refractive index with respect to the dilute gas model. By assuming that the charge carriers in the medium are unbound, and hence the spring constant restoring force, Ks is zero, we can develop a good approximation to the case of a conducting medium, called the Drude model. Following the notes and using Ks = 0, show that the dielectric function for a conductor is ǫr ≈ 1 − ω 2 p ω2 . (1) (You have seen the dielectric function last year, but it is also shown in the square brackets of Eq. 4.28 in the new notes). Hence or otherwise show that electric field propagating through a conductor has the form E = E0 exp(iωt) exp  −z q ω2 p − ω2 c   (2) Explain the behaviour of this field and give for the skin depth of the conductor. 10 marks Question 2: In lectures, we considered second harmonic generation as a consequence of a second order nonlinearity, but there are other effects that can be exhibited by such materials. Consider a bichromatic field incident on a χ (2) material. By treating this case in general, determine the frequencies and intensities of the fields generated by the material relative to each other (but not to the pump field). You may assume that the medium is optically thin and not resonant at any wavelength. 10 marks Question 3: For the case of a dilute gas, use the form of the real part of the refractive index to determine the refractive index for small frequencies around resonance, i.e. ω ≈ ω0, in terms of the detuning, δ. You should find nR ≈ 1 + ω 2 p δ ω0(4δ 2 + γ 2) (3) The group velocity, vg for light travelling in a medium is vg = c n + ω dn dω (4) where c is the speed of light, n is the real part of the refractive index, and ω is the angular frequency of the light. Show that the group velocity of a wavepacket for small detunings around the resonance is vg = c(4δ 2 + γ 2 ) 2 γ 2(γ 2 − ω2 p ) . (5) Given that ωp is typically the largest frequency scale in this problem, what does this say about the group velocity at resonance? 10 marks Page 1 of 2 Semiconductors and Lasers Question 4: Draw a suitably labelled band diagram showing just the valence and conduction bands for a hypothetical indirect band gap semiconductor with m∗ e > m∗ h . Only consider one dimension. Ensure that you identify the conduction and valence bands, as well as the band gap and Fermi level. Indicate the minimum phonon momentum required to mediate electron-hole recombination, and explain why a minimum phonon energy is not required in absorption. 10 marks Question 5: Calculate the expected peak wavelength and spectral bandwidth (in units of wavelength) of the emission for both a GaAs and silicon LED at liquid nitrogen temperature (77 K) and room temperature (300 K). Which of these cases would you expect to result in the best emitter and why. 7 marks Question 6: Erbium fibre amplifiers are one of the most important elements in modern telecommunications. They work on an effective three state scheme with the 4 I15/2 as the ground state, 4 I11/2 as the upper state and the 4 I13/2 intermediate state. The pump wavelength is typically the 4 I15/2 −4 I11/2 transition with wavelength 0.98 µm, with gain observed on the 4 I13/2 −4 I15/2 transition at around 1.53 µm. Assume that the lifetime of the 4 I11/2 is around 1 µs and that all of the decay is to the 4 I13/2 state, whilst the lifetime of the 4 I13/2 state is 10 ms with all decay to the ground state. (a) Draw a suitably labelled energy level diagram for the operation of an erbium fibre amplifier showing all appropriate rates. Identify the pump and gain transitions. (b) Write down the condition for population inversion on the gain transition. (c) Now, assuming that the time for spontaneous emission on the pump transition is also 1 µs and the refractive index of erbium doped glass is n ∼ 1.5, what is the minimum spectral radiant energy density that could achieve population inversion on the gain transition. (d) Calculate the maximum quantum efficiency of the amplifier, which is the ratio of the output photon energy to input photon energy. 13 marks Page 2 of 2