1. Given the relation:
ω = C2β 3 4 (a) Derive an expression for the phase velocity. (2 Marks) (b) Derive an expression for the group velocity. (2 Marks) (c) Derive an expression for dispersion parameter D in terms of β. (3 Marks) Note: C is a constant.
2. (a) Using the deﬁnition of group velocity show that it can be expressed as:
dvp dω where vp,and ω arethephasevelocity,andangularfrequency,respectively.(3 Marks)
(b) Consider a digital transmission system using OOK. The received output voltage pulse has a Gaussian shape for binary “1” as follows: v(t) = Aexp − t2 2σ2! where A is the peak pulse amplitude and σ is the pulse width at 60.7% level. The pulse is ﬂat (= 0) for binary “0”. Derive an equation for the maximum bit rate such that the vertical opening of the eye diagram is more than 80% of A. You can assume that the tails from bits that are not adjacent are negligible. (7 Marks)
3. (a) A step index ﬁber has a numerical aperture of NA = 0.1. The refractive index of its cladding is 1.465. What is the largest core diameter for which the ﬁber remains single-moded at the wavelength of 1.3 µm? (3 Marks)
(b) A step index ﬁber has a core index of 1.453, ∆ = 0.002, and a core diameter of 10 µm. What is the cutoﬀ wavelength for single-mode operation of the ﬁber? (3 Marks)
4. Light traveling in air strikes the core area of a ﬁber at an angle θ = 34.5o, where θ is the angle between the incoming ray and the surface of the ﬁber. Upon striking the ﬁber, part of the beam is reﬂected and part of it is refracted. The angle between the reﬂected and refracted beams is 90o. Given that the refractive index of cladding is 1.44 and the core radius is 30 µm: (a) what is the refractive index of the core? (3 Marks) (b) what are the numerical aperture and the maximum acceptance angle im of this ﬁber? (3 Marks) (c) If the ﬁber is immersed in water (nw = 1.33), what is the new maximum acceptance angle? (3 Marks)
5. Show that in the absence of material dispersion (i.e. when numerical aperture (NA) is independent of ω), the group velocity is given by the following expression
dV dβ where c is the speed of light and a is the radius of the core. (3 Marks)
6. In a 1.55-µm lightwave system, 120 µW optical power is launched into a ﬁber of length 8 km. The power at the output of the ﬁber is 3 µW. Determine: (a) The total loss (in dB) through the ﬁber assuming there are no connectors or splices. (3 Marks) (b) The total loss (in dB) for a 10 km optical link using the same ﬁber with splices at 1 km intervals, each giving a loss of 1 dB. (5 Marks)
7. Consider a binary digital communication system with received signal levels m1 and m0 for a 1 bit and 0 bit, respectively. Let σ2 1 and σ2 0 denote the noise variances
for a 1 and 0 bit, respectively. Assume that the noise is Gaussian and that a 1 and 0 bit are equally likely. In this case, the bit error rate is given by
Qm1 −Td σ1 +1 2
QTd −m0 σ0 , where Td isthereceiver’sdecisionthreshold. Showthatthevalueof Td thatminimizes the bit error rate is given by (7 Marks) Td = −m1σ2 0 + m0σ2 1 +rσ2 0σ2 1h(m1 −m0)2 + 2σ2 1 −σ2 0ln(σ1/σ0)iσ 2 1 −σ2 0 .