S-108.3110 OPTICAL COMMUNICATIONS
Round 1: Optical fibers
07.11.2007
exercise 1. Modes of an optical fiber.
We consider a step-index multimode fiber. The cut-off wavelength for the LP01 fundamental mode
is 1 m. Calculate the Number of modes supported by the fiber at the wavelength of 0.8 m.
exercise 2. Modes and refractive indeces of an optical fiber.
We consider a step-index fiber with a core radius of 4 m and cladding refractive index of 1.45. For
what range of values of the core refractive index the fiber will be single-mode for all wavelengths
between 1.2 and 1.6 m.
exercise 3. Dispersion and the angle of acceptance.
A step-index multimode fiber has a core diameter of 62.5 m. The cladding refractive index is 1.45.
The inter-modal dispersion is 10 ps/m. Find its acceptance angle. Calculate the maximum bit-rate B
for transmission over a 20 km distance.
S-108.3110 OPTICAL COMMUNICATIONS
Round 2: Nonlinear effects in fibers
14.11.2007
exercise 1. Effective length of an optical fiber.
Power along the optical fiber at distance z can be expressed as
P  z   P0 e z ,
where α is the loss of the fiber. Simple model defines effective length as
L
P0 Leff   Pz dz .
0
Calculate the effective length as a function of fiber length L and fiber loss .
Fiber 1 has loss of 0.18dB/km and length of 10 km. Fiber 2 has loss of 0.8dB/km and length of
20km. Calculate the effective length for both fibers.
exercise 2. Self-phase modulation(SPM).
Nonlinear refractive index causes phase shift that is dependent on the strength of the so called
nonlinear parameter defined as
n
 2 0,
cAeff
where n2 is the nonlinear refractive index, w0 is the angular frequency, Aeff is the effective area of
the fiber core and c is the speed of light. Simple model gives the following relation between the
maximum nonlinear phase shift, fiber input power P0, nonlinear parameter and effective length.
max  P0 Leff
Assume that γ is 2 W-1/km, P0 = 10mW, α = 0.2dB/km. What is the length of the fiber when the
nonlinear phase shift is 1? What is the needed length when P0 = 100mW?
exercise 3. Stimulated raman scattering(SRS).
Raman scattering can be useful (Raman lasers, amplifiers,..) or harmful (energy transfer,..). Like
Brillouin scattering, also Raman scattering has a treshold level. Raman effects can be generally
neglected below the treshold level. An approximate equation to present the treshold level is
g R P0Crit. Leff
 16 ,
Aeff
where gR is the Raman gain coefficient. Assume that the fiber is long i.e. αL » 1, gR = 1x10-13m/W,
α = 0.2dB/km and Aeff = 50 µm2. Calculate the Raman treshold? Will SRS occur in optical systems?
What if the radius of the fiber core is halved?
S-108.3110 OPTICAL COMMUNICATIONS
Round 3: Fiber-Optic Components
21.11.2007
exercise 1. Finesse of a Fabry-Pérot filter. Demo(no points)
For a Fabry-Pérot filter the Finesse is defined as
FSR
F
 FP
where FSR is the free-spectral range and ΔνFP is the full-width-at-half maximum of the passband.
Show that
 R
F
1 R
R is the mirrors reflectivity. Assume that F >> 1.
exercise 2. Fabry-Pérot filter.
We consider a Fabry-Pérot filter with free-spectral range FSR=1500 GHz and finesse F=100.
Calculate the mirrors reflectivity R and the distance between the mirrors. What is the width of the
passband? Assume identical mirrors.
exercise 3. Coupler.
We consider a 50/50 coupler which outputs are connected to each other through an optical fiber of
length L. Calculate the power exiting at port 1 and 2.
1
2
S-108.3110 OPTICAL COMMUNICATIONS
Round 4: Fiber-Optic Components
28.11.2007
exercise 1. Finesse of a Fabry-Pérot filter.
For a Fabry-Pérot filter the Finesse is defined as
FSR
F
 FP
where FSR is the free-spectral range and ΔνFP is the full-width-at-half maximum of the passband.
Show that
 R
F
1 R
R is the mirrors reflectivity. Assume that F >> 1.
exercise 2. Erbium Doped Fiber Amplifier(EDFA).
The small signal-gain of an Erbium Doped Fiber Amplifier of length 5 is equal to 20 dB for an
input signal of 0.1 mW. If the signal power is 1 mW, the gain starts saturating and drop to 10 dB.
Calculate the saturation power of the amplifier. Assume that the gain shape is 1.
exercise 3. Add/drop resonator.
We consider a ring resonator add/drop made from 3dB (50/50) couplers. What should be the length
of the ring resonator so that a wavelength of 1.5 m is dropped (i.e. reflected) by the resonator.
Assume that the ring resonator is made of fiber with refractive index n=1.5.
S-108.3110 OPTICAL COMMUNICATIONS
Round 5: Transmitters and Receivers
05.12.2007
exercise 1. Quantum efficiency of LED. Demo(no points)
Show that the external quantum efficiency of a surface emitting LED can be approximated by
1
 ext 
,
2
nn  1
where n  3.5 is the refractive index of the semiconductor. Assume that the fraction of reflected
intensity at the surface between the semiconductor material and air is
 n 1
R 0   

 n  1
for all angles between zero and the critical angle of total internal reflection. What is the numerical
value of ηext? (Hint: Assume no absorption)
2
exercise 2. Photodetector.
A 1.55-µm digital communication system operating at 1 Gb/s receives an average power of
40 dBm at the detector. Assuming that "1" and "0" bits are equally likely to occur, calculate the
number of photons received within each "1" bit.
exercise 3. Quantum Efficiency and Responsivity.
The internal quantum efficiency of a detector, in this case, may be approximated by
 

 int  sin 2 

2
 0 
when 1200nm < λ < 1600nm and λ0 = 1350nm.
Calculate the responsivity of the detector at 1,3 and 1,55 µm when the thickness of the active area is
200 µm and absorption coefficient is 12[mm-1]@1300nm and 20[mm-1]@1550nm. The refractive
index of the detector is n = 3,5. Assume reflection as in exercise 1.
S-108.3110 OPTICAL COMMUNICATIONS
Round 6: Fiber optic measurements
12.12.2007
exercise 1. bit error rate (BER).
Bit error rate is defined as the ratio of errorneous bits to the total number of received bits. To
measure the BER of certain signal, several errorneous bits need to be received to get “statistical
reliability”.
Assume that the bit rate of the signal is 155 Mb/s and 100 errorneous bits need to be received. What
is the expected measurement time when the BER is 10-10 (ITU-T standard for almost error free
signal). What if the BER is 10-14 ? Use suitable unit (sec,hr,day,yr,…) to express the time.
exercise 2. Fabry-Perot cavity modes. Demo(no points)
Show that the resonant frequencies fn of a Fabry-Perot cavity satisfy fn = f0 +nΔf, n integer, for some
fixed f0 and Δf. Thus these frequencies are spaced equally apart. Note that the corresponding
wavelengths are not spaced equally apart.
How many wavelengths will fit in 3,1cm cavity, which could be the length of ECD laser cavity, in
order to attain resonance if λ=1,55μm?
What should the cavity length be if as many wavelengths of 1kHz sound waves is wanted to
resonate? (speed of sound is 344m/s)
exercise 3. Integrating sphere(ISP)
Throughput of an integrating sphere can be estimated with the following formula
Pout Aphotodiode



,
Pin
Asphere 1   1  f 
where Asphere is the area of the integrating sphere, ρ is the reflectivity of the sphere wall and f is the
ratio of the total area of the ports to the total area of the sphere.
Calculate the throughput of the sphere when the diameter of the photodiode is 1 mm, diameter of
the sphere is 50 mm, diameters of the input and ouput ports are 5 mm and the reflectivity of the
sphere wall is 0,98. (HINT: Asphere = 4πr2 )
What happens to Pout/Pin if the diameter of input and output ports is decreased to 4,9mm (~2%
change)?
What about if the reflectivity of the sphere  experiences a change of same order of magnitude, i.e.
0,98  0,96 (~2% change)?
Calculate the effects of these changes on output power?
Fig 2: Operational principle of an integrating sphere.