4.  Slab waveguide [solution]

 

The numerical aperture is the sine of the maximum angle of incidence for a guided wave.  Here we have an asymmetric waveguide, with GaAs on one side and air on the other.  Since the critical angle increases as the cladding index increases, the numerical aperture will be determined by the GaInAs/GaAs interface.
 




The maximum entrance angle is 43°.
 
 

The minimum thickness for a propagating wave is that which just allows a single mode.  The order (m) of the highest-order allowed mode is given by
 
 



so we want to determine the thickness d1  for which mmax = 1 (recall that there is no m = 0 mode in an asymmetric waveguide) and d2 for which mmax = 2.  The waveguide will be single-mode for d1 < d < d2.  For m1 = 1.00, m2 = 3.37, m3= 3.30, the asymmetry term .  Thus we have
 
 


 
 




For a film thickness d < 540 nm there will be no guided wave, for 540 nm < d < 1492 nm there will be only a single propagating mode.  For d = 100 mm we have mmax  < 105.555, so 105 modes will propagate.  The smallest reflection angle from the interface is that of the highest mode, which is essentially the critical angle

.  The spread in arrival times is   for a distance of 1 cm.  This implies a communication frequency no greater than 4.3
 

GHz if the distance to be travelled is 1 m.  If we use HeNe laser light (l0 = 633 nm) we would expect the number of allowed modes to be greater for a given thickness, since mmax ~ 1/l0.  However, that assumes that the light propagates at all.  In fact the Ga0.67In0.33As  would absorb the light very strongly, since its bandgap is Eg = 0.97 eV which corresponds to l = 1.27 mm.  Light of wavelengths shorter than that is strongly absorbed, with the energy going into transitions between electronic energy states (creation of electron-hole pairs).  If we cap the waveguide with GaAs we make it symmetric, so the m = 0 mode is now allowed.  For single-mode operation that will be the only mode, so we want m < 1 and thus d < 950 nm.  The number of modes for d = 100 µm is unchanged (the asymmetry term makes a very small contribution at large values of m), so the arrival time spread is also unchanged.

For SiO2 in air we let m1 = m3 = 1.00 and m2 = 1.45. , so all entrance angles will produce guided waves.  The waveguide is symmetric, so
 
 

the m = 0 mode is allowed and there is no minimum thickness.  The thickness dm that will just support a mode of mode number mmax is given by

  and d1 = 619 nm, so the maximum thickness for single-mode propagation is a bit smaller than in the semiconductor case.  If d = 100 mm there

 

are 161 allowed modes, a few more than before.  The smallest reflectance angle = critical angle = 43.6°.  The arrival time spread is 150 ps for 1 cm, implying a maximum transmission frequency of 66.7 MHz.

To reduce the pulse spreading at a given wavelength and thickness we reduce the number of allowed modes, which requires reducing the index difference by putting a cladding with index m1 obeying 1.45 > m1 > 1.  This is of course what is done in real fibers.
 

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