Can a supersonic fighter incorporate a convergent nozzle or does it always have to be a convergent-divergent nozzle?
Short answer: Yes.
A convergent nozzle will not allow supersonic exit speeds of the combustion gasses, but due to their high temperature their speed of sound is considerably higher that that of the surrounding air. For example, at 700°C the speed of sound in air is 625 m/s. Since thrust is mainly determined by the difference in entry and exit speeds of the air flowing through an engine, a higher speed than flight speed is required for positive thrust. Low supersonic flight speeds are entirely possible with a convergent nozzle.
Long Answer: Better not.
Once the aircraft crosses the sound barrier, drag increases only slowly since the drag coefficient actually drops. If the design is meant to fly supersonically, which implicates a lot of design adaptions, it makes sense to go for higher supersonic speed; however, this requires both an adjustable intake and an adjustable convergent-divergent nozzle. Both increase efficiency, dramatically so at higher Mach numbers. The simple pitot intake of the F-16 is just good enough for Mach 1.6 (above that its efficiency becomes outright awful), but still its F-110 engine has a con-di nozzle.
With an adaptable convergent nozzle the exhaust gasses can be accelerated up to their speed of sound, but not more. A narrowing of the flow path will accelerate subsonic flow, but only to the speed of sound. Only a divergent flow path will then accelerate the flow further to supersonic speed. A convergent nozzle will be shorter and lighter, but means throwing away a good deal of useable energy with the hot exhaust gasses. In most cases it would be more efficient to convert the pressure energy of the exhaust gasses into more thrust by accelerating them to supersonic speed.
Convergent nozzles only make sense if the aircraft is designed only for short, limited supersonic dashes, but will spend almost all flying time at subsonic speeds. Examples are the Northrop F-5E or the Panavia Tornado.
Can a supersonic fighter incorporate a convergent nozzle?
It actually exists.
The Panavia Tornado has a maximum airspeed of Mach 2.2 at 9000m altitude and has variable geometry convergent nozzles (although they work only for the afterburner, while the turbine's nozzle is convergent-divergent):
Photo from Airliners.net
There is a school of thought that contends that the convergent nozzle is an inefficient version of the convergent -divergent nozzle. The jet velocity will rise to a maxima at the throat of the nozzle (smallest area) and then exit the nozzle and expand again against atmospheric pressure.
You will find that a fixed convergent nozzle would not be used because a fixed nozzle of any sort will be less efficient than a variable geometry nozzle. The ability to tune the nozzle for local atmospheric pressure, airspeed, and engine throttle setting is key to efficient generation of thrust.
Could it be used? Yes However the circumstances where it is the best option are probably so limited as to make it never the right choice.
Pilot here, as well as an engineer who made it most of the way to an Airframe and Powerplant technician license at one point in life and worked a lot on nozzles including at the jet Propulsion Laboratory (very fun).
If you want to understand nozzle shapes a fun introduction is to read about the shape that inspired modern rocket (and some jet) nozzles, the de Laval nozzle.
Let's keep in mind that what we want (ignoring for the moment optimization of air volume to feed combustion) is HIGH SPEED in the OUTGOING JET EXHAUST.
In a SUBSONIC flow the air remains compressible and a converging subsonic nozzle increases velocity while pressure and temperature decrease. (There are actually about a dozen subcases here that I'm sure others will yell at me on, so consider that P and T decrease to be an inexcusable simplification on my part to avoid another page of explanation.) So for a slower exhaust final speed, a converging nozzle makes perfect sense.
The ideal situation for SUPERSONIC flow is converging-diverging (see the de Laval again) because, for isentropic flow, the converging portion builds energy up to the point of the sonic speed. This sonic point is then put right at the "throat" where converging becomes diverging and the diverging part turns heat and pressure into more speed up to fairly high values of supersonic. Note that a modern jet engine is fairly complicated in air flow and the "throat" may be deep inside the engine and not look like a rocket engine throat at all.
For these reasons we would expect to see converging final nozzles on subsonic aircraft like airliners and some fighter jets, while supersonic jets usually have diverging nozzles. If you see a converging nozzle on a supersonic-capable fighter look at it carefully to see if it is articulated for changing shape in flight - which is pretty darn common.
Having said all that, aerospace designers are very clever and injecting fuel (afterburners) can cause non-isoentropic flow if designed for it. So I won't go out and say there aren't any converging nozzle supersonic nozzle jets, but at least now you know why they would be the exception and not the rule :)
Please feel free to post questions if I skipped something that should be addressed...
For a supersonic plane, a convergent nozzle with decelerate the flow. This is required only at the compressor, where flow velocity should be slower, but after that to get more thrust, the flow needs to be accelerated .
Now, one can argue that if a supersonic flow enters a converging nozzle, it would accelerate so subsonic speed (at about Mach 1 ) after which the convergent nozzle would accelerate the flow. But this would never achieve any Mach above 1 for the flow, because as soon as the flow tops Mach 1, it would be decelerated by the convergent part of the nozzle once again. So, the maximum exit Mach one could expect to get off a convergent nozzle is 1 (whether the input is supersonic or subsonic).
Now, for supersonic flight, it is necessary that the exit mach number is greater than 1. In such cases, one has to use the Convergent Divergent Nozzle (C-D Nozzle).
If some one wishes to build an engine which can accelerate a plane to supersonic velocities with subsonic (or transonic) exhaust velocity, then for sure one can use convergent nozzles only. But, there is a catch :
The thrust equation is:
Here for an engine in free stream Pe is approximately equal to P0. So for positive thrust, the terms containing Ve and V0 should give a positive answer. So to create an engine where Ve < V0 (because V0 is the free stream velocity which is supersonic, and the Ve is exit velocity which is assumed to be transonic in this case), one has to vent out a lot of Me at a high rate. This will mean that it is technically possible to create an engine which gives supersonic flight at transonic exhaust velocity, but such a configuration is going to cost you a lot of fuel.
Now, if you consider that the exit temperature is very high, lets say around 1200k, and lets say input temperature of free steam is about 300k, that means the exhaust velocity can be about Twice the Mach 1 velocity of air at 300k (without being supersonic), i.e. about 630 mps. If you input mach is about 1.2, that tranlate to input freestream velocity of 380 mps (apx). This means your exit has accelerated to 630 mps from 380 mps, just by the means of a convergent nozzle. But this still would require a lot of fuel to be burnt (because the thrust required usually for flying a plane a few thousand kilograms in weight is very high).
Hence, its possible, but at the expense of the efficiency of the aircraft