The amount of reaction mass needed to achieve a specific delta V increases exponentially by the ratio of the delta V to the rocket exhaust velocity. The exhaust velocities achieved by chemical rockets are considerably less than orbital velocity, which is why the fuel/payload ratio is so high for chemical rockets.

This also means that increasing the rocket exhaust velocity will
exponentially reduce this figure. However, the energy needed
to increase exhaust velocity also increases. Specifically,
thrust is given by `F = mv` for velocity `v` and mass flow
`m`, and power is given by `P = (1/2)mv^2 = (1/2)Fv`.
So for an exhaust velocity of 10 kps, 1 lb of thrust requires 2.27
kilowatts.

Nuclear and laser rockets are still thermal rockets, and are ultimately limited by the melting point of the engines. The main advantage of a nuclear rocket is not so much that it gets hotter, but that it uses hydrogen as its reaction mass. Because of its lower molecular weight, hydrogen will have a much higher exhaust velocity than any other gas starting from the same pressure and temperature (about 3 times higher than H2O).

The idea of firing a laser up the exhaust nozzel of a rocket engine is not well thought out, since it basically requires that the reaction mass be opaque inside the engine and transparent outside (not to mention the trajectory limitations). OTOH, attaching a laser target/heat exchanger to the side of a rocket would have all the advantages of a nuclear rocket with less weight.

For even more significant improvements, I prefer a microwave powered plasma engine.

A very large phased array could more than equal the aiming and focussing power of a laser, easily achieve the desired power levels, and is much more energy efficient. The rocket would have a relatively large (compared to the shuttle) delta wing which doubles as the rectenna array. This means that bottom of the wing must consist of a nonconducting material (into which the rectenna array is embedded) for one quarter wavelength in addition to the heat shields, topped by a (highly) conducting reflector.

The heart of the plasma engine is a linear induction motor operated in pulse mode. By operating in pulse mode the coils can be placed closer together and be tuned to resonate a higher frequencies as the plasma velocity increases.

Note: The induced currents should be sufficient to maintain the plasma temperature, but are not intended to create a plasma using ring discharge. Motors based on that approach are extremely inefficient.

It is hoped (assumed) that the plasma pulses (and the shock waves they generate) will help propel larger volumes cold gas through the induction motor. This would significantly increase thrust efficiency, since a major limitation on a plasma engine is the energy needed to generate the plasma.

Also, plasma is easier to generate in pulses, since the breakdown voltage for most gasses greatly exceeds the voltage needed to maintain a plasma arc. The usual approach is to place an inductor in series with the capacitor (or DC source) and spark gap, so that once the arc forms and current begins to flow, the voltage drop is switched from the spark gap to the inductor. The inductor can double as the secondary winding of a transformer to produce a high voltage RF initiating spark (easier to time accurately).

The amount of thrust generated depends not only on the output speed of the linear induction motor, but also on the mass of plasma generated and the speed at which it passes through the first (slowest) section of the motor. One might use multiple pulsed sources sequentially to increase the rate of plasma production.

The best plasma source might be a chemical rocket with an easily ionized additive (see NASA MAPX project). This would provide a continuous flow of ionized gas entering the first section of the linear induction motor at about Mach 1.

For maximum thrust, the coils ahead of the pulse should be running at max current in one direction and the coils behind the pulse should be running at max current in the other direction. This means that the coil needs to reverse the currrent from peak to peak (1/2 cycle) in the time it takes for the pulse to travel 2 coil widths.

Specifically, for a relatively constant acceleration `a`, the time
`t` it takes for the plasma disk to travel distance `d` is the given
by

(1/2)at^2 + vt - d = 0which has the solution

t = {\sqrt{v^2 + 2ad} - v \over a} .When

For a 10 meter long motor with a 10 kps exhaust velocity, the acceleration is 5x10^6 meters/sec^2 for 2 msec. For a 1 cm coil spacing, the resonance frequencies of the coils vary from 5,590 Hz (v=0) to 250 kHz (v = 10^4), a 45 fold increase.

To keep the induced EMF constant, the coil inductance should be adjusted as the frequency changes and the capacitance left constant. This will also keep the max voltage across the power transistors constant (the breakdown voltage of this component is a major limitation on the power output per coil).

The power output of the motor is limited by the energy stored in each coil/capacitor times the number of coils (1000 in the above example) times the number of pulses per second. To minimize vibration, a new pulse should be started each time one leaves the end, so that the pulse frequency should be a multiple of the transit time (2 msec or 500 Hz in the above example). Also, the pulse frequency must be less than the lowest cycle frequency (v=0), which is given by

f_c = {2V_s \over V_s + 4V_c} f_rwhere

Assuming a pulse rate of 500 Hz in the above example, a 250 mfd capacitor at 800 volts could handle less than 40 kilowatts, for a total motor output of less than 40 megawatts, producing less than 17.6 thousand pounds of thrust. Of course, how much of that power is translated into thrust depends on the strength of the MHD forces and conductivity of the plasma pulses, which limit the maximum mass flow.

Note: Pulses which travel slightly ahead of the point of maximum MHD force are stable, in that the leading edge is accelerated less than the trailing edge. Once plasma falls behind the point of maximum MHD force it will continue to fall further behind and eventually will no longer be accelerated at all. So increasing mass flow will increase thrust up to a certain point, then drop to nearly zero when the plasma can no longer keep up with the induction motor.

The power supply is connected to the coil by a power transistor and diode (in parallel) so that diode opposes the normal current flow. A second power transistor and diode can be connected between the coil and a capacitor. These prevent the capacitor from charging up from the power supply and wasting energy. How useful this is depends on the ratio of the power supply voltage to the max voltage generated by the coil & capacitor (squared).

The purpose of this design is to produce a coil current as shown below.

B /\ / | / | A/ |C ---------------> time | /E | / | / \/ DAt

At `B` the plasma pulse has just passed the previous coil.
At this point the first transistor is turned off and the second
turned on.
The coil begins to discharge through the capacitor via
the second diode.

At `C` the plasma pulse passes the coil. The magnetic
field produced by this coil is zero and induced EMF is at maximum.
At this point the capacitor is at its maximum voltage and
starts discharging back through the coil via the second power
transistor.

At `D` the pulse passes the next coil and the second
transistor is turned off, recycling the unused energy to the power
supply through the first diode.
At `E` the coil is totally discharged and the cycle is
complete.