We know very little about phasers -- they emit "nadion beams", but "nadions" are not known to modern science and little is told about them in the movies.
Yet, because light from phasers is seen in space, nadions are probably unstable and decay, producing light -- then one may speculate that the energy of the photons produced is proportional to nadion energy, and with this assumption the more blue the ray is the more energetic (faster?
more massive?
more excited?)
the nadions in the beam are.
What the impact of this fact to the destructive power of the beam is is an even more complex story.
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I would say that Phasers, just like Lasers, have a variety of different reasons why the color can be different.
In general, a higher-powered laser would probably be blue, but not necessarily.
The color of a laser depends on the material that makes it up, and a bit on the shape of the laser-emitting-area of the device.
The reason why blue would usually denote higher energy is that higher wavelengths of light have more energy than lower wavelengths, given the same amount of photons.
I expect that phasers are somehow related technologically to lasers, and that a similar phenomenon exists.
Of course, not knowing specifically how phasers work, it's hard to say.
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It may be that the phasers are intentionally colored a particular way for tactical reasons in the same way that soldiers wear particular uniforms to identify them as friend or foe, as well as to identify specialized soldiers from others.
Tracer rounds are often included in continuously firing weapons to help direct fire and monitor aim.
With this level of advanced technology, it seems like the ships of the Star Trek universe can color the beam however they like.
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I don't recall any canonical mention on colour.
If you watch ENT's pilot, Broken Bow , the phaser is introduced and the characters do express some confusion over whether the phaser is on stun or kill.
One likely explanation
(but I admit this is speculation) would be the source material/reaction used to create the beam.
Neon signs, for example, have various colours because they use different noble gases.
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It may have something to do with the resonance frequency of the phaser beam.
In
The Best of Both Worlds , Shelby orders Data to alter the frequency of the Enterprise phasers during their first battle with the Borg cube.
SHELBY:
Data, fluctuate phaser resonance frequencies.
Random settings.
Keep them changing.
Don't give them time to adapt.
When we see the result on-screen, the beam color does seem to fluctuate until they finally break free.
Also, from the Star Trek: Deep Space Nine Technical Manual : When phasers are fired by a ship with deflector shields active, the beam is frequency locked to the second-order harmonics of the shield emissions.
This prevents the beam impacting on the shields and overloading them, or rebounding back at the firing ship.
This could explain why different space-faring races might have different color beams - it's reasonable to assume that each type of shield technology would operate on a different frequency range, so phasers (and possibly disruptors) would have to operate on a corresponding frequency to match.
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Most likely, they are built using different technologies and parts.
For example, I'd imagine that they would use some kind of crystal lens to focus the energy, different crystal structures produce different visual properties.
This is not unlike light sabers in Star Wars... different parts.
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I would like to comment on Mark's technological reasons above.
If you assume these things are highly engineered, the color is just energy being lost from the beam.
So it should be undesirable from a weapons effects standpoint.
So the visible beam is probably a deliberately engineered side effect, so that the user can see where he has shot.
It would be nice to have different colors so as to be able to discriminate between different sources (such as us versus them), or different classes of weapons users (lightly armed crewmen versus full-on military forces, etc.)
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On Star Trek when a Starfleet ship fires its phasers, the color of the effect is usually red/orange, but sometimes blue (in TOS). This is also generally true for the phaser sidearms (with green sometimes being the color for the stun setting in TOS).  I realize the color is most likely an aesthetic chosen by a visual effects designer, but what would a reasonable technological explanation be?  For example could the colors indicate temperature or power variations?
What would be a reasonable technological explanation for why phaser beams are different colors?
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<star-trek><technology>
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Phasers may be different cvolors due to resonance frequency, different noble gases or different technologies and parts used in their construction. There is speculation as to why phaser beams are different colors. However, there is no canon reference to the reason for different color beams.
Phaser beams may be different colours for a number of reasons. It may be the frequency, the different materials used or the different technologies and parts. There is no canon reference to phaser color.