Spacecraft to travel at speed of light

[JohnDuke]

I don’t understand why it’s so hard, I have a learning disability where I can do and see mechanical things easier than most so my thoughts were, everything we have seen as kids like 20,000 Leagues under the Sea, Star Trek & lots of other things are a reality now so why is this so hard?

[Snark]

Hello, and welcome to the forums!

Can you please clarify the intent of this post? Are you talking about KSP gameplay, or are you talking about real-life science and engineering?

[JohnDuke]

5 hours ago, Snark said:

Hello, and welcome to the forums!

Can you please clarify the intent of this post? Are you talking about KSP gameplay, or are you talking about real-life science and engineering?

Real Science 

[Snark]

2 hours ago, JohnDuke said:

Real Science 

Ah, okay.  In that case, moving to Science & Spaceflight, then.

Going back to your original question:

9 hours ago, JohnDuke said:

I don’t understand why it’s so hard, I have a learning disability where I can do and see mechanical things easier than most so my thoughts were, everything we have seen as kids like 20,000 Leagues under the Sea, Star Trek & lots of other things are a reality now so why is this so hard?

It's hard because the laws of physics say so. 

Specifically, relativity says that,

1. you can't reach the speed of light
2. even getting close to it requires stupefyingly huge amounts of energy

That's the physics side of it.  From the engineering side, the combination of the tyranny of the rocket equation, coupled with the lack of an energy source powerful enough, means that it's just not practical.

 

[Espatie]

Well this is about to get moved to another forum, but...

 

Basically you can't go at Lightspeed because Physics says No.

 

You are thinking of velocity in purely newtonian-mechanical terms, with a linear relationship between the force applied and the resulting velocity (in, of course, a totally frictionless environment).

 

That's not actually how the universe works, is the problem. Energy input has diminishing returns in velocity.

 

This is a fiendishly complicated subject which a lot of very very smart people have been working on for decades and they still don't know all the answers.

[Snark]

2 minutes ago, Espatie said:

This is a fiendishly complicated subject which a lot of very very smart people have been working on for decades and they still don't know all the answers.

Actually, it's a relatively straightforward subject, in which all the relevant equations have been fairly well understood since Einstein's day, and they pretty much do know all the answers.  They've got the equations.  Wanna go such-and-such speed?  You'll need such-and-such energy.

So the problem isn't that it's still poorly understood.  The problem, rather, is that it is well understood, and what we understand is that it's really freaking hard because the numbers are just astronomical.

[Espatie]

I was more thinking of the underlying structure-of-the-universe implications than the physical level TBH.

[cubinator]

If we figured out a way to get a lot of antimatter as fuel for some kind of engine, we might be able to make a really fast ship. But we don't know how to do that yet.

[Deddly]

26 minutes ago, cubinator said:

If we figured out a way to get a lot of antimatter as fuel for some kind of engine, we might be able to make a really fast ship. But we don't know how to do that yet.

But it still would not reach the speed of light, according to the theory of relativity. 

Even Star Trek and a lot of other works of science fiction recognise this problem. Even if you have an almost infinite amount of energy and thrust, you can't get better than 99.99(add as many 9s as you like) % of the speed of light, because to travel at the speed of light, you need an infinite amount of energy, which is why they need to invent imaginative ways to travel quickly, like warp speed, hyperspace and wormholes. 

So the question really is: what prevents humans from developing warp speed or quantum slipstream or traveling through wormholes? And the answer there is really that there is nothing stopping us, except for the problem that none of these are actually proved to exist. They are, instead, inventions of science fiction to make a good (or bad) story. Whether or not we ever invent something that enables faster-than-light travel remains to be seen, but for now it is merely wishful thinking, sadly. 

(All except for Galaxy Quest, of course, because that was actually a historical documentary, but we were never able to get a proper look at the propulsion technology beyond the stompers) 

[HebaruSan]

It all started with the observation that light always seems to move at the same speed, no matter the direction or how fast you're moving. In the same way that sound is a wave traveling through the air, it was thought that light might be a wave traveling through something, then speculatively called the luminiferous aether. If this is true, then if the earth is moving relative to the aether, we should be able to observe light moving faster in some directions than others, the same way sound waves take longer to reach you if you're moving away at close to the speed of sound. But the speed of photons (light particles) is always exactly the same, so the luminiferous aether had to go.

Here's the problem. Let's say your goal is to get up to light speed, so you send out a photon, measure its speed, and then use some immensely powerful engine to accelerate to half that speed in the same direction (0.5c). Then you send out another photon chasing the first and measure its speed, and it's receding from you just as quickly as the first one did (which, by the way, is also still traveling away from you at light speed if you measure it again)! In effect, you've gotten no closer to light speed. In fact, no matter how long you accelerate, you'll still measure all photons as traveling away from you at light speed. If this sounds crazy and impossible to you, don't feel bad; it took 18 years for physicists to figure it out.

When all is said and done, the key is mass. You have mass, so you can only travel sub light speed. Light has no mass, so it can only travel light speed. All massless particles travel at "light speed," indicating that it's not a special property of light as much as it is a property of the universe.

[DDE]

22 hours ago, JohnDuke said:

everything we have seen as kids like 20,000 Leagues under the Sea, Star Trek & lots of other things are a reality now

They aren’t.

Mostly because they either were not intended to be scientifically accurate in the first place (Star Trek) or they got their science wrong - we don’t have 1700 t all-electric submarines with a maximum speed of 50 knots and operating depth of 16 km (nor is there even such depth to be found on Earth), not to mention the Nautilus would never submerge without approximately 21 m3 of lead ballast (cf. A. Grossman), just as the Stalin-era Soviet Union never built the Pioneer, a submarine with a supercavitating drive propelled by hydrolox rockets, armed with FLIR-enabled aerial scout drones and a sonic disintegrator beam that reduces Japanese cruisers to dust.

It’s an oft-repeated, annoying myth that sci-fi has any more predictive power greater than astrology does.

Edited December 31, 2018 by DDE

[wumpus]

If you want to see the issues involved in possible near-speed-of-light travel (not to mention the physics of "reaching" the speed of light), then you need to be familiar with the rocket equation (playing KSP will get a deeper understanding of this than anything else).  Once you realize exactly what a delta-v of 30,000,000 m/s means (only 10% of the speed of light, since the equation doesn't deal with issues of speed vs. momentum changes as you approach the speed of light) and exactly how much fuel that requires *normal* rockets require 99.99% of their mass going to fuel, engines, and plumbing.  Speed of light starships require even more fuel, and that fuel had better involve nuclear power.

The rocket equation is Delta-v=Isp*g*ln(m_full/m_empty)   Note the ln in the equation.  That means that increasing delta-v is *hard*.  Just going to orbit requires around ~9k m/s (KSP is intentionally easy to make a better game and get into space faster) and typically requires multiple stages to get the delta-v required.  Dawn holds the record for delta-v (as far as I know) with over ~20k m/s delivered to the payload (~10k by the boosters to escape velocity and then an additional ~10k by the spacecraft itself).  Of course it used solar power which wouldn't be available between the stars (or even much past Mars).

Going around the rocket equation doesn't help much either.  You can generate momentum from energy via something as simple as an LED light (any light will do, but LEDs are efficient), but they have probably the worst energy efficiency of any "rocket" engine.  Perhaps some balance between ion thrusters and cyclotrons (to maximize output momentum).

There's also the issue of how hard any spacedust will hit your starship at near-c travel.  When each atom becomes a beta particle, shielding is critical.  And that shielding goes into "Mempty", ruining your rocket equation.  And don't forget to double your delta-v if you want to stop and land at any of those stars.

Edit: on the last page [as of the end of 2018, might not be true now] of the "random science facts thread" Bill Phil computes that it would require a billion tons of fuel to accelerate a single ton of spacecraft to .2c (presumably also works for getting to .1c and slowing down again before you arrive) with an Isp of 3000s.  Since Dawn's record-breaking 10km/s delta-v "only" had an Isp of 3100s, this is bad news for anybody considering trying to get to near light speed.

 

Edited January 1, 2019 by wumpus
included link to calculation on going near c

[DDE]

What @wumpus fails to add is that Newton's domain ends at about a few percent of C, and every additional m/s starts requiring more and more energy. The energy needed to accelerate any amount of mass to lightspeed equals infinity.

Even with all the power in a billion universes, you won't have enough to reach C.

1 hour ago, wumpus said:

Going around the rocket equation doesn't help much either.  You can generate momentum from energy via something as simple as an LED light (any light will do, but LEDs are efficient), but they have probably the worst energy efficiency of any "rocket" engine.

To be exact, 1 N per 300 MW of energy input. Assuming perfect efficiency all around - whereas lasers tend to stay under 30% efficiency, et cetera.

[Green Baron]

Citing from the thread @wumpuslinked:

"According to Ziolkovsky, assuming an exhaust velocity of 5200m/s, to accelerate a mass of 1 ton to 0.2c takes 10²⁶⁰⁵⁷ tons of fuel. [snip]

This is ludicrously more mass than the entire observable universe contains (~10⁵¹ tons).

Assuming an ion thruster with 50km/s v exhaust still needs 10⁵²¹ tons of propellant, which still is ~470 magnitudes off the entire mass in the obs uni."

@p1t1o has then done the same thing for an exhaust speed of 3000km/s, which would need a mere 1 billion tons to accelerate 1 ton to 0.2c.

----------

So, here are your numbers @JohnDuke. Pls. be aware that the calculation does not work for relativistic speeds. Things get worse the closer we come to c. But anyway, we don't even get close to that realm, not even with future tech, which may one far day, if all goes well, get us in the area of 500km/s v exhaust. Forget Star Trek and whatnot, they are nice to watch from a relaxed position in an armchair but have, in contrary to some saying, no real world relevance at all.

[Bill Phil]

1 hour ago, DDE said:

What @wumpus fails to add is that Newton's domain ends at about a few percent of C, and every additional m/s starts requiring more and more energy. The energy needed to accelerate any amount of mass to lightspeed equals infinity.

Even with all the power in a billion universes, you won't have enough to reach C.

To be exact, 1 N per 300 MW of energy input. Assuming perfect efficiency all around - whereas lasers tend to stay under 30% efficiency, et cetera.

Well you can get more momentum per photon if you use a photonic laser thruster. Those are generally more efficient up until a few percent of the speed of light. Building one is much more difficult though.

[Snark]

53 minutes ago, Bill Phil said:

Well you can get more momentum per photon if you use a photonic laser thruster. Those are generally more efficient up until a few percent of the speed of light.

Can you clarify, please?  This doesn't sound like a thing, to me.

The energy of a photon is given by E = mc².  The momentum of a photon is given by p = mc.  Since m = E/c²,  that means that the momentum of a photon is E/c... and therefore the momentum per energy is 1/c.  Momentum per time is force; energy per time is power.

So, the thrust of a photon beam is equal to the power divided by c.  So, for example, as @DDE states above, it takes 300 MW of power to generate 1 N of thrust.

That's not for a specific "type" of photon; it doesn't care about the wavelength or whatever.  All you need to know is the power and the speed of light, and that'l tell you the thrust.

So, regarding your point,

1 hour ago, Bill Phil said:

more momentum per photon if you use a photonic laser thruster

...first, momentum per photon depends only on wavelength (being a laser or not won't change that); and, second, momentum per photon isn't what you care about anyway.  What you care about is total momentum in the beam for all the photons, and that's simply a function of the beam power as described above.

1 hour ago, Bill Phil said:

more efficient up until a few percent of the speed of light

You seem to be claiming that "efficiency" changes based on how "fast" the ship is going?  How is that?  The thrust developed by a photon beam is simply equal to the beam power divided by c.  Note that this doesn't care about how fast the ship is going.

[Bill Phil]

1 hour ago, Snark said:

Can you clarify, please?  This doesn't sound like a thing, to me.

The energy of a photon is given by E = mc².  The momentum of a photon is given by p = mc.  Since m = E/c²,  that means that the momentum of a photon is E/c... and therefore the momentum per energy is 1/c.  Momentum per time is force; energy per time is power.

So, the thrust of a photon beam is equal to the power divided by c.  So, for example, as @DDE states above, it takes 300 MW of power to generate 1 N of thrust.

That's not for a specific "type" of photon; it doesn't care about the wavelength or whatever.  All you need to know is the power and the speed of light, and that'l tell you the thrust.

Force is a change in momentum over a change in time. Photon sails are given momentum through an exchange with photons. But those photons still have momentum. Unused momentum. It took a lot of energy to create those photons and they ultimately impart very little momentum to the sail. But if you can somehow exchange more momentum between the photon and the sail, you can impart more momentum to the sail. More momentum exchanged per photon. Of course those photons lost energy, but not much. If the sail is configured to reflect the photons to another reflector, with a tight beam collimation, and that reflector can reflect back at the sail with a tight beam collimation, then more momentum can be extracted from each photon. Eventually the photon will exhaust its useful momentum, but new photons can be input into the beam. And since each photon transfers more momentum than they would have otherwise, it's significantly more energy efficient.

A photon's energy is given by E = hv, where v is the frequency of the light, and h is the Planck constant. A photon's momentum is indeed E/c, but also hv/c. After the photon imparts energy to the sail, it still retains energy. Most of the energy that was used to create the photon is wasted in a conventional photon sail system. This energy can still be used by continuously reflecting the photon back and forth between two mirrors. As mentioned earlier, the photon's useful momentum/energy will be exhausted, but new photons can be added to the beam. But since these new photons are only replacing the recently lost photons, there can be far fewer photons added to the beam per unit time. Lower power for a similar effective thrust. Basically there are numerous overlapping photon beams going in opposite directions, repeatedly reflecting off of either end. One end being the source, the other being the photon sail.

The problem is that as the speed of the sail increases relative to the speed of the source, the reflected photons take more time to cross the distance and relativistic doppler shift changes the relative energy of the photons to the photon sail. To keep it up more power has to be added, and eventually it loses effectiveness and becomes as effective as a standard light sail.

2 hours ago, Snark said:

So, regarding your point,

...first, momentum per photon depends only on wavelength (being a laser or not won't change that); and, second, momentum per photon isn't what you care about anyway.  What you care about is total momentum in the beam for all the photons, and that's simply a function of the beam power as described above.

Yes, momentum per photon depends on wavelength. But that photon exchanges momentum and the wavelength changes, so it still retains momentum. In a conventional sail this photon is just reflected and ignored. In a recycling photon propulsion system such as a Photonic Laser Thruster, this photon is reflected back towards the source, exchanges momentum with the source, and is reflected back towards the sail. This cycle repeats as long as practical. Thus momentum per photon is quite important, as the goal is to maximize the use of this momentum.

2 hours ago, Snark said:

You seem to be claiming that "efficiency" changes based on how "fast" the ship is going?  How is that?  The thrust developed by a photon beam is simply equal to the beam power divided by c.  Note that this doesn't care about how fast the ship is going.

Efficiency of a recycling photon propulsion system in terms of energy used at the source versus energy the sail gains. In conventional photon sail systems the efficiency is quite low, as enormous amounts of energy are spent while small thrusts are generated at the sail. But the efficiency of photon rockets increases as the relative velocity approaches the speed of light. A recycling photon propulsion system is more efficient at lower speeds but eventually conventional photon propulsion systems overtake the efficiency of recycling photon systems. This happens at some percentage of the speed of light, dependent on the individual system design. This could be as high as 20% of the speed of light or as low as a few percent of the speed of light. Relative to the photon source.

The thrust from a single photon beam where each photon is used once, yes. But for a recycling photon propulsion system, the thrust is twice the power times the number of times the photons are reflected divided by c. 

This is still pretty hard to do, but it could be quite useful for beamed propulsion systems.

Here's an demonstration:

Wikipedia article:

https://en.wikipedia.org/wiki/Photonic_laser_thruster

It's not going to be easy to do at all, but could be worth pursuing.

[Ultimate Steve]

I may be very wrong, but IIRC you seem to be moving through time faster the faster you go, to the extent that if you were, for some reason, travelling at the speed of light, from your perspective any trip you took would be instantaneous. So if you got on a spaceship and reached 99.99% of the speed of light, from your perspective you would be going FTL because of this time dilation. From Earth's perspective you would not be going FTL.

[Snark]

5 hours ago, Bill Phil said:

photonic laser thruster

Ah, okay.  Thanks for the video and the link.

Physics explanation is nice, but not necessary in this case-- I get the physics, it was the terminology that was what I stubbed my toe on.  The problem (for me at least) is that "photonic laser thruster" is about the worst name ever that I've seen for a technology-- it's both insufficiently explanatory and highly misleading.

Terminology rant in spoiler. 

Spoiler

For someone who hasn't heard the term before (i.e. me), even if they're familiar with the concept (also me), the thought process on seeing that goes something like:  "Hmmm, okay, 'thruster', that means it's a technology for moving things.  'Photonic', okay, that means we're talking about photons' momentum, and not a traditional rocket or ion drive or something.  'Laser', makes sense, it means we're getting the photons from a laser and not from, say, sunlight."

In other words, it sounds like what you're talking about is just another name for a "photon drive", i.e. you've got a spacecraft with some high-energy source on it, which has a big aft-pointing laser to generate thrust.

Since I hadn't heard this particular term of art, and since it sounds so clearly like it just means "using a laser for thrust", I didn't have any reason to go look up the term.  My post was based on that (incorrect) assumption about what you were talking about.

Yes, I know, it's not your fault that I hadn't heard the term, nor is it your fault that it's a terrible (IMO) name for what it means.  Might be handy, though, when introducing it into a conversation, to always include a link, just so folks don't do what I do and think you mean something different. 

Anyway, to quote the relevant snippet from the Wikipedia article,

Quote

The thruster, invented by Young K. Bae differs from other solar sail and laser propulsion thrusters in that an amplification process is used, in which the incident beam is re-used by being reflected by a stationary mirror, with an amplification stage at each reflection.^[2][3][4][5] Because of the recycling of energy, the photonic laser thruster has been demonstrated to be more energy efficient than other laser-pushed sail concepts.

Okay, sure, that makes sense-- the idea is that you save mass on the spacecraft by leaving the laser and energy source at home, and you boost the power (a lot) by using multiple reflections to get many momentum transfers out of each photon.

That's great as far as it goes... and perhaps could be handy over limited distances (e.g. going to Mars or something).  However, it seems to me to be not super practical when we're talking about the topic of this thread, which is getting spacecraft up to high fractions of lightspeed.  That's because it's going to take a very long distance to accelerate up to a significant percentage of c, and I don't see how the multiple-reflections thing is going to be practical over really long distances.  How the dickens would you be able to maintain focus for that many reflections?  There's beam spread, and although you can presumably have a really big, well-collimated source back home, the spacecraft is going to be limited in how accurately it can send every photon back to the original reflector.  I'd be astonished if you can get even two or three round trips with any significant percentage of recycling, let alone the thousand cycles proposed in the article.

Going back to the Wikipedia article again,

Quote

To date the experimental tests of the photonic thruster are limited to laboratory-scale distances. The maximum range of operation for a photonic laser thruster, PLT, is yet to be established. Bohn^[28] demonstrated a 1 km-long laser resonator similar to the PLT cavity, which is an active cavity, in 1995 and proposed that such resonators could scale to 100 km. Recently, 4-km  Fabry–Pérot cavities, which are passive cavities, but share the same intracavity power multiplication principle with the PLT cavity, have been demonstrated in LIGO for gravitational wave detection with an intracavity multiplication factor of 280 and an intracavity laser power on the order of 100 kW. Based on these results and the state-of-the-art technologies in precision optics, the PLT cavity length over 1,000 km is promising. Further studies need to be performed to determine whether the PLT cavity could be scaled for astronomical distances.

I gotta say I'd be pretty skeptical that this would be practical over interstellar distances, which is what we're talking about if you want to get up to the kinds of speeds that this thread is about.

 

 

33 minutes ago, Ultimate Steve said:

I may be very wrong, but IIRC you seem to be moving through time faster the faster you go, to the extent that if you were, for some reason, travelling at the speed of light, from your perspective any trip you took would be instantaneous. So if you got on a spaceship and reached 99.99% of the speed of light, from your perspective you would be going FTL because of this time dilation. From Earth's perspective you would not be going FTL.

Weelllllll, not quite. Yes, at very high fractions of c, there are time dilation effects... but there's also distance contraction.  From the perspective of someone on the ship, you do not appear to be going FTL.  Instead, you appear to be going the exact same speed that a "stationary" observer would see-- but to you, the distance you travel appears shorter.

So, let's suppose that you take a trip on a spacecraft at 0.994987 c, i.e. about 99.5% the speed of light.  At that speed, the Lorentz factor (i.e. amount of dilation) is 10.  Let's suppose you take a trip over a distance of 1 light year as measured by a "stationary" observer back home on Earth.  What does an Earth-based observer see, versus what do you see?

• Earth-based observer:  You're traveling at 0.994987 c.  You take just a smidgeon over 1 year to travel one light-year's distance.
• Observer on the spacecraft:  You're traveling at 0.994987 c.  You take just a smidgeon over 1/10 of a year to travel 1/10 of a light-year's distance.

The two observers agree on how fast the ship is traveling.  They disagree about how far it went or how long it took.

[Ultimate Steve]

4 minutes ago, Snark said:

32 minutes ago, Ultimate Steve said:

 

Weelllllll, not quite. Yes, at very high fractions of c, there are time dilation effects... but there's also distance contraction.  From the perspective of someone on the ship, you do not appear to be going FTL.  Instead, you appear to be going the exact same speed that a "stationary" observer would see-- but to you, the distance you travel appears shorter.

So, let's suppose that you take a trip on a spacecraft at 0.994987 c, i.e. about 99.5% the speed of light.  At that speed, the Lorentz factor (i.e. amount of dilation) is 10.  Let's suppose you take a trip over a distance of 1 light year as measured by a "stationary" observer back home on Earth.  What does an Earth-based observer see, versus what do you see?

• Earth-based observer:  You're traveling at 0.994987 c.  You take just a smidgeon over 1 year to travel one light-year's distance.
• Observer on the spacecraft:  You're traveling at 0.994987 c.  You take just a smidgeon over 1/10 of a year to travel 1/10 of a light-year's distance.

The two observers agree on how fast the ship is traveling.  They disagree about how far it went or how long it took.

Okay, then. Dang, this is making my head spin... So they would both observe the spacecraft going 1 light year in 1 of their local years (time to distance traveled ratio remaining the same, so no magical speeding through time) but time is passing 10x faster for the guy on the ship relative to Earth time?

[DDE]

33 minutes ago, Snark said:

How the dickens would you be able to maintain focus for that many reflections?

Grasers.

Wait, that's an interplanetary weapon system, not a propulsion beam.

[Snark]

22 minutes ago, Ultimate Steve said:

Okay, then. Dang, this is making my head spin... So they would both observe the spacecraft going 1 light year in 1 of their local years (time to distance traveled ratio remaining the same, so no magical speeding through time) but time is passing 10x faster for the guy on the ship relative to Earth time?

No.  They both observe the spacecraft going the same speed.  For the guy on the spacecraft, the entire universe looks "squashed", like a giant sat on it, along the axis of travel.  So what looks like 1 light year to the "stationary" observer looks like only 1/10 of a light-year to the guy on the ship.

• The Earth-based observer sees the ship go 1 light year in 1 local year.
• The guy on the ship sees it go 1/10 light year in 1/10 of a local year.

[Ultimate Steve]

1 minute ago, Snark said:

No.  They both observe the spacecraft going the same speed.  For the guy on the spacecraft, the entire universe looks "squashed", like a giant sat on it, along the axis of travel.  So what looks like 1 light year to the "stationary" observer looks like only 1/10 of a light-year to the guy on the ship.

• The Earth-based observer sees the ship go 1 light year in 1 local year.
•  The guy on the ship sees it go 1/10 light year in 1/10 of a local year.

Oh. That almost makes sense. Thanks!

[Green Baron]

The PLT needs a lot of energy and a mass in the vicinity to push against. It would work in the inner solar system, between a pair of vessels. If this can ever be used for more than station keeping or stretching a tether is unclear. The achievable thrust is too low to propel a massive human capable ship. It cannot be used in interstellar space because there is nothing to convert energy from (except untapable vacuum energy).

 

8 minutes ago, Snark said:

• The Earth-based observer sees the ship go 1 light year in 1 local year.
• The guy on the ship sees it go 1/10 light year in 1/10 of a local year.

But the ship observer cannot see (but knows because he follows our forum) that time on earth passes 10 times faster than time on board (it lies behind). He cannot see because a message sent to him from earth is heftily red shifted.

Edited December 31, 2018 by Green Baron

[magnemoe]

2 hours ago, Bill Phil said:

Force is a change in momentum over a change in time. Photon sails are given momentum through an exchange with photons. But those photons still have momentum. Unused momentum. It took a lot of energy to create those photons and they ultimately impart very little momentum to the sail. But if you can somehow exchange more momentum between the photon and the sail, you can impart more momentum to the sail. More momentum exchanged per photon. Of course those photons lost energy, but not much. If the sail is configured to reflect the photons to another reflector, with a tight beam collimation, and that reflector can reflect back at the sail with a tight beam collimation, then more momentum can be extracted from each photon. Eventually the photon will exhaust its useful momentum, but new photons can be input into the beam. And since each photon transfers more momentum than they would have otherwise, it's significantly more energy efficient.

A photon's energy is given by E = hv, where v is the frequency of the light, and h is the Planck constant. A photon's momentum is indeed E/c, but also hv/c. After the photon imparts energy to the sail, it still retains energy. Most of the energy that was used to create the photon is wasted in a conventional photon sail system. This energy can still be used by continuously reflecting the photon back and forth between two mirrors. As mentioned earlier, the photon's useful momentum/energy will be exhausted, but new photons can be added to the beam. But since these new photons are only replacing the recently lost photons, there can be far fewer photons added to the beam per unit time. Lower power for a similar effective thrust. Basically there are numerous overlapping photon beams going in opposite directions, repeatedly reflecting off of either end. One end being the source, the other being the photon sail.

The problem is that as the speed of the sail increases relative to the speed of the source, the reflected photons take more time to cross the distance and relativistic doppler shift changes the relative energy of the photons to the photon sail. To keep it up more power has to be added, and eventually it loses effectiveness and becomes as effective as a standard light sail.

Yes, momentum per photon depends on wavelength. But that photon exchanges momentum and the wavelength changes, so it still retains momentum. In a conventional sail this photon is just reflected and ignored. In a recycling photon propulsion system such as a Photonic Laser Thruster, this photon is reflected back towards the source, exchanges momentum with the source, and is reflected back towards the sail. This cycle repeats as long as practical. Thus momentum per photon is quite important, as the goal is to maximize the use of this momentum.

Efficiency of a recycling photon propulsion system in terms of energy used at the source versus energy the sail gains. In conventional photon sail systems the efficiency is quite low, as enormous amounts of energy are spent while small thrusts are generated at the sail. But the efficiency of photon rockets increases as the relative velocity approaches the speed of light. A recycling photon propulsion system is more efficient at lower speeds but eventually conventional photon propulsion systems overtake the efficiency of recycling photon systems. This happens at some percentage of the speed of light, dependent on the individual system design. This could be as high as 20% of the speed of light or as low as a few percent of the speed of light. Relative to the photon source.

The thrust from a single photon beam where each photon is used once, yes. But for a recycling photon propulsion system, the thrust is twice the power times the number of times the photons are reflected divided by c. 

This is still pretty hard to do, but it could be quite useful for beamed propulsion systems.

Here's an demonstration:

Wikipedia article:

https://en.wikipedia.org/wiki/Photonic_laser_thruster

It's not going to be easy to do at all, but could be worth pursuing.

Thanks, note that the star-shot project will try to launch an very small payload perhaps .2c plans of doing with with reflecting the laser a lot of times. 
The problem is that it will require 30-100.000 g acceleration. Yes this can be done with artillery shells but they are centimeter thick steel structures inside an even stronger tube for 20 meter maximum. Not an foil getting pushed by an laser array, an error of 0.1% would give an 30-100 g force on that part. 
Now the idea makes way more sense for accelerating heavier stuff slower as your main problem is distance to target and shear forces. Something heavier can easy use an more sturdy sail and as its accelerate slow it stays in range longer.