# B Is it possible to collimate monochromatic light with a taper?

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1. Nov 23, 2016

### norrrbit

I am ignorant about laser physics, but i have this idea of collimating monochromatic light with an optical fiber/cable that is cone shaped: at inlet the cone has a wide diameter (several hundred microns - like an ordinary optical fiber, which might be used as a light source), on the other end it narrows to almost light wavelength diameter (e.g. 1.1 wavelength) and furtter for some distance it keeps this small diameter constant (the tail). I imagine that light would be squeezed in such a device to be almost unidirectional, with good beam quality. It would also greatly increase optical power density at its end, which would probably make it difficult to operate continuously. Maybe using hollow, cryogenically cooled, small cone angle and long fiber would help.

My question is: practical challenges aside, are there some fundamental physics laws that make such a device pointless? Will be the outgoing light squeezed into narrow, good quality and much more intense beam? I have a feeling that my intuition is naive in some way, but I don't know how.

2. Nov 23, 2016

### Staff: Mentor

There may be some way to do it with a graded index material maybe, but with a uniform index material, I don't think you can do what you want. Just draw a ray-tracing diagram in 2-dimensions, showing the path of a ray that reflects off of the inside surface of the tapered material. How does the angle of incidence change as the ray reflects successively off of the top and bottom taper surfaces? Do you see what happens with several reflections?

3. Nov 23, 2016

### Henryk

Yes, there is a fundamental physics law that tells it is impossible. It is the second law of thermodynamics.
Basically, you are trying to squeeze radiant power into a smaller area and solid angle, that is, increase irradiance.
If you could do that, you could build a system that transfers radiant energy from a source at lower temperature to another body at higher temperature.
For any optical system, the irradiance of the image is never greater than that of the source.

4. Nov 23, 2016

### Staff: Mentor

interesting. Could you post some links and references for that? I'd like to learn more about the fundamental limitations for this. That's my intuition, but it's better to understand the fundamentals, IMO...

5. Nov 24, 2016

### norrrbit

@berkeman Thanks for your time. Yes, I think I get it. With shrinking the pathway, light rays approach to being perpendicular to fiber walls. I guess that after passing the acceptance angle of the fiber, they will be dispersed. Right?

6. Nov 24, 2016

### norrrbit

@Henryk Thank you for answering. But I do not understand your point. Isn't it that lens are doing just that: squeezing radiant power into a smaller area?

7. Nov 24, 2016

### davenn

not in the same way, with a lens, the ray paths are focussed to a smaller point
in your cone, any rays that hit the side of the cone will get dispersed

I would be very surprised if you get any form of collimation from a cone system as you have described

8. Nov 24, 2016

### Henryk

berkeman,

I don't have any link but just think about it. Suppose you have a black body of a surface area S2 and temperature T1. Now, collect all the radiation and focus on a smaller body of surface area S2 at the temperature T2. The second body will receive all the radiation from the first body that is $\sigma S_1 T_1^4$ but will emit $\sigma S_2 T_2^4$. If $S_2 < S_1$, the second body will receive more radiation power than it emits even if its temperature is higher thant $T_1$!! You can repeat the same argument if both body are white except a narrow spectral range and emit/absorb only in a finite solid angle.

norrbit,

You are right, if the size of the image is smaller than the object, a lens will squeeze light into smaller area BUT the light will come from a larger solid angle.
So, yes, in principle, you can squeeze light into a smaller area but the beam will be proportionally more divergent (or convergent).

Now, going back to your idea of a fiber with narrowing diameter. The fiber will contain the light only if the incidence angle at the core/cladding interface is larger than the critical angle required for the total internal reflection. If you put the taper, that angle will decrease causing some light to escape.

Henryk

9. Nov 24, 2016

The radiance (brightness) with perfect imaging stays the same between object and image. It can not be increased. This is known as the "brightness theorem" in optics. If you change the image size and make it smaller, the rays making up the image necessarily have a larger divergent angle. Basically $P=LA \Omega$ is conserved where $P$ is power and $L$ is the radiance(brightness). $A$ is the area and $\Omega$ the solid angle of the rays making up each point on the image. $\\$ Highly coherent beams work under a slightly different set of rules. For these, there is a diffraction limit, for which a beam going through a narrow slit necessarily has a divergence to it, based upon diffraction principles. A collimated collimated beam can be made to have a very narrow divergence angle by making it sufficiently wide. Basically the equation $\Delta \theta=\lambda/b$ defines the limitation, where $b$ is the beam width and $\Delta \theta$ is the divergence(In radians).

Last edited: Nov 24, 2016
10. Nov 24, 2016

### cosmik debris

I'm probably missing something obvious but isn't collimating a laser done every day in optics labs with a lens and a pinhole?

11. Nov 24, 2016

Yes. However, the OP is wondering if it is possible to funnel the collimated beam into something like a fiber optic and still have it be collimated. Perhaps a partial answer to this question is oftentimes a collimated beam is brought to a focus with a lens and injected into a fiber. It is able to propagate through the fiber so long as the cone of the focussed beam lies within the acceptance cone for the fiber.

12. Nov 25, 2016

### Cutter Ketch

The size of the beam at a waist times the angle the beam makes leaving that waist is a conserved quantity. It is called etendue, beam quality, or brightness. You can look it up under any of those terms. The size and angle are conjugate variables and together constitute a phase space for the light. How the beam fills that phase space is how disordered the beam is i.e. entropy. You can trade beam size for angle, but you can't reduce the size of the phase space used, you can't change the product of the two, You can't lower the entropy without throwing away some light. It is easy (often unavoidable) to make the beam quality worse.

Devices similar to what you describe are used all the time. They are called tapered prisms and they are very common in pump optics for lasers and in solar concentrators. Tapered prisms use total internal reflection to squeeze the pump light down to the size of the gain medium, sensor, or solar cell. (Fiber optic tapers are similar). However, in squeezing the beam down they don't decrease the angles. They increase the angles in accordance with the beam quality constraint, and actually usually quite a lot worse. Kaleidoscopes are good at making the beam quality worse. (which confusingly makes all numerical measures of beam quality bigger so "increasing the beam quality" usually means worse, not better).

Picture a collimated beam going into a tapered prism. The part of the beam in the middle doesn't reflect. The edges of the beam are reflected to fold over onto the center of the beam. However in so doing, those parts reflect at twice the taper angle. If you keep going and let the light reflect again on the other side of the taper the angle will double. In just a few reflections the light will actually reflect back up the cone! So even with increasing the angle there is a limit to how much you can squeeze.

Someone asked if a lens wasn't doing the same thing you describe. Note that to make a small spot at the focus, the lens puts a large cone angle in the beam. If this sounds like the diffraction limit, that's because it pretty much is the same thing for the same reason. And just like the diffraction limit, there is a minimum beam quality which is called a diffraction limited beam or M^2 =1 (m^2 is a well defined measurable quantity often called "the beam quality") or sometimes related to the theoretically lowest possible transverse mode in a laser cavity and called TEM00.

Others mentioned laser collimators. These do the opposite. They are telescopes that make the beam bigger and in increasing the beam diameter decrease the angular content.

Someone already noted that a pin hole achieves improved beam quality by throwing away light.

I'll throw in one parting shot. The transverse disorder is not the only phase space. I just found a paper that shows you can take the disorder in the transverse dimension and stuff it into the longitudinal dimension thus doing the apparently impossible: making the beam quality better ... by making the longitudinal modes worse. I haven't looked at it carefully yet.

13. Nov 27, 2016

### cosmik debris

So I was right, I have missed something completely obvious, thanks. :-)

14. Nov 28, 2016

### Andy Resnick

I think you mean "For any loss-less optical system, the radiance at the image is never greater than from the source."

15. Nov 28, 2016