For starters, the waist radius is denoted $\omega_0$, which is the radius at the $1/e^2$ intensity as pointed out before, and the divergence half-angle is denoted $\Theta$. For this particular device those values are known. See page 5 of the guide, it notes the location of the beam waist is typically designed to be close to the output surface of the laser. I will be assuming that this is a real Gaussian beam and I thus won't be working with aperture. Again, Gaussian beam, not diffraction limited, and highly imperfect. I'll still be forced to take the reported "beam diameter" to be the representative of the $1/e^2$ radius, as well as being at the waist. This could be fairly wrong, but only by the same factor that my answer will be.

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The manufacturing process for anti-reflection coatings presents significant limitations. Most techniques cannot accommodate the deposition of AR coating on large-scale surfaces.

The aperture size of the laser is not the same as the waist size. If the beam is more or less collimated, then the aperture will still be larger, because the waist radius is usually defined in terms of the radius at which the intensity drops to $1/e^2$ of its peak value. If the beam is cut off by an aperture at that radius, then even if it were close to diffraction-limited, it certainly wouldn't be anymore. So, apertures are always larger.

What temperature could this reach? I'll just assume dissipation from blackbody radiation for now (back of the envelope).

Sol-gel chemistry processing is one of the most commonplace techniques for creating anti-reflection coatings and lenses. It uses metal oxides and organic solvents to condense the compounds into an inorganic polymer bond. [5] Standard sol-gel techniques include meniscus coating, dip coating, and spin coating.

Of course, the properties of an anti-reflection coating directly influence its useful lifespan. In particular, optoelectronic devices like camera lenses and touchscreens require the best anti-reflective coating possible. Ideally, the coating should have broadband, ultrathin thickness, and non-iridescent properties. [3]

5. Raut, H. K., Ganesh, V. A., Nair, A. S., & Ramakrishna, S. (2011). Anti-reflective coatings: A critical, in-depth review. Energy & Environmental Science, 4(10), 3779–3804. https://doi.org/10.1039/c1ee01297e

Note that if you know the $M^2$ and measure the divergence of a beam, then you can calculate the waist radius. We are going to do that now. Suppose the laser pointer beam is nearly collimated: you measure a divergence of 0.3 milliradians, about 0.017 degrees. Then the waist size is

2. Burghoorn, M., et al. (2013). Single layer broadband anti-reflective coatings for plastic substrates produced by full wafer and roll-to-roll step-and-flash nano-imprint lithography. Materials, 6(9), 3710–3726. Retrieved August 25, 2022, from www.ncbi.nlm.nih.gov/pmc/articles/PMC5452668/, 10.3390/ma6093710.

As you can see, anti-reflective coatings offer modern-day technology a world of opportunities for improving products, efficiency, and our quality of life. At Korvus Technology, we’re proud to be the leading source for deposition systems in the UK. To learn more, check out our blog or contact us online.

In addition, thinking of a Gaussian beam as being "straight" is not quite correct. There is always a waist, always a Rayleigh range less than infinity, and always a nonzero divergence angle.

Also, it is important to realize that there is no difference between an unfocused and a focused Gaussian beam. Refocusing a Gaussian beam with a lens just moves and resizes the waist.

Let's say that I have a laser beam of some given power that starts with some diameter $D_0$ at the point of emission and increases to $D_f$ at some distance $r$ away. Would this be sufficient information to imply a limit to the power per unit area (W/m^2) that could be obtained through focusing and what would that be?

Now, I am having great difficulty with this problem, and I'm still in disagreement with what ptomato has written as well as many other things I find online. My question, however, is still an objective and definable one, which is more or less "can the above laser cut steel with the appropriate lens?" Anonymous Coward gave a good reference, which is a guide to Gaussian optics. This document uses 1983 Sidney A. Self, Focusing of spherical Gaussian beams as its primary reference. The equations are the same throughout and are congruent to what you find in the prior Wikipedia links and such.

The literature goes into definition of several other values, as well as an expression for $\Theta$ itself. See page 3 of the guide to get the following equation without the $M^2$.

Physical and chemical vapour deposition are two other common manufacturing methods and require using complex deposition systems like the HEX Series. Etching is another conventional technique, but it uses selective surface ablation to achieve the desired AR coating. [3,5]

Anti-reflective coating and anti-glare lenses have dozens of practical uses for modern-day technology thanks to their unique properties. However, that doesn’t mean manufacturing AR coatings is easily accessible or affordable for the masses. As with any delicate and complex manufacturing process, there are certain limitations to consider.

Through thin film and vacuum deposition technology, you can apply an AR coating to an object’s surface (like that of a standard lens), reducing light reflections and eye strain. [3] Anti-reflection coatings also depend on their refractive index to minimise light loss on lens surfaces. [1,4,5]

$$ w_0 = \frac{M^2 \lambda} {\pi\Theta} = \frac{1.5 \times 671 \times 10^{-9}} {\pi \times 3 \times 10^{-4}} \approx 1\,\text{mm}. $$

The temperature that a solar death ray can produce is limited due to the solid angle of the sun itself. Entropic arguments dictate that you can't focus the sun's light to create temperatures higher than the sun's surface, and sure enough, the size of the dot where its focused can't be minimized beyond a certain limit. Lasers offer a very potent beam of light, but it would seem to me that it is also highly organized beam of light, since all of the photons in the beam are traveling in very nearly the same direction.

Once light passes through the air and meets a medium, the Fresnel equations can determine the amount of light reflected and transmitted, depending on the refractive indices. [1,3] The following equation defines the fraction of reflected light:

I just wanted to throw that out there. The guide has equations as to how to calculate the lens equations as well as the magnification factor, but I can't figure out how the equations imply that you can't focus this to a finite dot with a sufficiently large lens. I know this is not possible, so I had no choice but to apply a high school level approach to the problem.

You need a lens with a very short focal length. This gives you the largest convergence. Note that the more convergent the beam, and the smaller the waist size, the smaller the Rayleigh range is. That is, the beam radius will get very small, but it won't stay very small, it'll get bigger very quickly as you move away from the focus. (The Rayleigh range is the distance over which the beam radius increases by $\sqrt{2}$.

A “V” anti-reflection coating follows the same transmission and light reflectance principles as a single-layer coating. However, it undergoes optimisation to improve performance within a small niche of wavelengths. [1] The name derives from its high refractive index, creating a “V” shape that curves over multiple wavelengths. The centre arcs around each design wavelength (DWL). [5]

The highest numerical aperture you could dream of finding--with a good oil-immersion lens--is in the neighborhood of 1.5 (a bit more or less, I don't know). (Without oil-immersion, the numerical aperture is always less than 1.) So we get:

Korvus Technology LtdUnit 1 Barnes Wallis Court,Wellington Road, Cressex Business Park,High Wycombe,United KingdomHP12 3PS

Generally, anti-reflection coating applications have two purposes (besides eliminating reflections): to improve an object’s aesthetic or efficiency. [2] Regarding aesthetics, applications include anti-glare glasses, picture glass, and electronic displays.

Nanostructured lenses with AR coatings that have a gradient to increase the refractive index have effective anti-reflection properties. However, the nanostructures in the topcoat are a double-edged sword as they decrease the mechanical strength of the coating. [3,4,5]

Next, we’ll summarise the different manufacturing processes for anti-reflection coatings and lenses. These processes fall under two primary categories: conventional techniques and non-conventional techniques. [5] Of course, cutting-edge equipment – such as the HEX Series deposition system we manufacture – is necessary for creating anti-reflection coatings. Conventional techniques include top-down and bottom-up technologies. [3,5]

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Again, I'm going to act like a high school physics student, pretend I don't even know what wavelength is, and use geometry. I'll assume the angle the furthest beams are deflected at is 45 degrees, and denote the things about the area it focuses on as $dot$.

Let's try a more realistic example with some numbers. Take your red laser pointer with $\lambda$ = 671 nm. Laser pointer beams are often crappy, but not so crappy as you might think, if they are single-mode. Let's assume that this particular laser pointer has an $M^2$ ("beam quality parameter", which is the beam parameter product divided by the ideal beam parameter product of $\lambda/\pi$) of 1.5. A quick Google search didn't give me typical $M^2$s of red laser pointers, but this doesn't seem to me to be too much off the mark.

However, the inherent differences and bonds between the coating’s thin layer and the front and back surfaces of the substrate impact durability, hardness, strength, refraction, and reflectability. [1,3] Therefore, most anti-glare coatings are vulnerable to abrasion, which can pull off the coating on the lens surface. Thermal cycling and solvents can also cause stress or damage to the bond. [5]

or $\Theta\approx$ 6 degrees. Applying the formula once more to calculate the waist yields a waist radius of 3.2 microns, which is quite small indeed.

Below is my illustration of a beam of 100% collimated rays (I'm talking about Newtonian physics here), and the first vertical line is some hypothetical perfect lens, that would converge perfectly on the next vertical line. If the rays coming in were perfectly collimated then the rays would exactly converge on a single point, but we know they don't because of the divergence angle.

Now suppose you focus your collimated beam with a 1 cm focal length positive lens, which is quite a strong lens. The beam's new waist will be at the lens's focal length. That means you can calculate the divergence half-angle: it is the smaller acute angle of a right triangle with legs 1 mm and 10 mm. So,

If you’ve ever squinted reflexively after a bright sunbeam reflected off your windshield, you probably wished for a pair of sunglasses with an anti-reflective coating on the lenses to cut the glare. While light reflection is necessary for objects like mirrors, it causes absorption in glasses, telescopes, and lenses. However, depositing a special coating on the object’s surface (as in anti-reflective lenses) reduces reflections and glare, improving visual acuity. [1]

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Question: Let's say that I have a laser beam of some given power that starts with some diameter $D_o$ at the point of emission and increases to $D_f$ at some distance $r$ away. Would this be sufficient information to imply a limit to the power per unit area (W/m^2) that could be obtained through focusing and what would that be? What lens characteristics and approaches would someone look for in order to do this with a laser pointer?

The equation calculates the index of refraction for an optimal AR coating that will reduce reflections off the surface. [1,5]

A Gaussian beam is a beam where if you shine it on paper, the intensity has a Gaussian distribution. Easy enough. The Divergence half-angle is the angle it's diverging at, given a few qualifiers, from http://www.rp-photonics.com/beam_divergence.html

Ar coating

So you can focus a 645nm beam (if ideal gaussian) to an area of roughly $\pi w_0^2 \gtrsim 0.05\mu \mathrm{m}^2$. If the power is 5 mW, I get $100 \,\mathrm{ GW}/\mathrm{m}^2$, or 100 million times brighter than daylight.

The path length of the incident light will differ, reducing destructive interference. Many applications require single-layer anti-reflection coating, including photodiodes, lasers, and solar cells. However, the reflection dip in a single-layer anti-reflection coating makes it unfeasible for displays, lenses, and glasses. [3]

The anti-reflective coating cost varies based on the manufacturing process, necessary equipment, intended use, surface substrate, etc. [2] However, we’re happy to answer questions regarding the cost of anti-reflection coatings and how they can add value to your business.

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Far field is a point far to the right on this image where the width of the beam is close to the black lines. Some other mechanism prevents it from converging at a point in the middle because, of course, we could have already dismissed that as theoretically impossible.

Optical coating process

The problem is that they seem to keep giving equations for a diffraction limited case, saying nothing to clearly identify that's what they're talking about. In a diffraction limited case of course you need the wavelength, but my anticipation is that a $12.50 laser will not be diffraction limited.

You would also need to know the beam radius at the waist, so you could calculate the beam parameter product. Then, to get the minimum spot size, you would need to refocus the beam so that it is maximally convergent. The absolute limit is the fictitious divergence half-angle of $\pi/2$, or 90 degrees, although in practice the theory breaks down for half-angles of more than 30 degrees (this number is from Wikipedia) since the paraxial approximation stops being valid. For an ideal beam at this impossible opening half-angle, this gives you a minimum waist radius of $2\lambda/\pi^2$. So yes, it does depend on the wavelength.

A multi-layer AR coating contains multiple microscopic layers to improve performance and minimise reflection to less than 0.1% of incident light. Each thin layer is deposited onto the surface substrate to increase the destructive interference, maximising transmission. [3,5]

The parameters you have given are sufficient for calculating $\Theta$, but only if $r$ is large enough so that the points at which you measure the diameter are in each other's far field.

However, other applications like telephoto lens material, light-emitting diodes, and solar cell panels require AR coatings that maximise efficiency. [2] An anti-reflective lens coating that improves vision is also ideal for increasing available light transmission, enhancing contrast, eliminating ghost images, and sharpening visible focus.

1. Bauer, G. (n.d.). Anti-reflection coatings. PVEducation. Retrieved August 25, 2022, from https://www.pveducation.org/pvcdrom/design-of-silicon-cells/anti-reflection-coatings

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At Korvus Technology, we’re the UK’s premier source for thin film manufacturing, and over 25 organisations, universities, and brands trust our HEX Series deposition system. In this article, we’ll explain anti-reflection coatings, including different types, how they work, limitations, common uses, and more.

In practice, for a diode laser, your best bet is to first use a weak cylindrical lens to get the spot more round [all diode lasers have a built-in cylindrical lens, but in my experience the spot is still slightly elliptical], and then use a strong microscope objective lens (oil-immersion if possible) to focus as tight as possible. Also, put a two-lens telescope before the objective lens to either expand or shrink the beam so that it exactly fills the entrance aperture of the objective lens. Actually, maybe you shouldn't use an oil-immersion lens, you might burn the oil! Don't try this at home...

An anti-glare coating works by splitting light waves into two reflections. The split creates destructive interference, causing the light waves to cancel each other partially or entirely. [4] How the light waves travel and behave through mediums and interfaces determines how the AR coating will work. [5]

3. Keshavarz Hedayati, M., & Elbahri, M. (2022). Antireflective coatings: Conventional stacking layers and ultrathin plasmonic metasurfaces, a mini-review.” Materials 9(6), 497. https://doi.org/10.3390/ma9060497

A "safe" laser pointer might have a power of 1 mW. The peak intensity is equal to $2P/\pi w_0^2$, so before the lens the peak intensity is about 600 W/m^2. After the lens it is about 100000 times larger.

Micro-replication is another type of non-conventional manufacturing process. It involves a roll-to-roll process replicating nanostructures on a thermoplastic film surface, such as PVC. The photo-aligning technique is another method that minimises transmission to 99.1%. [5]

Some manufacturers use non-conventional techniques when creating an anti-reflective coating. Lithography falls under this category and consists of patterning the substrate surface with microscopic features. [5]

“V” AR coatings are for highly specialised applications that single- and multi-layer coatings are unsuitable for, like high-frequency lasers. Other applications include high index lenses, anti-reflective glasses with UV protection and less glare, digital microscopy, fibre optics, engraving, and more. [5]

I'm also interested if this is the mechanism used for laser cutting. Note that the Wikipedia article introduces the subject with Laser cutting is a technology that uses a laser to cut materials. Pardon the internet speak, but this makes me want to facepalm.

Furthermore, chemical vapour deposition or sol-gel chemistry creates a durable, strong AR coating. However, the process is prohibitively expensive, particularly for multi-layer stacks. Additionally, multi-layer filters are highly sensitive to variations in the refractive index and coating thickness. [3,4,5]

Another common limitation occurs with quarter-wavelength anti-reflection coatings. To lower the refractive index, manufacturers must use a porous coating material, which occurs in a single processing step. However, the coating’s porous nature reduces its strength and could make it more vulnerable to contamination. [3,4,5]

The mechanical and chemical properties of anti-reflective lens coatings make them invaluable for modern-day applications, including anti-glare glasses, lasers, display screens, optic lenses, and solar panels.

I should note that the Wikipedia has an article that kind of sort of addresses this but I'll be forthcoming that I don't understand it and find it confusing. Well, it mentions wavelength, which I recognize would limit the W/m^2 power if the beam was perfectly straight, but I'm interested in the limit due to manufactured imperfection of the laser, so unless somebody is over-complicating this (which is fine as long as you answer the question first), I don't think the answer should have anything to do with the wavelength. I mean, the wavelength should be sufficiently small compared to the size of the dot that could be created, correct me if this is the wrong way of thinking of it.

4. Nave, R. (n.d.). Anti-reflection coatings. HyperPhysics. Retrieved August 25, 2022, from http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/antiref.html

I'm okay with this value. I don't expect it to get a good laser. Next, the Rayleigh range, $z_R$ is a rough value beyond which the far lens approximation is valid. I think it might need a $M^2$ on it, but I really don't know.

A single-layer AR coating may only become anti-reflective at a single wavelength, typically in the visible middle. [4] When depositing single-layer quarter-wavelength AR coatings, they can reduce surface reflectivity for incidence angle and limited wavelengths. [3]

There is a limit to how small you can focus an ideal single-mode laser beam. The product of the divergence half-angle $\Theta$ and the radius $w_0$ of the beam at its waist (narrowest point) is constant for any given beam. (This quantity is called the beam parameter product, and is related to the $M^2$ beam quality measure you may have heard of.) For an ideal Gaussian ("diffraction-limited") beam, it is:

Most manufacturers switch between a low and high refractive index when depositing layers. Generally, anti-reflection coatings with multiple layers provide stronger broadband performance. However, the cost of manufacturing multi-layer anti-reflection coatings is prohibitive. [5] These coatings are more sophisticated than single-layer coatings and essential for optical applications, like lenses, astronomy, and aerospace telemetry. [1]

The waist is the thinnest point of the beam. Usually this point is inside the laser cavity, or outside the laser if there are focusing optics involved, which there often are. So still, the answer to your question is no. You are not missing the definition of $\lambda$; rather, you are comparing your minimum waist radius to the value of $2\lambda/\pi^2$ that I said was "impossible". I called it impossible, because to make a beam converging that strongly, you would need a lens with a focal length of zero!