Greatest mathematical holographic explanation ever?

[Ed Wesly]

Lots of people on social media are touting this as the end-all and be-all of all holo videos: https://youtu.be/EmKQsSDlaa4?si=x-w_2ahvBvxkJ7St

[Din]

Not too bad.
There was a bit of handwaving. You can only see the entire image from a cut piece of the original hologram if the image was far enough away, and the cut piece was large enough, relative to the image. In other words, the angle subtended by the image at some point at some distance from the film must be equal to the angle subtended by the film at the same point. If the image is of height i, and the piece of film is of dimension p, then the defining characteristic is the i/p must be less than the distance of the image from the film. So, for example, you cannot see a 10' tree a foot away through a hole half an inch wide.

There were also a few errors. Thomas young did not use a 'monochromatic source" for his two-slit experiments. "monochromatic" sources did not exist in 1801, and don't exist today. A monochromatic source would have to have an infinitely thin linewidth. Also, it's a common error to assume that the two-slit experiment "proved" that light is a wave. I think this source of error arises because De Broglie's hypothesis that particles had wavelike characteristics was proved with a two-slit experiment. Huygens (1629 - 1695) had already postulated a wave theory. Grimaldi (1618 - 1663) had already discovered diffraction (a wave effect) and Malus (1775 - 1812) had discovered polarisation. What Young did was to alter the polarisation of the light going to one slit. If the polarisation of light going through one slit was perpendicularly polarised to the other, the interference disappeared. This showed that light was a transverse wave - a fact not known till then.

The radius of the nth ring is:

r(n) = nλD + n²λ²/4

Benton gives a different formulation in his MIT notes (as well as the phase footprint of other geometries):

Benton.jpg (25.18 KiB) Viewed 717257 times

[lobaz]

Lots of people on social media are touting this as the end-all and be-all of all holo videos: https://youtu.be/EmKQsSDlaa4?si=x-w_2ahvBvxkJ7St

Definitely agree with that rating. I have never seen any video explaining principle of holography better, not even close. Are there any simplifications or imperfections? Indeed they are, any explanatory material must simplify which leads to imperfections. The important point here is that it is not oversimplified and provides a good physical insight of the matter. Moreover, it covers many subtle points necessary to understand in order to begin with computational holography. In fact, when I was teaching computer generated holography, I used similar reasoning as presented here. I guess students would welcome such a supplementary video.

[Din]

The problem I find is that this is not the "greatest mathematical holographic explanation ever" A lot of display holographers got angry (and I'm being polite here!) whenever I presented mathematics on this forum, repeatedly stating (yelling?) "We don't need math in holography". These same people are calling this the "greatest mathematical holographic explanation"??? Can anyone here follow the mathematics, specifically the Taylor expansion in one step of the derivation of the grating equation? There are people whose hearts go aflutter when they see a mathematical expression, depending on who states the mathematics. The Pythagorian theorem is true regardless of your skin colour or religion, but not to some people apparently. A scalar equation cannot determine the value of a vector quantity, unless you think there should be no math in holography and still think this is the "greatest mathematical holographic explanation ever'. Without naming names, I mentioned that my professor in College was Nobel Laureate (Abdus Salaam - Nobel 1979 for the Electroweak Theory, ). One person who said 'there should be no math in holography' went on to say that the Nobel prize was worthless!!!!!!!

Where do I begin?
The idea that the phase profile of a point is a zone plate and the idea that the phase variations are being recorded was brought up in the video by TJ for the Encyclopaedia Brittanica ( https://www.youtube.com/watch?app=deskt ... Mlkc&t=38s ) in 1972.

The idea that the pattern and it's reconstruction is based on the fact that the equations are similar is only partially true. It's only true for thin transmission hologram - thin being based on the idea that Λ/d <~ 0.1. (I hope no one now says we don't need math in holography!). The actual reason that holography 'works' is that the classical wave equation, derived from Maxwell, is a linear equation; the linear sum of any number of solutions is itself a solution. If the two solutions R = exp(ikx) and O = exp(ik'x) are both solutions, then the sum is also a solution and, if you take the modulus of the sum, you get a sinusoidal variation of intensity due to the phase difference k-k'

In the last few minutes of the video (~ 43 mins), he "explains" the object wave by: R.|R + O |² = R(R+O)(R*+O*) = <terms in R, R* and O*> + RR*O
and then, like a magic act - "that's why you see the image! How many people understanding the "greatest mathematical holographic explanation ever" understood the complex algebra here?


In this video, the author derives the grating equation (details in the video):

λ = √(L² + x² = 2xd) - √(L² + x²) ~ 1/[2(L² + x²)][2xd]

But, this is a Taylor expansion to the first order. All the derivations of the grating equation I've seen are not based on the first order Taylor expansion.

Then there seem to be two common myths:
1. If you cut a small piece of the hologram, you can see the entire image in any small piece, no matter how small. This is not true. The entire image must subtend the extreme angles with respect to the size of the cut piece. If the image is not far enough away, or the piece is too small, you won't see the entire image. So, it's only true for a transmission hologram of a (relatively ) small image far enough away. If you cut a small piece of a Denisyuk, you will not see the entire image.
2. "Holography" meant the "whole image" and so the film captures the 'entire object'. Not true. You don't capture the back of the object. What Gabor meant by 'the whole image' is that both phase and intensity were captured. the word 'whole' had nothing to do with a pictorial image.

There are other videos explaining at this level, or better. For those who don't follow mathematics ( and most people here don't - 'there should be no math in holography'), there's this: https://www.youtube.com/watch?v=0ics3RVSn9w

[BobH]

It's only true for thin transmission hologram - thin being based on the idea that Λ/d <~ 0.1.

YOUR math is the only math in question. The same math that led you to argue that a transmission hologram is "thin" if Q= <10.

[Din]

It's only true for thin transmission hologram - thin being based on the idea that Λ/d <~ 0.1.

YOUR math is the only math in question. The same math that led you to argue that a transmission hologram is "thin" if Q= <10.

So, perhaps you can criticise 'My math"?
I claimed that, in the video, the derivation of the grating equation is given by the Taylor expansion to the second term:
λ = √(L² + x² = 2xd) - √(L² + x²) ~ 1/[2(L² + x²)][2xd]

But, this is a Taylor expansion to the first order.


Since you seem to adept enough at mathematics to criticise "my maths", perhaps you can either determine the first order term in "Bob Hess math", or, give me the next order in "Bob Hess math". I'll even give you a hint. The Taylor expansion in standard math in standard mathematics textbooks (you know, the kind you've never opened) to three orders is:

f(x + δx) - f(x) = δxf'(x) + {(δx)²/2!}f''(x) + {(δx)³/3!}f'''(x) + ...{(δx)ⁿ/n!}fⁿ(x)

So, what does "Bob Hess math" give as the the third term? You do understand the question, seeing as how you criticise "my math", as if it's somehow different from standard math? You can differentiate, right? I'm sure you can differentiate, as you're such a paragon of higher mathematics! Or, does "Bob Hess math" not need basic calculus?

I'm also curious how "Bob Hess math" derives the grating equation? Perhaps you could enlighten us.

Or, are you ignorant and clueless about any mathematics and are spewing your usual diatribes? Sincere criticism requires knowledge of the subject being criticised.
"The truest characters of ignorance are vanity and pride and arrogance" Samuel Butler

[BobH]

So is a thin hologram Q=<10 or Q=<0.1?

[Din]

Since you ask:
"Any hologram in which the thickness of the recording material is small compared with the average spacing of the interference fringes can be classified as a thin hologram" P. Hariharan, "Optical Holography" ( my bold). In other words, follow this carefully as it might get complicated for your brain, the fringe spacing relative to the material thickness. It's not the absolute thickness - it needs to be compared to the material thickness. In standard math, not Bob Hess math, this means that 'thin' is classified by the criterion Λ/d is small.

"...thin holograms...correspond to Raman-Nath diffrraction regime..."
"Small values of Q correspond to thin gratings. However this criterion is not always adequate"
"...the transition boundaries between the various regimes depend on modulation amplitude as well as the value of Q. It is possible to allow for this by the introduction of another parameter P [Nath (1938) "Diffraction of Light by Supersonic Waves", Proceedings of the Indian Academy of Sciences]:
P = (λ/Λ)²/(δn)" P. Hariharan, "Optical Holography" (my bold).

""A classification places <holography> in either the thin or thick (volume) grouping. This distinction is made according to whether the modulation extends through large or small distances compared to the spatial interval of the grating" "Handbook of Optical Holography" ed H. J. Caulfield (my bold). Inother words, the distinction between thin and thick is based on whether Λ/d is small.

There are other references, some of which declare a thin hologram as one capable of dispersion. Since dispersion is the criterion by which rainbow holograms "work", if you can make a rainbow hologram on any medium of any absolute size, it's classified as "thin".By the way, thinking that a 16 micron hologram cannot be thin is showing ignorance. The absolute thickness of any medium must be based on a relative dimension. For example, your limited experience of silver halide may make you think 16 microns is "thick", but I've had to coat a 100 micron emulsion in DCG, which is much thicker than 16 microns (except in Bob Hess math). I've also recorded on a 1 cm thick Lithium Niobate crystal, which is 10,000 microns, which is much thicker than 16 microns (except in Bob Hess math).

[BobH]

So is a thin hologram Q=<10 (as you stated) or Q=<0.1?

Are transmission holograms necessarily "thin" (as you stated)?

Is Covestro's 16 micron material a "thin" material with visible light (again as you stated)?

No explanation necessary.

[Din]

So is a thin hologram Q=<10 (as you stated) or Q=<0.1?

It depends, specifically on the spatial frequency. I assume you can read. I was quoting Hariharan, so I'll repeat Hariharan:
""Small values of Q correspond to thin gratings. However this criterion is not always adequate"
And, once again:"However this criterion is not always adequate"

Are transmission holograms necessarily "thin" (as you stated)?

Yes. The distinction is made dependent on the spatial frequency. If the Bragg condition is not met, then there is dispersion. The Bragg condition is given by equation 14 in Kogelnik, reproduced:

cos(φ - θ) = K/(2β)

where φ and θ are the angles between the K and ρ vectors, K is the grating vector and β is the propagation constant. If this condition is met, there is no dispersion, and the hologram is considered 'thick'. If this condition is not met, then there is no Bragg selectivity, and dispersion exists. In this case, the hologarm is considered 'thin'.

Is Covestro's 16 micron material a "thin" material with visible light (again as you stated)?

Firstly, I did not state that a 16 micron material is necessarily 'thin'. I said that an absolute figure is invalid, since an absolute figure is meaningless. I know such sophisticated thinking must be difficult, do try and pay attention. Secondly, it depends on what you mean by "visible light". Light itself cannot be seen, only the interaction of light with matter. However, all 'light', when it interacts with matter is visible, assuming that the word 'light' refers to electromagnetic radiation in the range 400nm - 700nm, as determined by Maxwell's equations ( http://hyperphysics.phy-astr.gsu.edu/hb ... maxeq.html ). If the hologram on 16 micron material shows dispersion, it's 'thin', as stated above. If it meets the Bragg condition cos(φ - θ) = K/(2β), then it's thick. One more time: the defining condition is determined by the spatial frequency.

By the way, I don't see anywhere in the video that's the subject of this post any statement about "thin" or "thick" holograms. Therefore, this insistence on the difference between "thin" and "thick" has off topic, and is simply an extension of your hate-filled cyber-bullying. You clearly don't understand the point of dispersion relations or Bragg conditions.

If there are any more posts on this "thin" and "thick" differentiation, I will report it as bullying and hate messaging.
I repeat:

If there are any more posts on this "thin" and "thick" differentiation, I will report it as bullying and hate messaging.