[FRIAM] [WedTech] Diffracton: minding the gap

Sun Aug 15 08:46:19 EDT 2021

Congratulations, Steve! These moments of insight are rare and wonderful.

On Sat, Aug 14, 2021, 9:57 PM Stephen Guerin <stephen.guerin at simtable.com>
wrote:

>  Ed,
>
> Yes, that's how I'm seeing it.
>
> For others, Ed's Step function is what I was calling Rect pulse function
> which is Fourier dual of the Sinc function. Attached is an .mp4 recording
> of Ed description of the relationship of gap width, frequency, and the
> amount of spread of the Sinc function which is the diffusion pattern
> observed. I recorded the mp4 from the illustration/animation"  I linked
> to earlier
> <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/>
> .
>
> And I think it's pretty cool to think of the gap as a sampler. I suspect
> this is a well-known idea in optics/physics and old hat to many of you but
> it's exciting to come onto ideas like this for oneself :-)   I can almost
> hear John Zingale saying, "of course, it's just the convolution theorem
> <https://en.wikipedia.org/wiki/Convolution_theorem> applied to a square
> wave". In the past I would nod and frankly, my eyes glaze if I can't ground
> it in a microscopic understanding that guides my intuition.
>
> Or if given this amazingly deep statement I came across as I'm searching
> for connecting sampling and diffraction -  "*the diffraction pattern of
> an object is the Fourier Transform of the object*" from here
> <http://www.sci.sdsu.edu/TFrey/Bio750/FourierTransforms.html>, it finally
> makes sense to me.
>
> And I can practically hear Steve Smith and our dear and late Fred
> Untersher calling out, "that's how we've been describing holograms to you
> for 20 years and you always nodded like you understood".
>
> Or Ed who brought Pradeep Sen into our world saying what do you think I
> was showing you with Dual Photography
> <https://graphics.stanford.edu/papers/dual_photography/>, you idiot?
>
> And Alvy Ray Smith, again Ed bringing into our office,  saying that's
> what a Pixel is <https://youtu.be/dvHDXUV7hmQ>! it's not a little square
> nor gaussian point sample, it's Kotelnikov Sampling (Nyquist-Shannon
> Sampling Theorem),
>
> or potentially worse is Eric Smith and Roger Critchlow shaking their heads
> saying "you're just confused and making connections that aren't there". :-)
>
> --------------------------
>
> Now even after having said this, I *still* want to know how the
> diffraction is happening using only the interaction rules in the model.
> Obviously, there are no Sinc or Rect functions in the code, nor Fourier
> transforms explicitly coded. All these wonderful explanations above are
> emergent properties from the model I would call a macroscopic explanation
> and description. If nothing else perhaps I learn a better phrase for the
> level of explanation I'm asking for when you trace an algorithm and
> understand where the emergent property comes from. (BTW, I think I have a
> micro answer and will put it in my response to Alex).
>
> -S
>
> _______________________________________________________________________
> Stephen.Guerin at Simtable.com <stephen.guerin at simtable.com>
> CEO, Simtable  http://www.simtable.com
> 1600 Lena St #D1, Santa Fe, NM 87505
> office: (505)995-0206 mobile: (505)577-5828
> twitter: @simtable
> z <http://zoom.com/j/5055775828>oom.simtable.com
>
>
> On Sat, Aug 14, 2021 at 3:57 PM Angel Edward <edward.angel at gmail.com>
> wrote:
>
>> I hope someone can check out the analysis below.
>>
>> If you look at the gap as a sampler, you can do the following analysis
>> using Fourier methods:
>>
>> A gap is a window on a continuous function. A perfect gap is a step
>> function multiplying the continuous function.
>>
>> In the Fourier domain, the Fourier transform of the continuous function
>> on the input side of the gap is convolved with the Fourier transform of gap
>> (the step function).
>>
>> The Fourier transform of a step function is a sinc (sin(ax)/(ax))
>> function.
>>
>> The width of the main lobe of the sinc is inversely proportional to the
>> width of the gap.
>>
>> Consequently, the smaller the width of the gap, the more a given
>> frequency is distorted because the sinc is wider. Convolution applies the
>> sinc at each frequency of the input function.
>>
>> I think it gets more complicated when we add in sampling. If we take a
>> number of samples that is proportional to the width of the gap, then as we
>> make the gap smaller there are fewer samples, hence more reconstruction
>> issues which is the second, often overlooked, part of the sampling theorem.
>>
>> In the limit as the gap goes to zero width, there is no distortion to the
>> continuous function but in the digital world you could have only a single
>> sample.
>>
>> Ed
>> __________
>>
>> Ed Angel
>>
>> Founding Director, Art, Research, Technology and Science Laboratory (ARTS
>> Lab)
>> Professor Emeritus of Computer Science, University of New Mexico
>>
>> 1017 Sierra Pinon
>> Santa Fe, NM 87501
>> 505-984-0136 (home)   edward.angel at gmail.com
>> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>>
>>
>> On Sat, Aug 14, 2021 at 10:17 AM Stephen Guerin <
>> stephen.guerin at simtable.com> wrote:
>>
>>> At yesterday's Virtual Friam I asked a question on diffraction and said
>>> I would send more background.
>>>
>>> The gist of my question is:
>>>
>>> *Even though I completely understand the micro-level rules that generate
>>> diffraction in the wave model described below,  I still don't have an
>>> intuition **how** the gaps in an obstacle have the emergent effect of
>>> diffracting waves when wavelengths >= gap width. Can anyone help?*
>>>
>>>
>>> Background:
>>> The question arose from my mentoring NM School for the Arts high school
>>> students in the NM Supercomputing Challenge
>>> <http://nmsupercomputingchallenge.org/> where the students simulated
>>> spatial acoustics by appropriating Saint-Venant equations used for shallow
>>> water waves to instead model acoustic pressure waves. We wrote a
>>> Netogo agent-based model with Python extension for reading / writing
>>> the sound files and simulating spatial acoustics.
>>>
>>> <image.png>
>>>
>>>
>>> The students explored the effects of different room configurations on
>>> acoustics.
>>>
>>> One configuration of interest was a wall gap illustrated below in the
>>> top right under Madelyn's video below. The wall gap is hard to see on right
>>> side.
>>>
>>> <image.png>
>>>
>>> They simulated microphones in Netlogo by recording amplitudes at a patch
>>> (red dot below in top-right visualization of room) and simulated speakers
>>> (hard-to-see blue dot below red dot on other side of wall) by driving
>>> amplitudes at a patch from the time series of amplitudes in  .wav files
>>> (recordings of a singer and viola performance). They could hear, and
>>> through Fourier analysis, see the gap acting as a low-pass filter on the
>>> acoustic signal. ie, only the low frequencies were "bending" around the
>>> wall to reach the microphone.
>>>
>>> You can see and listen to this effect and the spectrogram visualization at
>>> time 33:11 in their presentation <https://youtu.be/61p97NWJiQ8?t=2117>.
>>>
>>> <image.png>
>>>
>>> It took me a few weeks after their presentation in the NM Supercomputing
>>> Challenge - they got second place - to connect the low pass filter behavior
>>> to the concept of diffraction. Had this been a light model and I saw the
>>> rainbow effects I would have clued in much faster.  Their presentation was
>>> a month after finals and they added this epilogue in the presentation
>>> above to identify the effect as diffraction.
>>> <https://youtu.be/61p97NWJiQ8?t=2761>
>>>
>>> Their presentation included this physical wavepool video demonstration
>>> <https://youtu.be/BH0NfVUTWG4> which was helpful to me to begin to
>>> understand the diffraction relationship with frequency and gap width.
>>>
>>> Note: my question is not about "describing" the behavior with
>>> macroscopic equations or geometric models but fundamentally how does the
>>> gap become a point source ala Huygens Principle at the micro-level of the
>>> patches interacting with the emergent waves. To help with the distinction,
>>> I consider this interactive model
>>> <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/> a
>>> great macroscopic description of the phenomenon that nicely illustrates the
>>> relationship of frequency and gap width but doesn't help me interpret the
>>> micro-level interactions giving rise to the diffraction effect in our
>>> simple shallow-water model.
>>>
>>> The students describe the details of the shallow water model at this
>>> point in their presentation <https://youtu.be/61p97NWJiQ8?t=870>:
>>> <image.png>
>>>
>>>
>>> Here is my simplified Netlogo wave model
>>> <https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html> of
>>> the same shallow water equations without the acoustics. It's set up to
>>> explore double slit but you can change it to single slit and mess with
>>> frequency and gap and watch the wave propagations, diffractions and
>>> interference patterns
>>> https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html
>>> <image.png>
>>>
>>> As a related aside, with some follow-up discussions with Ed Angel and
>>> Steve Smith I am also trying to understand how the gap might be considered
>>> a sampling function on the signal. My intuition is that the diffraction of
>>> the wave creates a spreader Sinc function and the gap is Rect
>>> function which are Fourier duals. In some way, i see Nyquist-Shannon
>>> Sampling Theorem
>>> <https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem> related
>>> to the gap. Note that diffraction creates a spreader function on the back
>>> wall in single gap experiments and the gap may be considered a Rect pulse
>>> when smaller than the wavelength.
>>>
>>> <image.png>
>>>
>>>
>>>
>>> _______________________________________________________________________
>>> Stephen.Guerin at Simtable.com <stephen.guerin at simtable.com>
>>> CEO, Simtable  http://www.simtable.com
>>> 1600 Lena St #D1, Santa Fe, NM 87505
>>> office: (505)995-0206 mobile: (505)577-5828
>>> twitter: @simtable
>>> z <http://zoom.com/j/5055775828>oom.simtable.com
>>>
>>>>
>> - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
> un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> FRIAM-COMIC http://friam-comic.blogspot.com/
> archives: http://friam.471366.n2.nabble.com/
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://redfish.com/pipermail/friam_redfish.com/attachments/20210815/6b016077/attachment.html>