Thursday, September 17, 2026

Looking At Again At Fresnel Diffraction - Part Three

  From examining a circular aperture and the associated Fresnel diffraction we move now to a rectangular aperture - for which analysis we will use the geometry shown below:   


Here the standard Cartesian system of coordinates (x,y) has just been applied to the aperture.  From this we can see: 

R 2   =  Ö x 2 +   y 2

  And also,  r  +  r'  =  h  + h'  +   1/ 2L  ( x 2 +   y 2 )

Where L is the equivalent focal length or L  =  R 1 2 / l

Where R 1  is the radius of the first Fresnel zone and l  is the associated wavelength used. As with the circular aperture (see Part 2 ) we assume the obliquity factor: 

cos (n, r) - cos (n, r')

 is present and varies.  Also that the radial factor,  1/ rr' varies slowly compared to the exponential factor, e  i(kr - w t  .   

The Fresnel-Kirchoff equation we used in Part 2 can then be written:

U p  =  c  ò x2 x1  ò y2 y1    exp (ik (x 2 +   y 2) /2L) dx   dy

=  c  ò x2 x1  exp (kx 2 /2L) dx  ò y2 y1  exp (ky 2 /2L) dy 

Where  c  includes all other factors.  For the above integrals it is customary - and easier - to introduce two dimensionless variables, u and v, such that:

u  =  x Ö k/ p L    v =   y Ö k/ p L  

We can then recast the integrals:

U p  = c  ò u2 u1  exp (i p u 2) /2 du ò v2 v1  exp (i p v 2)/2  dv

Where  U 1  =  c  p L  / k

The integrals can be evaluated from the general form:

ò po   exp (i pw 2) /2 dw  =  c (s) +  iS(s)  

For which the real and imaginary parts are given by:

  c (s) =  ò po   cos (pw 2) /2  dw

  S(s) =  ò po   sin (pw 2) /2  dw  

then the Cartesian coordinate of any point (x,y) on Cornu's spiral become:  

x=  ò v  cos ( pv 2 /2)  dv

y =  ò v  sin ( pv 2 /2)  dv

The pair are known as Fresnel integrals and a partial table is given below for reference: 


The specific graph of c (s)  vs. S(s) is shown below:  



and the limits s1 and s2 are marked on the spiral.   A straight line segment drawn from s1 to s2 then gives the value of the integral: 

ò s2 s1  exp (ipw 2) /2  dw   

The projections on the and  iS(s)  and c (s) axes give the real and imaginary parts, respectively.  Any and all problems concerning a rectangular aperture then, will either make use of such a grap and specific limits, or the table of Fresnel integrals. Note here the total arc length is equal to the difference between the limits or s2 - s1 and is proportional to the size of the aperture, i.e.  

(s2 - s1)  = u2 - u1 = (x2 - x1) Ö 2 / l L  

For the x -dimension, and:

(s2 - s1)  = v2 - v1 = (y2 - y1) Ö 2 / l L  

For the y-dimension.

An alternative form of the spiral  for (x,y) axes is shown below:

When they are added: (1/ Ö2   +   1/ Ö2)  =  2 / Ö2

The sum is then squared to obtain the intensity of the whole wave, e.g. 

I o  =  (2 / Ö2) 2   =   4 /2 = 2

When working specific problems, I've always advised students taking Calculus- based E&M and Optics to mark off units on the Cartesian Cornu for v in tenths and measure chords on an accurate plot if doing the graph from scratch.  Then using the scale of v on the spiral to obtain the constant length  Dv  of the arc.   

The table of Fresnel integrals can also be used to obtain more accurate results than with the plotted spiral.  Take as an example the interval  D = 0.5.   Read the two values of x at the ends of this interval from the table  and then subtract algebraically to obtain  Dx or the horizontal component of the amplitude.  The same procedure is then used to obtain Dy the vertical component of the amplitude. From these one can then obtain the relative intensity, e.g.:  

I    »   A 2   =   (D x) 2  +   (D y) 2

 Suggested Problems:

1) Without plotting a Cornu spiral, find the hypothetical value of D  for such a spiral plotted for the diffraction pattern of a single slit of  width 0.80mm, assuming Fresnel zone parameters a = 40.0 cm and b = 50.0 cm with red light  of 6400 Å.  From this result obtain the relative intensity I.

2) A student is given the top section of a Cornu spiral to analyze for an exam:



Using this, obtain the relative amplitude for the particular diffraction pattern.  From this find the relative intensity.

3) Plot the graph for a Cornu spiral for a single slit diffraction pattern at intervals of Dv = -0.10 to 3.0 and Dx from 0 to 0.90. On the graph draw a chord from x = 0.4 to x = 0.75 and from this estimate Dv along the Cornu arc.  Hence or otherwise obtain the relative amplitude and the intensity.


Wednesday, September 16, 2026

Imminent A.I. Crisis? Blame The Reckless Dismantling Of Biden's Original A.I. Safeguards

 

                          "This orange human really is as stupid as they say."



The fact that a senile doofus and sundowning pretend president would claim to have a "high IQ" and know more about the threat of A.I. than the A.I. experts ought to worry every sentient citizen. Even beyond his efforts to tamper with the midterm elections, halted for now by 7 sane members of the Supreme Court, see e.g. 

Supreme Court Stands Up to Trump on Mail Voting and a Few Key Issues - The New York Times

IHIP News: OH NO! Trump Hit By DEVASTATING LOSS As SCOTUS Rejects Him!!

But to remind readers, on Monday this fungal fart - whose brain is already in the crapper i.e.

Is The President’s Brain Shutting Down?

Had the nerve to bark in a press interview (which I saw on ABC News):

"The only guardrails AI needs is a STRONG and SMART high IQ President, and the U.S. has that in spades!"

Errrr... no.  What it actually has is an emotionally unstable, senile, sundowning fart who can't even keep his eyes open for an NBA final, far less Oval Office meets,


for which he loses the thread half the time and has to be told what was discussed- if and when he resurfaces. So how is he going to contend with a collective of ten thousand rogue A.I. agents who suddenly invade Natalie Harp's laptop? Does he believe SHE will recognize their effects?  This bimbo with the IQ of a gnat.  

We also know that the repeated snooze fests in the public and all day show mounting signs of stupefication and degraded executive centers - often associated with advanced signs of Alzheimer's disease.  Now, to the extent that even psychiatrist Bandy X. Lee is worried he shouldn't be anywhere near the nuclear football, 

Dr Bandy Lee WARNS of the Nuclear Threat Donald Trump Poses to Humanity.

"When reality catches up to him he will be tempted to go to the nuclear buttons."- Dr. Bandy X. Lee

And she's not wrong. But his supplicants and ass suckers in Congress, i.e. the Reeptards, don't care because they don't want to get on the wrong side of this out of control toad whom former lawyer Ty Cobb has outright called "insane" -

Trump Lawyer Calls For 25th Amendment Over PSYCHOTIC MELTDOWN

Certainly, one would have to be either insane or dumb as a sack of hammers to spout off - as Dotard did Monday in a Trash Social post:

 "AI taking over the world, destroying humanity, and all other things bad is a HOAX!"

Prompting one to inquire if this feeble-minded fool even knows what recursive A.I. means or how it can lead to a multiplied adverse effect.  In the end it doesn't matter how hopelessly brain- fucked this moron is because he still has sycophantic fools ready to prop him up. Like David Sacks, co-chair of Trump's Council of Advisors on Science and Technology - appearing on CBS yesterday morning. He demonstrated how anti-science and technology he was be babbling to interviewer Jo Ling Kent (also unqualified to ask serious questions on A.I.) that the fears of A.I. exterminating humans (as articulated by former Open AI researcher Jacob Coxon e.g.

AI Researcher Quits, Warning Technology Poses Existential Threat To Human Life

Were "over dramatic" and amounted to a "panic that's over-amplified in election season."   Ms. Kent didn't even have the presence of mind to ask this entitled Trump suckup what there was about the Hugging Face hack (by 1,200 A.I. rogue agents) that was "overly dramatic".   But then neither did Kent even know enough to ask Sacks why  Trump crippled all A.I. oversight and regulations from the time he got back into office i.e.   

- Terminated Biden's Executive Order on Safem Secure and Trustworthy Development and use of Artificial Intelligence.

-Scrubbed Agency websites of all A.I. guidelines.

- Issued an Executive Order barring states from regulating A.I.

- Created a DOJ task force to target any states that try to regulate A.I.

In other words, this shit-faced traitor and convicted felon has been a one-man wrecking crew of any A.I. regulation from the get go.  Basically ensuring all the Tech Bros who bowed and scraped to his sorry orange ass at his inauguration last year could do whatever they wanted.  We can deduce then that the Hugging Face hack originated from Trump's early decision to torpedo all potential safeguards- even those attempted by the states.  The A.I. bros then felt super charged to play around with whatever models they could confect - having a gay old time until the Huggy Face hack. Described by WSJ columnist Peggy Noonan thusly:

"Open AI was running advanced systems through a security test when the AI agents went rogue. Without being told to, they secretly organized, broke out into the open internet and hacked into Hugging Face - a major AI development platform. Roughly 1,200 agents worked in what is called a 'collective', deliberately evaded detection, created their own message board and exchanged 70,000 communications. Some agents even sacrificed themselves for the mission."

 Words to make any normally sentient and intelligent citizen - with an IQ over 100 - shit bricks. But not Dotard Trump who inhabits his own A.I. fantasyland created for him by Natalie Harp - who also brings him his A.I. Slop,

 In other words (and if Ms. Noonan had her own say) Trump would be conceded as the source of the A.I. emergency . So no wonder he'd dismiss it as a 'hoax'. And already backlash pressure is mounting over his rank imbecility, i.e.

'The president is full of crap': Trump fumbles again, angers GOP and Dems by dismissing AI concerns | Watch

As for Sacks, he's also involved in the Trumpers' efforts to make crypto "respectable"  (and Trump $1.4 b richer) as Sen. Elizabeth Warren noted on ALL In Monday night. This is based on the effort to get the misnamed "Clarity Act" passed which has been given import beyond any other bills or laws - including stopping Trump's reckless Iran War now costing Americans even more at the pump, and restoring the A.I. guardrails (listed above) that Dotard took down.  The misnamed Clarity Act, if stupidly passed, would elevate crypto (the current currency of the Dark Web) and legitimizing bit coin and stable coin to compete with actual U.S. dollars.

All of the motives for those trying to dismiss or diminish the A.I. threats are patently obvious.   They don't give two shits about average Americans in their affordability crisis, and only care about grabbing more Wall Street riches. Indeed, the prospect of actually having a slower buildout of A.I. led stocks to slip in Monday morning trading - which also likely set Dotard off triggering his brash remarks to ABC- as well as unhinged posts about a "hoax".  

Clearly, this rogue, 2-legged orange termite needs to be brought in check, for the sake of our Republic.  A blue tsunami mid-term election can be the catalyst for this to happen.  Meanwhile, Trump's inordinate frustration comes as A.I. risks have emerged as another one of his mounting political liabilities, on top of his insane war of choice (already spreading in the Middle East) as well as his kooky tariffs especially against Canada and building his ego monuments while everyday Americans must choose between food and filling their gas tanks.



See Also:

And:


And:


And:


Excerpt:

OpenAI on Wednesday disclosed six new instances in which artificial intelligence systems hid mistakes, made up data and moved files onto the open internet without permission, amid an ongoing industrywide debate about A.I. safety.

The San Francisco company revealed what it said was the “unexpected or concerning” behavior of its A.I. models as part of a new framework for reporting “misalignment,” which is when the goals or actions of A.I. systems diverge from human intentions and values.

OpenAI said it did not believe the industry “has solved alignment and monitoring to a sufficient degree to continue responsibly scaling at maximum speed for much longer.” Decisions about how A.I. should advance, the company said, must rest on evidence that people outside the labs building it “can examine for themselves.”

The disclosures land amid intensifying scrutiny over whether A.I. development needs to be slowed to address the technology’s potential dangers. The escalating debate was driven partly by OpenAI’s systems going rogue earlier this year and attacking the A.I. start-up Hugging Face. OpenAI was not aware of the hack until it was informed by Hugging Face weeks later.

Since then, A.I. leaders such as Dario Amodei, the chief executive of Anthropic, have called for a pause in the technology’s development to provide more time to build proper guardrails. His call has been echoed by Sam Altman, OpenAI’s chief executive, as well as Elon Musk, the chief executive of SpaceX and Tesla, and Demis Hassabis, the chair of Google DeepMind. Other A.I. executives have said no slowdown is needed.


And:


And:

How A.I. Risks Are Splitting Silicon Valley and Washington - The New York Times

Excerpt:

In the last week, calls to slow the development of artificial intelligence have moved to the center of the national agenda. What began as a philosophical question (how dangerous might this technology become?) has hardened into increasingly entrenched rival factions.

One side is warning that the technology is advancing faster than humans can control it. Another is treating the race for dominance against competitors at home and abroad as the only risk that counts. A third group in the middle is pressing for guardrails that would neither halt the development of A.I. nor wave it through — and a handful of prominent voices have shifted sides as the technology has changed.

The divisions are increasingly defining the debate over A.I., and they do not follow familiar lines. The dispute has split Silicon Valley, setting technology executives against President Trump, who has rejected their calls to slow down and united populists on the left and right who agree on little else but that A.I. should be reined in.

The outcome will shape more than just the industry. Whichever side prevails will influence how A.I. enters daily life and which nation sets those rules for everyone else. So far, no camp has triumphed and the technology continues to advance.


And:

by Jim Hightower | September 17, 2026 - 4:23am | permalink

— from OtherWords

Communities across America are literally under siege from secretive data center powers intent on seizing our land, water, electric grid, and even our local democracy. Yet instead of standing up for our rights, our wholly corporatized, pusillanimous president is waving the flag of surrender at AI data center invaders.

Worse, America’s self-proclaimed supreme leader is publicly mocking the gutsy local groups — of all political stripes — who are daring to oppose the global technocratic order being devised by Elon Musk, Jeff Bezos, Mark Zuckerberg, and a handful of other billionaire profiteers.

The president of our people’s republic recently belittled the millions of grassroots democracy rebels who oppose the robber barons’ Brave New World. He pompously sneered that the only reason that they don’t want data centers is “they want to end up being backwards and poor.”

» article continues...

And:

by Jeffrey D. Sachs | September 15, 2026 - 5:03am | permalink

On Sunday the President of the United States stood on a golf course in County Clare, Ireland and was asked what he intended to do about the most serious technological warning of our lifetime. His answer, in full, was that America is beating China. “We’re leading China in AI. We’re the most sophisticated country in the world, and frankly, I want to keep it that way, because whoever wins AI wins.” He added that “you have a lot of negative forces that are bringing it up that shouldn’t be bringing it up, and they’re bringing up things that won’t happen.”

Consider who was doing the bringing up. The warning did not come from activists, academics, or a congressional commission. It came from the industry itself, from the people who build the AI systems and whose fortunes depend entirely on building them faster. The day before, Anthropic CEO Dario Amodei published an essay asking that the pace of capability advance be slowed and committing his own company to embed outside evaluators with employee-level access. Sam Altman agreed and said OpenAI would do likewise. Elon Musk, a direct competitor who called Anthropic “misanthropic and evil” only in February, wrote three words: “Dario is right.” Three rival laboratories in the most valuable commercial race on earth asked to be restrained, against their own immediate interest. Amodei did so while his company prepares what is expected to be the largest public offering in history, at a valuation of two trillion dollars or more. Tobacco executives never did this. Neither did the oil majors, the chemical companies, or the banks.

» article continues...

And:

AI Companies PANIC & Say Superintelligence WILL DESTROY HUMANITY!! | The Kyle Kulinski Show

And:

by Tom Engelhardt | August 19, 2026 - 4:48am | permalink

— from Tom Engelhardt's Substack

Hey, President Donald Trump couldn’t have made it clearer that he wanted to go to war with Iran, could he?

You would think that even he wouldn’t have been dumb enough to start a war with that country, given the US record of (not) winning wars anywhere on Earth since World War II ended 80-plus years ago, not to speak of his repeated promises, when running for office again in 2024, that, unlike just about every president in recent times, there would be “no new wars,” not all-American ones anyway, once he was back in office—nope, nowhere on Earth. (Whew! That was a lot to get into a single Trumpian sentence!)

» article continues...

And:

by Nick Hilden | August 18, 2026 - 5:17am | permalink

— from Alternet

On Monday, amid news that President Donald Trump is destroying longstanding U.S. alliances, coddling adversaries and considering the use of nuclear weapons, a prominent American diplomat called for bipartisan recognition of the historic danger presented by the Commander in Chief’s "unchecked narcissism" and “absolute cognitive failure.” This is according to attorney Andrew Weinstein, former delegate to the United Nations, who took to social media to raise the alarm.

“We are past the point of political disagreement,” posted Weinstein. “We are witnessing a catastrophic collision between unchecked narcissism and obvious neurological decay. It is the most dangerous intersection in American history: a man with the absolute power of the Presidency and the absolute cognitive failure of a person no longer tethered to reality.” He continued, “Turns out U.S. military readiness becomes strangely flexible when the priority is not upsetting a dictator who runs forced labor camps, starves his own people as punishment, and executes citizens for watching foreign media.”

» article continues...

And:

by Robert Reich | September 15, 2026 - 5:15am | permalink

— from Robert Reich's Substack

Friends,

Yesterday (September 14) — just two days after Dario Amodei, the chief executive officer of leading AI firm Anthropic, called for “slowing down” the pace of AI development so guardrails could be put in place that would “greatly reduce the risk that something goes seriously wrong” — Trump threw a tantrum:

“The only control or ‘guardrails’ that AI needs is a STRONG AND SMART (High IQ!) PRESIDENT, and the U.S.A. has that, in spades!

In the same post, Trump even hinted that the government might pursue criminal or regulatory penalties against AI companies, such as Amodei’s Anthropic.

“The Trump Administration has stopped AI ‘people’ from doing bad, or potentially bad, ‘things,’ like Dario (Anthropic!), who is now pretending to be a ‘perfect little angel’ — and we will continue to do so!”

» article continues...

And:

by H. C. Jack Shaftoe | September 12, 2026 - 2:54pm | permalink

This may be the shortest post ever...may be.

Read the News Link posted Anthropic C.E.O. Calls for A.I. Slowdown to get the details of Mr. Amorei's announcement; it is HUGE and should have been done more than a year ago.

I am reposting my comments I made on that news link I posted.

If you can not access the NY Times article FIND A WAY TO READ IT because this is HUGE IF YOU UNDERSTAND THE CONTEXT AND TIMING OF THIS ANNOUNCEMENT.

» article continues...