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Python3, Molecular Dynamics, Mdanalysis, Plotly.
show_hist=False, show_rug=False, colors=colors ) fig.update_layout(legend=dict(itemsizing='constant')) fig.update_layout( legend=dict( orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1.2, font = dict(family = "Arial", size = 45), bordercolor="LightSteelBlue", borderwidth=2, ), | medium | 5,425 |
Python3, Molecular Dynamics, Mdanalysis, Plotly.
legend_title = dict( font = dict(family = "Arial", size = 45)) ) fig.update_traces(line=dict(width=10)) fig.update_xaxes( ticks="outside", tickwidth=3, tickcolor='black', tickfont=dict(family='Arial', color='black', size=50), title_font=dict(size=46, family='Arial'), title_text='Angle (θ)', | medium | 5,426 |
Python3, Molecular Dynamics, Mdanalysis, Plotly.
ticklen=15, range=[0, 95], tickvals=[0,30, 60, 90, 120] ) fig.update_yaxes( ticks="outside", tickwidth=3, tickcolor='black', title_text='Probability Density', tickfont=dict(family='Arial', color='black', size=50), title_font=dict(size=46, family='Arial'), ticklen=15, tickvals=[0.012, 0.024] ) | medium | 5,427 |
Python3, Molecular Dynamics, Mdanalysis, Plotly.
fig.update_layout( title_text="Dipole Vector wrt E-Field Distribution", title_font=dict(size=40, family='Arial'), template='simple_white', xaxis_tickformat = 'i', bargap=0.2, height=600, width=1400 ) Here we go: Interesting that we can start to align systems with the electric field and reduce their | medium | 5,428 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
We often face non-linear effects as researchers or data scientists. In traditional linear regression, this is a straightforward enterprise to handle in most circumstance. In logistic regression, in contrast, this is different because of the inherent non-linearity of the logistic (sigmoid) function. | medium | 5,430 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
That is, we must distinguish between the intrinsic non-linearity of the logistic regression model and the (possible) non-linear associations between the variables in the data. This post takes you through the particulars in this regard. As usual, I eschew symbols, equations, tables, and abstract | medium | 5,431 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
reasoning anyway I can for the benefit of presenting results in graphs and plots. Data and context The data up for analysis pertain to 1,033 Norwegian students answering a questionnaire on various tourism choices. The questions refer to the longest trip (in days) the students made during the summer | medium | 5,432 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
months, and the dependent variable is the general destination for the trip in question: a foreign trip (coded 1) or a domestic trip (coded 0). (Research of mine based on other variables in these data have appeared in Tourism Management, Tourism Economics, and Tourism Management Perspectives.) Key | medium | 5,433 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
information about the dependent variable appears in Figure 1. We note that making a foreign trip (in blue) was the most typical destination choice for both student sexes. More precisely, about 61 percent of the male students and 68 percent of the female students made this choice. (For the total | medium | 5,434 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
student sample, the foreign trip percentage was 66.) Figure 1. Trip destination by students’ sex and ‘sun-fun’ travel motivation: linear regression Figure 2 shows the association between making a foreign trip and students’ sex and ‘sun-fun’ travel motivation based on linear regression. The motive | medium | 5,435 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
variable ranges from 0 to 100 points, where 0 suggests “of no importance” and 100 suggests “of all importance.” The regression coefficients for the student sex variable and the travel motive variable are, respectively, 0.07 and 0.005. Both are significant at the 5% level at a minimum. (If you are | medium | 5,436 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
rusty on linear regression, please see my primer here.) Figure 2. The figure tells a translucent story. Female students (the blue line) are in general somewhat more inclined to make a foreign trip than male students (the red line). The more important insight, however, is the strong effect of the | medium | 5,437 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
travel motive: the more importance the ‘sun-fun’ motive has for the trip, the larger the probability of the trip being to a foreign destination. Yet we notice the problem of the linear regression model in the top right of the figure. That is, some females have more than a 100 percent probability of | medium | 5,438 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
choosing a foreign trip. This, as we all know, is absurd — probabilities are, by definition, bounded between 0 and 100 percent. For this reason, we should employ logistic regression rather than linear regression on these data. As I do next. (You find my primers on logistic regression here and | medium | 5,439 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
here.) Trip destination by students’ sex and ‘sun-fun’ travel motivation: logistic regression Figure 3 is the logistic regression-equivalent of Figure 2. Figure 3. Three things merit attention in the figure. First, we no longer have the problem of probabilities above 100 percent or below 0 percent. | medium | 5,440 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
Second, the travel motive variable appears to have a non-linear effect on the probability of making a foreign trip. Third, the sex difference in the probability of making a foreign trip appears to be contingent on where we are on the travel motive scale on the x-axis, as in an interaction effect. | medium | 5,441 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
The key word in the two last sentences was, yes, “appears.” These are neither non-linear effects nor interaction effects! No, they are the “pure and simple” mathematical consequences of the non-linearity inherent in the logistic (sigmoid) regression model. As such, we should not interpret the | medium | 5,442 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
associations shown in Figure 3 in terms of non-linearity or interaction effects … Trip destination by students’ sex and ‘sun-fun’ travel motivation: non-linear logistic regression To find out about any possible non-linearity in the association between the travel motive and the trip destination | medium | 5,443 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
choice in the data, we must add a quadratic variable to the model specification. Such a quadratic variable (the travel motive variable multiplied by itself) is added to the logistic regression model yielding the predictions depicted in Figure 4. (This quadratic variable’s regression coefficient has | medium | 5,444 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
a negative and significant effect.) Figure 4. Based on Figure 4, however, we can now make the claim that the ‘sun-fun’ travel motive has a non-linear effect on the probability of making a foreign trip. We can also extend the regression yielding Figure 4 to account for a possible interaction effect | medium | 5,445 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
between students’ sex and the travel motive. Yet since this model produces non-significant interaction effect, I see no reason to do it. Takeaways Logistic regression analysis produces non-linear effects as defaults in the sense that the statistical machinery of logistic regression is non-linear in | medium | 5,446 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
nature. To identify a non-linear effect in a logistic regression model, however, one must go the extra mile and incorporate solutions for handling non-linearities just as in traditional linear regression. About me I’m Christer Thrane, a sociologist and professor at Inland University College, | medium | 5,447 |
Data Science, Data Visualization, Logistic Regression, Tourism, Non Linear.
Norway. I have written two textbooks on applied regression modeling and applied statistical modeling. Both are published by Routledge, and you find them here and here. I am on ResearchGate here, and you also reach me at [email protected] | medium | 5,448 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Chichen Itza, Mexico (1986). Photo by author. A ‘Solution’ to Our Predicaments: More Mass-Produced, Industrial Technologies. Got into one of those social media discussions with someone yesterday morning. The post I was commenting upon is, unfortunately, no longer available and I failed to take a | medium | 5,449 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
screenshot of it when I originally commented. However, it was from the Globe Content Studio, a content marketing group of the Canadian newspaper The Globe and Mail. It was advertising content on the importance of new technologies to address climate change and global carbon emissions. Edit: The post | medium | 5,450 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
popped up again in my Facebook feed and I was able to grab a screenshot this time: These two images, I believe, are relevant to the conversation that evolved after my original comment: I have to admit that I’m not sure what this other person thought I was advocating besides wanting to curtail our | medium | 5,451 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
pursuit of industrial technologies to address atmospheric overloading (and other symptom predicaments of ecological overshoot) but perhaps some readers can discern something I am unable to see. Keep in mind that I share this dialogue as I have previously to provide a glimpse into the variety of | medium | 5,452 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
opinions, perceptions, and stories that are being circulated over social media and elsewhere regarding our predicaments and how they might, or might not, be addressed. Without further ado, here is the conversation and please note that I have copied verbatim and not corrected typos/grammar/etc.. Me: | medium | 5,453 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Complex, industrial technology is what has helped to create our ecological overshoot predicament. More of it only exacerbates the dilemma. Stop marketing the illusion that it can ‘solve’ anything. WS: Steve Bull the world is not out to “solve”. That is why the Global Energy Transition is called…a | medium | 5,454 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
transition. What part of that is so difficult yet to grasp. Are you interested in the problem or just dismissing it? Me: WS, Perhaps you don’t understand the difference between a predicament and a problem. Ecological overshoot is an example of the former — there is no ‘solution’ apart from a | medium | 5,455 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
correction via Mother Nature. WS: Steve Bull unless we slow the rate of acceleration by reducing and restricting burning. Exactly what the world has agreed to do. The run away acceleration of warming of the planet and the oceans is a PROBLEM no matter how articulate you try to spin it. So spare me | medium | 5,456 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
your Bull. Me: WS, Please peer behind the greenwashed curtains of said energy ‘transition’ being pushed by the mainstream media and politicians. Look at the work of Dr Bill Rees, Dr Simon Michaux, Derrick Jensen, Alice Friedemann, Dr Nate Hagens, Max Wilbert, Erik Michaels, and many others. | medium | 5,457 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Attempts to scale up non-renewable, renewable energy-harvesting technologies and their associated products will exacerbate the symptoms of overshoot including atmospheric sink overloading through hydrocarbon use (all of such technologies rely heavily upon them, and they have simply been additive to | medium | 5,458 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
human energy use over the decades — they have not reduced hydrocarbon use in the least). To say little about the continued destruction of ecological systems through their production, maintenance, and end-of-life reclamation/disposal. There is nothing green, clean, or sustainable about them. WS: | medium | 5,459 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Steve Bull Good grief. More deflective nonsense. So what do you suggest is to be done. Think I will stick with the 250+ scientists from 60+ countries and their collective 3 year study that aligns with NASA and the WHO and MIT reports on the troposphere where 75% of ghg gases reside elevating the | medium | 5,460 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
ceiling and trapping earth radiated and human induced heat in the lowest level of the atmosphere causing escalation in record heat events…record fires and fire seasons that are full month longer than 100 years ago. Record advancing drought and record hurricanes in frequency and intensity to the | medium | 5,461 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
extent of “rapid intensification” one day intensity increases. Record hottest years ever recorded and record warming of the oceans. Plain English talk about about the escalation of extreme weather records which 2010–2019 saw the most records broken of any decade in recorded history which was also | medium | 5,462 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
the hottest decade ever re order and likely both the records and the heat will be broken this decade and the next. Over 580 months without a single below average month for the planet for global mega surface temperatures. All is easily verifiable. I will check the work of the names you mentioned if | medium | 5,463 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
their names are not on my list of debunked contrarians. Your opinion is very well articulated but still reads as just opinion. You value it..I don’t. I prefer facts. Me: WS, I don’t disagree with the predicament created by hydrocarbon burning and subsequent atmospheric sink overloading. But I | medium | 5,464 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
return to my general thesis: it is our technology (that has been supercharged by the leveraging of hydrocarbons) that has led us to our overshoot predicament. Yes, reduce hydrocarbon use but this necessarily includes almost all modern, mass industrial processes including all those required to | medium | 5,465 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
produce non-renewable, renewable energy-harvesting technologies and their associated products. More technology (that requires industrial processes) is no ‘solution’. WS: Steve Bull As it is not solvable stating something is not a solution is redundant al…”I don’t disagree but” is just more | medium | 5,466 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
selection no matter how articulate. Reduction and restriction of emissions across all modes of transportation and burning for energy is the only practical direction which is the agree upon global direction. The rest of you commentary is just dismissive deflection and I believe you know that. You | medium | 5,467 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
can baffle people with BS but it is little more than a veiled vested interest in the status quo. Necessity fuels innovation and the debate is really over so I will take your point but don’t really see the point of it other than dismissive deflection. Me: WS, We will have to agree to disagree then. | medium | 5,468 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
The laws of thermodynamics (especially pertaining to entropy) and the biological principles of ecological overshoot trump what us naked, story-telling apes wish or hope for, especially as it pertains to supposed human ingenuity and our technological prowess. Here’s a recent paper by Dr Bill Rees | medium | 5,469 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
that might help inform you on these issues: https://www.researchgate.net/publication/353488669_Through_the_Eye_of_a_Needle_An_Eco-Heterodox_Perspective_on_the_Renewable_Energy_Transition WS: Steve Bull I do t need to be informed so don’t be condescending . Theory is irrelevant in terms of the facts | medium | 5,470 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
the drive the global direction that is necessary to attempt to slow down the rate of accelerating warming or planet and oceans leading to exponential decadal increase in disaster costs and economic loss and the potential tipping point collapses of multiple feedback loops. You ever been in a | medium | 5,471 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
disaster Steve? Theory is rather irrelevant Me: WS, So, you’re interested in just the facts but refuse to read more widely the researchers who have a different story to tell than those who support your perspective? You accuse me of supporting more of a status quo path when I am suggesting a | medium | 5,472 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
significant reduction in technology but you are arguing for replacing that technology with other industrial technology — which is much more a status quo path. And all the while you are saying that I am deflecting…sounds more like you’re projecting your behaviour onto me. Again, we must agree to | medium | 5,473 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
disagree on this. Enjoy the remainder of your day, I have better things to do than continue to engage in what is increasingly a pointless debate. WS: Steve Bull at least I am debating facts and not theory and perspective. We definitely disagree as the debate is really over and actions have been | medium | 5,474 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
agreed upon. I don’t have a perspective Steve. I have only a decade of research and following of weather records and climate altering extreme weather events. While you ponder your perspective your children if you have them and their children will have to live through devastating life threatening | medium | 5,475 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
extreme events the likes that have never been seen other than cataclysmic events. You break an ice cube into smaller pieces and the melting pendulum cannot be stopped. So either it continues to get to hot to live in some places with wet bulb temperature potentials …or…the unstoppable melting slows | medium | 5,476 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
or stops the currents that regulate climate. We simply waited too long debating the warnings and now action is needed to try and slow it down..not stop it or reverse it. Theory and perspective are at this point completely irrelevant. So drop out of this pointless debate in your opinion. I am happy | medium | 5,477 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
to have the last word. Me: WS, In reading through our discussion I believe that we may be speaking past one another. I am and believe that I stated that I agree with the predicament of atmospheric sink overloading, which seems to be your position. Correct me if I am wrong. My initial comment was a | medium | 5,478 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
challenge of the approach being pushed to address this predicament: more mass-produced, industrial technology. It was not to deny nor deflect a concern for emissions. In fact, my point is that to reduce this consequence of human impacts upon our planet as well as the other planetary boundaries we | medium | 5,479 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
have broached (such as biodiversity loss, land system changes, biogeochemical flows, etc.), we need to be reducing our industrial technologies, significantly — especially because they all require the continued use of hydrocarbons (and exponentially increasing use if we attempt to replace much or | medium | 5,480 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
most of our current technologies). This perspective is not theoretical in nature as you suggest. It is factual. Modern, industrial processes cannot continue or expand without hydrocarbons, except perhaps on the margins in very limited ways. Want to mitigate atmospheric sink overloading (and the | medium | 5,481 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
other boundaries)? We cannot do it via massive expansion of technologies as is being marketed (by those who stand to profit from this, not surprisingly), we need to reduce human population, consumption, and complex technologies. WS: Steve Bull well we seem to have been cut of for some reason as I | medium | 5,482 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
cannot load the post of see your comments where you suggested I did know the difference between predicament and problem. Predicament is soften terminology to what is a problem and life threatening one at that. You still are theorizing and discussing philosophy of perspective. I think that is | medium | 5,483 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
deflection even if it is a predicament. It is not practical to stop technology or production at this point in time as action to drastically reduce burning is a practical action for the situation. Truthfully do you how a way to reduce population in any kilns of significant manner and do you know | medium | 5,484 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
anyone that will voluntarily sacrifice their lifestyle. Humanity is addicted to comfort and convenience and your they is not applicable for a large enough scale. So talking about is not changing what needs to change now to even slow down the rate of extreme weather. Or just for lost lives and homes | medium | 5,485 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
and entire towns but for the unsustainable quadrupling of extreme weather related disaster costs and economic loss. Politicians have to protect employment levels and that requires feeding the machine. We just have to do so without burning. Period. So you keep theorizing and I will debate facts and | medium | 5,486 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
current events. I have been doing this for a very long time and have seen the extent of regurgitated deflection sponsored by organized and funded misinformation campaigns with what about isms and cherry picked data and you tube contrarians. While you may be 100% right of what is needed it still is | medium | 5,487 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
deflection of the action necessary right now. It simply is not practical to stop the prosecution. Only innovate that so it better and and in the meantime we must agree to reduce and restrict emissions whenever and however possible. You are clearly more educated than me but education does not always | medium | 5,488 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
equate to acquired knowledge. Happy holidays. I don’t know whether I May internet is sketchy or once again I have been sensores which has happens many times as my views that may be considered wrong by many are disliked but many as well. Especially if I bring up what the militaries are doin got | medium | 5,489 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
prepare for the inevitable while the debate is allowed to be perpetuated. Which is what your entire dialog feels like to me. If you’ve made it to the end of this contemplation and have got something out of my writing, please consider ordering the trilogy of my ‘fictional’ novel series, Olduvai (PDF | medium | 5,490 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
files; only $9.99 Canadian), via my website or the link below — the ‘profits’ of which help me to keep my internet presence alive and first book available in print (and is available via various online retailers). Attempting a new payment system as I am contemplating shutting down my site in the | medium | 5,491 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
future (given the ever-increasing costs to keep it running). If you are interested in purchasing any of the 3 books individually or the trilogy, please try the link below indicating which book(s) you are purchasing. Costs (Canadian dollars): Book 1: $2.99 Book 2: $3.89 Book 3: $3.89 Trilogy: $9.99 | medium | 5,492 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Feel free to throw in a ‘tip’ on top of the base cost if you wish; perhaps by paying in U.S. dollars instead of Canadian. Every few cents/dollars helps… https://paypal.me/olduvaitrilogy?country.x=CA&locale.x=en_US If you do not hear from me within 48 hours or you are having trouble with the system, | medium | 5,493 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
please email me: [email protected]. You can also find a variety of resources, particularly my summary notes for a handful of texts, especially Catton’s Overshoot and Tainter’s Collapse: see here. It Bears Repeating: Best Of…Volume 1 A compilation of writers focused on the nexus of limits to | medium | 5,494 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
growth, energy, and ecological overshoot. With a Foreword and Afterword by Michael Dowd, authors include: Max Wilbert; Tim Watkins; Mike Stasse; Dr. Bill Rees; Dr. Tim Morgan; Rob Mielcarski; Dr. Simon Michaux; Erik Michaels; Just Collapse’s Tristan Sykes & Dr. Kate Booth; Kevin Hester; Alice | medium | 5,495 |
Climate Change, Ecological Overshoot, Predicament, Renewables, Technology.
Friedemann; David Casey; and, Steve Bull. The document is not a guided narrative towards a singular or overarching message; except, perhaps, that we are in a predicament of our own making with a far more chaotic future ahead of us than most imagine–and most certainly than what mainstream | medium | 5,496 |
Laser, Optics, Physics, STEM, Light.
It is the coherence of light that makes patterns like the one shown below possible. If you have spent any time around the concepts of wave optics, it is a topic you must have come across. Here is a quick refresher: for two light waves to interfere and produce a maximum, they must be coherent and | medium | 5,498 |
Laser, Optics, Physics, STEM, Light.
carry a phase difference equal to an integral multiple of the wavelength of the source. The resulting pattern is ethereal to look at and I have attached an image below: Figure 1: Interference patterns produced via a double slit experiment. As you progress in your studies from secondary school to | medium | 5,499 |
Laser, Optics, Physics, STEM, Light.
graduate level classes, you will come across the division of coherence into temporal and spatial coherence. This post will discuss these two concepts in detail. The aim of this article is not to regurgitate the pure theory developed over the years about the coherence of light. Rather, my focus will | medium | 5,500 |
Laser, Optics, Physics, STEM, Light.
be more experimental in nature, and I am hoping to provide you with a practical means of understanding how these concepts affect the interference pattern produced by a light source. Temporal Coherence We begin by considering a Michelson interferometer as shown below, consisting of a light source S, | medium | 5,501 |
Laser, Optics, Physics, STEM, Light.
a beam splitter G, two mirrors M₁ and M₂, the former of which is fixed in place while the latter is moveable forwards and backwards, and finally, the observer O. If you recall, most variations of the Michelson interferometer also keep a glass plate to compensate for the optical path difference | medium | 5,502 |
Laser, Optics, Physics, STEM, Light.
introduced while traversing the beam splitter, but that is irrelevant in this scenario. Here is the apparatus we have with us now: Figure 2: Michelson interferometer assembly. Normally, the mirrors are kept equidistant from the beam splitter G¹, and what we expect is an excellent interference | medium | 5,503 |
Laser, Optics, Physics, STEM, Light.
pattern at the bottom of the setup. But imagine what would happen once I started steadily moving the mirror M₂ away from the light source. Naturally, the contrast of the interference pattern would decrease and in fact, for ordinary light sources like a sodium lamp, as soon as the extra separation | medium | 5,504 |
Laser, Optics, Physics, STEM, Light.
becomes a few millimeters, the fringes vanish completely. So why does this decrease in contrast occur? The answer is where temporal coherence of light sources enters the picture. One thing you must keep in mind is the fact that no light source in this world can emit continuous, unbroken wave | medium | 5,505 |
Laser, Optics, Physics, STEM, Light.
trains. Instead, all practical sources emit electromagnetic radiation that looks something like this: Figure 3: Wave trains united by a light source; size exaggerated. Let us suppose that the average duration of the wave train that our source emits is given as τ seconds². It is evident that there | medium | 5,506 |
Laser, Optics, Physics, STEM, Light.
will be no predictable phase relationship between each of these wave trains. If the time taken by our light to traverse the distances from G to M₁ and M₂ is significantly smaller than τ, then our resultant pattern will have excellent contrast since it is produced by the interference of two wave | medium | 5,507 |
Laser, Optics, Physics, STEM, Light.
fronts emanating out of the same wave train. But once the temporal distance³ from G to M₁ and M₂ becomes larger than τ, the contrast will start to decrease and eventually, our pattern will vanish. This occurs when the interference starts to occur between two entirely separate wave trains with no | medium | 5,508 |
Laser, Optics, Physics, STEM, Light.
predictable phase relation. This phenomenon is exactly what temporal coherence is. No light source can emit an infinitely continuous wave train and thus, even light from a lone source only remains coherent for a particular amount of time. In the example above, the time τ is referred to as the | medium | 5,509 |
Laser, Optics, Physics, STEM, Light.
coherence time of the light source, and the length of each wave train l=cτ is known as the longitudinal coherence length. Spatial Coherence To talk about spatial coherence, we turn our attention towards the famous and ubiquitous Young's Double Slit Experiment. In its original form, this remarkable | medium | 5,510 |
Laser, Optics, Physics, STEM, Light.
setup was able to demonstrate the wave nature of light and later, it even gave rise to a definitive demonstration of the dual nature of matter. Above all, the probabilistic nature of quantum mechanics can also be described via this experiment, but that is not our target for now. If you recall, the | medium | 5,511 |
Laser, Optics, Physics, STEM, Light.
original experiment utilized a single point-source. We will use two point-sources instead, viz. S and S’. Here is a diagram: Figure 4: The double slit setup we will use. Here, S and S’ are two point-sources which have no defined or predictable phase relationship. S₁ and S₂ are the two slits, | medium | 5,512 |
Laser, Optics, Physics, STEM, Light.
separated from the plane of the light sources by a distance D, whereas T represents the screen, on which we have selected an arbitrary point O. Note that as far as our interest goes, the areas away from O where the distances S₁O and S₂O differ significantly have no value. If there were only a | medium | 5,513 |
Laser, Optics, Physics, STEM, Light.
single light source, namely, S, then the interference pattern would be easy to predict. However, due to the contribution from S’, the pattern we see will be washed out and fuzzy. Part of the maxima and minima due to S will coincide with that due to S’, leading to a blurred picture. And when we | medium | 5,514 |
Laser, Optics, Physics, STEM, Light.
start to move S’ away from S, the contrast will decrease even further. There will finally come a point at which the maxima of one will coincide with the minima of the other and vice versa, leading to an effectively blank screen. Let us fix S’ here, separated from S’ by l. Now, we try to estimate | medium | 5,515 |
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the separations S₁S’ and S₂S’. Figure 5: Exaggerated view of the sources and slits. In the figure above, applying Pythagoras’ theorem yields the following relations: Equations 1, 2: The separations between the second source S’ and the two slits. The approximate expressions may be arrived at by | medium | 5,516 |
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considering D >> d, l. For the fringes to disappear, the path difference between the light waves that arrive from S’ at the two slits must equal half the wavelength of the source, i.e., λ/2. Thus, we have S’S₂ — S’S₁ = ld/D = λ/2. — (3) In other words, if instead of two point-sources, we had a | medium | 5,517 |
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single extended source from S to S’, it would no longer remain coherent beyond a certain distance l, approximately given as λD/2d. This is an effective way to understand what spatial coherence is. The light waves emitted from various parts of an extended light source will not be coherent and | medium | 5,518 |
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spatial coherence is a measure of the distance up to which, appreciable coherence may be achieved in such a source. Finally, from equation (3), you can infer that a good interference pattern would be obtained when the following condition is satisfied: l << λD/d, or d << λD/l — (4) The angle | medium | 5,519 |
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subtended by the light source at the plane of the slit can be approximated as l/D, and we represent it by θ. Let lₛ = λ/θ — (5) The quantity lₛ = λ/θ is known as the lateral coherence width of the given light source. It depends inversely on θ and thus, decreases with the length of the source. | medium | 5,520 |
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Conclusion In short, temporal, and spatial coherence may be understood by using a Michelson interferometer and the Young Double Slit Experiment, respectively. They may be defined as follows: Temporal coherence is a measure of the time period for which, light emitted from a source remains coherent. | medium | 5,521 |
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It is measured longitudinal to the beam and is related to the interval during which the light source in question emits a continuous wave train. Spatial coherence is a transverse measurement. It describes the length up to which an extended light source can go before the waves from its extreme ends | medium | 5,522 |
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become incoherent. The concept of coherence plays a major role in a variety of optical phenomena and is also one of the major factors that differentiate an ideal source from a practical one. An ideal source would have infinite temporal and spatial coherence. Further, a LASER light has much better | medium | 5,523 |
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temporal and spatial coherence as compared to other sources, so much so that it doesn’t need to be passed through a pinhole to create a coherent source (unlike normal sources). If you enjoyed reading this article, consider dropping me a tip via the link provided in the end. And for learning more | medium | 5,524 |
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about LASERs, I recommend the excellent LASERS: Fundamentals and Applications by Thyagarajan and Ghatak. Footnotes As mentioned, there is usually a transparent glass plate placed before the mirror M₁ to compensate for the fact that the light travelling towards it goes through the beam splitter | medium | 5,525 |
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twice. However, in this case, that is not really essential. Instead, when I say that the mirrors are equidistant from the beam splitter, I mean that the optical path traversed by the beams is the same, regardless of the geometrical path. Naturally, the time would be within the nanoseconds range. | medium | 5,526 |
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