I am writing a graduate physics thesis on diffraction limited vision, and did some research. It is my understanding that the factor of 1.22 is given by x1/pi, where x1 is the first solution to the differential Bessel Function of the first kind (which looks like a sinusoidal curve decaying proportionally to 1/sqrt(x). The complicated Bessel Function is used to describe the dark fringes that arise in an airy disk. Thats the best I can do, I see differential equations, and want to run and hide.
the 1.22 comes from the diffraction pattern you see. Basically, if you look at a double slit, you get this diffraction pattern that just oscillates indefinitely (cos^2). But with the circular slit, the higher order fringes die off in intensity, and the dying off is controlled by a function called a bessel function (think of it as a decaying cosine). The part we see the most is the primary maxima, and we can say that the size of the maxima is the distance to the first minima, which is determined when the bessel function passes through 0 in its oscillation. The value where it passes through zero (divided by pi or something in the calculations) is where the 1.22 comes from
Yay, Bessel Functions and Airy disks! You are awesome, Kaitlyn. Do not run and hide, but rather stay and fight. We need you. I particularly like your clear depiction of how the first Bessel looks. You would make a fine teacher. Thanks for the explanation. I'm sure it will suffice for my high school students!
Google Bessel function circular aperture and you might find a website that derives the intensity of diffracted light through a circular aperture as a function of angle theta, which should give you something like I(theta) = I(0) (2 J_1(x)/x)^2, where x = kr sin theta, where k is the wave number 2 pi / lambda and r is the radius of the circular opening, which is half the diameter d. Then the first minimum of the diffraction pattern is when I(theta) = 0 for the smallest value of theta; but I(theta) is only 0 if J_1(x) is 0, and the smallest root of that Bessel function is at x = 3.832; but x = kr sin theta = 2 pi / lambda * d / 2 * sin theta = approximately d pi theta / lambda, so you get theta = 3.832 / 3.142 lambda / d = 1.220 lambda / d :D
Ok, so you want to know why there is 1.22, I want to know why there are sin symbols? What is the point of that? What is the point of 1.22? The whole point is the diameter of an object that is narrow to expand, which is gap distance from furthest to nearest to you as you are stationary, so why need sin symbols and 1.22?? That would be enough just you and the gap diameter of the light itself expand or narrows, or an object that narrows or expand, so the whole point of Rayleigh Criterion is expanded of two edges of an object, so the object expand, so the objects become two, which you can see the details in between like in moon which you can't see the many dots, but closer, you can see many dot, same with hair, you can't see the individual until you get close, so the whole point is diameter of an object that is two outer edges, to inner two edge, and between two inner edges which it is a diameter. So, what matters is you are stationary, and the object is moving, so it requires one stationary, or you can be dynamic, but the object in front of you is stationary. Or, the gap diameter of two objects is stationary, so you are moving toward, or you are stationary, and two objects are stationary as a gap diameter in between and two objects is moving thru space. So, the point of this formula is narrow and expands light, so it will be used to calculate how far the objects, and how wide of a diameter of two objects, and how much time of the gap diameter will decrease or increase in time. So, or it is about primary about your eye as a source how far wide of your eye or it is connect as a horizontal line which one part of your eye to another part of an object, same with other eyes, so both top and bottom level has an interact, which it shows triangle above and bottom, which triangle is how much diameter is between and how narrow and wide is the gap diameter? It has the potentials for understanding things from the measure, but how it works from the basic process.
If your trying to see an object 100,000 miles away, that is 25 miles across with the naked eye - 1) is this even possible? and if so, is it resolution ( which to me is the eyes power or ability to resolve that image) or angular resolution and what causes the difference? - I'm not a math guy and since we know math is only one type of tool to describe a function and there are many others try explaining this to me without the math. Thanks.
Do the trigonometry. Draw a line from you to the object, label that line's length (100,000 miles), draw another line from you to the edge of the object, and call the angle theta. Since I believe the minimum angle goes from edge to edge of the object, the angle you are looking for is twice the value of theta. Now, draw a line from the end of the first line to the end of the second, and label this line 12.5 miles (half of the width). Now you need to know the angle. You know the length of the side adjacent to the angle as well as the length of the side opposite, so you can just solve this by taking the inverse tangent of 12.5mi/100,000mi. Now multiply the result of that by two and if that value is less than the minimum angle for the average wavelength of visible light, then you can't discern it. This is why stars appear to be dots in the sky. This might not be the correct solution for discerning a single object, but I think it at least still works.
Fun fact everyone. That 1.22 funny enough is the same number globe believers claim for earth curvature. 1 22 x the square toot of the observers height in feet. We arent seeing earth curvature or boats over a horizon. We are ACTUALLY experiencing the raliegh criterion. Food for thought. Earth is flat.
I am writing a graduate physics thesis on diffraction limited vision, and did some research. It is my understanding that the factor of 1.22 is given by x1/pi, where x1 is the first solution to the differential Bessel Function of the first kind (which looks like a sinusoidal curve decaying proportionally to 1/sqrt(x). The complicated Bessel Function is used to describe the dark fringes that arise in an airy disk. Thats the best I can do, I see differential equations, and want to run and hide.
I'm a class 12th student.... I still got your point
"What fun..." *CRUSHES PAPER*
Just when I was desperate to find a good video on Rayleigh criterion.... I found you!!!!
the 1.22 comes from the diffraction pattern you see. Basically, if you look at a double slit, you get this diffraction pattern that just oscillates indefinitely (cos^2). But with the circular slit, the higher order fringes die off in intensity, and the dying off is controlled by a function called a bessel function (think of it as a decaying cosine). The part we see the most is the primary maxima, and we can say that the size of the maxima is the distance to the first minima, which is determined when the bessel function passes through 0 in its oscillation. The value where it passes through zero (divided by pi or something in the calculations) is where the 1.22 comes from
nightreaper999 BESSEL FUNCTIONS! Woo! Thanks.
i love the manic energy coming from this video
Yay, Bessel Functions and Airy disks! You are awesome, Kaitlyn. Do not run and hide, but rather stay and fight. We need you. I particularly like your clear depiction of how the first Bessel looks. You would make a fine teacher.
Thanks for the explanation. I'm sure it will suffice for my high school students!
I'm surprised so few people watch your videos. This is like the equivalent for patrickjmt for math.
Thank you for being the best physics teacher on RUclips!
Google Bessel function circular aperture and you might find a website that derives the intensity of diffracted light through a circular aperture as a function of angle theta, which should give you something like I(theta) = I(0) (2 J_1(x)/x)^2, where x = kr sin theta, where k is the wave number 2 pi / lambda and r is the radius of the circular opening, which is half the diameter d.
Then the first minimum of the diffraction pattern is when I(theta) = 0 for the smallest value of theta; but I(theta) is only 0 if J_1(x) is 0, and the smallest root of that Bessel function is at x = 3.832; but x = kr sin theta = 2 pi / lambda * d / 2 * sin theta = approximately d pi theta / lambda, so you get theta = 3.832 / 3.142 lambda / d = 1.220 lambda / d :D
Thanks so much for this! It made me laugh in the middle of my physics mental breakdown. And it answered my question , so that's great, too.
Fantastic video!
omg the baby....XD Thank you for the vid!!
i love these videos! super helpful and entertaining :D
I love it . ^.^
The factor 1.22 appears due to circular aperture of the objective lens
I hope you recycle all that paper
아주 훌륭한 영상입니다.
good
Ok, so you want to know why there is 1.22, I want to know why there are sin symbols? What is the point of that? What is the point of 1.22? The whole point is the diameter of an object that is narrow to expand, which is gap distance from furthest to nearest to you as you are stationary, so why need sin symbols and 1.22?? That would be enough just you and the gap diameter of the light itself expand or narrows, or an object that narrows or expand, so the whole point of Rayleigh Criterion is expanded of two edges of an object, so the object expand, so the objects become two, which you can see the details in between like in moon which you can't see the many dots, but closer, you can see many dot, same with hair, you can't see the individual until you get close, so the whole point is diameter of an object that is two outer edges, to inner two edge, and between two inner edges which it is a diameter. So, what matters is you are stationary, and the object is moving, so it requires one stationary, or you can be dynamic, but the object in front of you is stationary. Or, the gap diameter of two objects is stationary, so you are moving toward, or you are stationary, and two objects are stationary as a gap diameter in between and two objects is moving thru space. So, the point of this formula is narrow and expands light, so it will be used to calculate how far the objects, and how wide of a diameter of two objects, and how much time of the gap diameter will decrease or increase in time. So, or it is about primary about your eye as a source how far wide of your eye or it is connect as a horizontal line which one part of your eye to another part of an object, same with other eyes, so both top and bottom level has an interact, which it shows triangle above and bottom, which triangle is how much diameter is between and how narrow and wide is the gap diameter? It has the potentials for understanding things from the measure, but how it works from the basic process.
If your trying to see an object 100,000 miles away, that is 25 miles across with the naked eye - 1) is this even possible? and if so, is it resolution ( which to me is the eyes power or ability to resolve that image) or angular resolution and what causes the difference? - I'm not a math guy and since we know math is only one type of tool to describe a function and there are many others try explaining this to me without the math. Thanks.
Do the trigonometry. Draw a line from you to the object, label that line's length (100,000 miles), draw another line from you to the edge of the object, and call the angle theta. Since I believe the minimum angle goes from edge to edge of the object, the angle you are looking for is twice the value of theta. Now, draw a line from the end of the first line to the end of the second, and label this line 12.5 miles (half of the width). Now you need to know the angle. You know the length of the side adjacent to the angle as well as the length of the side opposite, so you can just solve this by taking the inverse tangent of 12.5mi/100,000mi. Now multiply the result of that by two and if that value is less than the minimum angle for the average wavelength of visible light, then you can't discern it. This is why stars appear to be dots in the sky.
This might not be the correct solution for discerning a single object, but I think it at least still works.
What school do you teach at?
Fun fact everyone. That 1.22 funny enough is the same number globe believers claim for earth curvature. 1 22 x the square toot of the observers height in feet. We arent seeing earth curvature or boats over a horizon. We are ACTUALLY experiencing the raliegh criterion. Food for thought. Earth is flat.
0:32 "What -fun- a waste" of paper. PS- Great explanation except for that _stupid_ no.
Explanation of 1.22 constant : ruclips.net/video/f2Wuo--pdoE/видео.html
i hate these people wasting paper
matiss ramma I was actually about to comment on the same.....😀😀
wth dont waste paper
so funny, ha ha ha, "''AS If "" , "" ChEaP ThRilLs "" , show this comment to ur girlfriend..u r so deserving of it.. CHEAPO..