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Wave Optics
Interference Diffraction
Both effects rely on the fact that when a wave passes through an aperture (opening), the wave spreads out. How much spreading depends on the relative size of the aperture (often labeled "d" or "b") compared with the size of the wavelength
This can be observed with any waves---light, sound, water, gravity, ….you name it. A good video for diffraction of water waves---
https://www.youtube.com/watch?v=BH0NfVUTWG4
And
https://www.youtube.com/watch?v=IoyV2dljw18
Notice in the video, when the aperture gets small (or comparable ) relative to the wavelength Then--there is a lot of wave spreading
We will look at two basic effects of waves--
In that case, the aperture acts almost like a point source with waves emanating in all directions.
We end up assuming that we have sufficiently point sources
of light, and that the light from each source is in sync (synchronized) --and that the light (or any wave) has the
same frequency.
Overall--this means we have coherent light--Easy to come by with a $5 wal-mart pen laser. The aperture is harder.
Notes 13-Wave Optics Page 1
Adding Waves
Making waves is easy, we just need a repeated disturbance in a medium.
Even a single pulse wave demonstrates what happens when two waves cross the same region--they add
https://www.geogebra.org/m/dJrTcxYd
When we add waves travelling in two dimensions (on a surface rather than just along a string) the picture gets cooler.
http://physics.bu.edu/~duffy/HTML5/2Dinterference_point.html
In the simulation above, the situation is similar to dipping finger in water and pulling up --repeat---but at two offset locations. The red regions show the water wave lifting up, and blue--depressed downward. The pattern forms an interference pattern.
If I look at a "wall"--say the top edge of the screen, there are regions
where there is no up/down motion to the wave (destructive interference "fringes"), and there are regions in between where there is
the BIGGEST UP DOWN MOTION OF BLUE/RED.---At such locations that is where the wave is "loudest" or "brightest".
Notes 13-Wave Optics Page 2
The big picture---we are going to start by adding waves from
2 "point" sources=Interference
So, the waves start someplace--and end up at a final location (like point
p on a wall). In the picture above, the waves have different wavelengths---and the two waves do add, but complicated. To see the
effects--we want waves with the same wavelength--COHERENT WAVES.
These two waves start somewhere with the same wavelength, and somehow- end up at the same endpoint--where the wave effect ADDS UP.
We may want to consider what happens at many such points say--along
a wall.
How do we get waves that started in the same place (so they are
synchronized---coherent) to wind up at the same location?
Diffraction through single slit much smaller than . Diffracted waves head in all forward directions.
Notes 13-Wave Optics Page 3
Diffraction through single slit much smaller than . Diffracted waves head in all forward directions.
Notes 13-Wave Optics Page 4
Now let those diffracted waves hit two additional slits.
Small slits that are closely spaced and that are narrow
The light passing through the slits DIFFRACTS so some light hits all locations along the wall or screen. We've picked a sample point p on the wall. There is a wave that reaches p along the lower path (red) and a second wave that reaches p along the upper path (blue).
Usually the slit spacing is so small between the two slits---that the red and blue line are very very nearly parallel, so the blue and red line are almost going at the same angle .
At point "p" the waves may add up--fully---effectively doubling their up/down motion (THIS HAPPENS AT SPECIAL PLACES AND IS CALLED CONSTRUCTIVE INTERFERENCE).
Or the waves at a different location may add --cancelling out completely--(THIS HAPPENS AT DIFFERENT SPECIAL LOCATIONS--CALLED DESTRUCTIVE INTERFERENCE).
In between there is partial cancellation.
What are the special conditions that make either Constructive or Destructive interference happen? If I know this, maybe I can figure out where those
locations along the wall are (y--special, or --special).
Discover gravity waves•Make better sound--like Dolby Atmos--or "beats"•Make lasers--•Find the coolest stuff ever--no really (Bose Einstein Condensate)•and much more•
Knowing these locations might just help me to:
Center of pattern
Notes 13-Wave Optics Page 5
and much more•
Notes 13-Wave Optics Page 6
OK--so let's take a closer look at that "double slit" picture. Remember, this cannot be drawn to scale.
The two path lengths (red and blue) are not the same. The lower path is longer. If one wave travels farther than the other, the waves get out of sync--- if a wave travels one wavelength farther (or two) the waves are back in sync again.
We must look at the PATH DIFFERENCE BETWEEN THE RAY FROM THE
LOWER PATH REACHING P, AND THE UPPER PATH REACHING P---The
difference between red and blue paths---labeled IT IS THIS PATH
DIFFERENCE THAT MAKES EVERYTHING HAPPEN.
The path difference is simply red line length, minus blue line length.
Since the paths are really (picture not drawn to scale---pull the wall out
a thousand times farther away)---nearly parallel, then the triangle on the left looks like.
Notes 13-Wave Optics Page 7
If Then---look at the two waves reaching p---
This may be millions of
cycles long---remember waves of light are very
short.
If the lower wave falls behind the upper (since it must travel farther) by
one wavelength--THEN-- most of the waves still line up, and add constructively. OK --there is one dangling off the front, and one off the
back---but no big deal since the wave train is really millions (billions and billions) of cycles long.
Again--IF THE PATH DIFFERENCE IS ONE WAVELENGTH (OR TWO,
THREE) THEN THE WAVES ADD CONSTRUCTIVELY
This only happens at specific locations--special angles---changing the
angle changes the length of "red" "blue" paths.
IF d sin (n) = m Then Constructive interference fringe exists at that special angle.
Notes 13-Wave Optics Page 8
Notes 13-Wave Optics Page 9
Fringes
The bright spots on the wall are called constructive interference fringes and may occur for m=0, ±1, ±2, ………….There is a maximum n that can occur
If the path difference is a whole number of wavelengths then we are looking at a bright spot. Different types of geometry tricks can give us this path difference instead of two slits.
Are the dark spots halfway between? Not quite, and we must be
careful how to count.
I LABEL MY DARK FRINGES WITH A DIFFERENT LETTER ---- "n" (RATHER THAN "m")
Notes 13-Wave Optics Page 10
For purposes of counting fringes---lets always look on the positive side---and remember the picture is symmetric.
If we always look on the + side (…"of life"……know my quotes….)--then the formula look like this.
What is observed on the screen. In this simulation you can change the
slit spacing (d) the slit size (it is not called D in this simulation) and the distance to the screen (also not called L in the simulation). You can also
change the wavelength.
https://www.geogebra.org/m/ZSbeWGbe
The graph below the simulation shows where you can see --along the
screen--brighter, darker spots, how that is impacted by 2 point sources--and also further impacted by the fact that our sources are not truly points of light (it matters).
Notes 13-Wave Optics Page 11
My hand drawn picture of what is happening (double slit)
At any point along the screen--(wall at the right) we have Intensity (brightness) as a function of position. We can describe position using either y or angle from the center.
For double slit, the pattern gets bright, dim, bright, dim. I have labeled a few interference fringes (interference maxima, minima) or also called interference orders.
I will always refer to fringes by their "fringe numbers " m for bright, and n for dark---as I have outlined.
So if we are told to go find the position (ym or m ) how do we do that?
We use two tools---the interference conditional statement, and
geometry. After that we may use trig or algebra.
Notes 13-Wave Optics Page 12
If given 500nm light passing
through the double slits. We are
given slits with d=10.0m and
the distance to the screen is L=2.00m.
Notes 13-Wave Optics Page 13
Interference Problem continued
Where is the third (m=3) bright fringe ….both y3 and 3 a)
What is the maximum constructive order appearing on the screen?b)
See picture to right…biggest path difference is , and the max "m" is
how many wavelengths fit inside the slit.
In this case the maximum order number is "20" at 90o. And I count that
as being on the screen. So the total number of bright spots on the screen is 41 (count it out---remember, 1,2,3,,,,,,,,20, and minus 1,
…..AND ZERO). The number of dark fringes on the screen is 40 Notes 13-Wave Optics Page 14
…..AND ZERO). The number of dark fringes on the screen is 40 (0,1,2,3……19) and double for both sides of pattern.
Notes 13-Wave Optics Page 15
How far apart are the fringes --say fringe zero and fringe
m=1……..OK here we will develop an algebra trick for small angles. Same two formula---but simplified for
small angles. Remember trig--for small angles
sin() ~ ~ tan() so our formula become
c)In that last problem:
We used the small angle approx which is valid near the center of the
pattern. Valid if m/d is small.
OK so we can find the positions /locations of various fringes. We can
measure how far apart they are (telling us slit size or sometimes the size of wire, or hair or other small stuff). Cool.
Now--can we do spectroscopy---that means, can we tell the difference
between lights of a different color you've heard tell of -----
What can I say---I enjoy quoting movies (you know this one--and it introduced technicolor).
Notes 13-Wave Optics Page 16
Let's use interference to consider the difference between
the location of a fringe with say red light and bluish light. For the same setup we had back a couple pages, where is
the first order fringe for each red light (650nm) and blue say 450nm
These fringes--or maximums will be 4.00cm apart, but look like this.
So, they separate, the light disperses--but I kinda see a smear. Not great.
There must be a way to make these peaks really sharp instead of
spread out ---with "half on" then "half off" duty cycle
We can do this. It's a simple adding trickMany slits !!!!!!
Notes 13-Wave Optics Page 17
Many slits---a Diffraction Grating
Parallel rays from each slit reach
an endpoint
The rays have a path difference of
d sin() ---neighboring rays (1,2 or 2,3 or 3,4)
If neighboring rays add constructively then they all add constructively.
If d sin () =m then bright spot.
Everywhere else---lets look. If we are just off of that maximum, then
ray 0 and ray 1 may have a path difference
that is just off of a whole number by a tiny bit.
But the path difference between rays 0 and 50 is much bigger. Imagine that the path difference between these rays is half a wavelength (if not 0 and 50, then 0 and 51 or some other number).
Then those two rays cancel, and so do 1,51, so do 2, 52, etc…….maybe a few dangling at end---but mostly rays cancel if off of bright spot.
Notes 13-Wave Optics Page 18
Single slit---width D--
There is one slit here. I have drawn many closely spaced points along the slit (really there are infinite number of points)---points much closer than wavelength.
If I cut the slit in half and consider a ray from top edge and middle--
(slit cut in two pieces)---then ask
IF
DESTRUCTIVE
WELL, THEN---ray 1,51 add destructive, 2,52, go to middle and bottom
edge of slits--we have added all rays. So ….
Special case--for a single slit, I can find the
dark spots.
Notes 13-Wave Optics Page 19
Notes 13-Wave Optics Page 20
The intensity vs angle for a single slit looks something like
this
If I make the slit wide---I get a large "nmax" out at 90 degrees.
If I make slit narrow--that pattern stretches out---and +90, -90 may go from (orange markers). If I make the slit even narrower--then the orange markers get closer to central peak---and screen looks nearly uniformly lit (point source of light).
Diffraction makes light smear out and ---decreases resolution of standard optical equipment (eyes to telescopes).
Notes 13-Wave Optics Page 21