wave optics we will look at two basic effects of waves...

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

https://www.youtube.com/watch?v=BH0NfVUTWG4

https://www.youtube.com/watch?v=IoyV2dljw18

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".


https://www.geogebra.org/m/dJrTcxYd

http://physics.bu.edu/~duffy/HTML5/2Dinterference_point.html

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.


Diffraction through single slit much smaller than . Diffracted waves head in all forward directions.


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


and much more•


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.


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.


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")


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).


https://www.geogebra.org/m/ZSbeWGbe

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.


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.


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

…..AND ZERO). The number of dark fringes on the screen is 40 (0,1,2,3……19) and double for both sides of pattern.


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).


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 !!!!!!


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.


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.


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).


wave optics we will look at two basic effects of waves...

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