Hello and welcome to this video for physics 132 on the idea of optical path length.
By the end of this video you should be able to know what an optical path length is,
and be able to calculate the optical path length in a variety of situations,
both using graphical counting methods and
situations with different materials.
To understand what optical path length is we have to go back to what may seem to be a very fundamental question, which is
when I say that something is 10 meters long,
what do I actually mean?
So,
here we have a figure of a Tarbosaurus, which is about 10 meters long.
What do we mean by that?
Well we mean that a Tarbosaurus is as long as 10 meter sticks end to end.
1 2 3 4 5 6 7 8 9 10.
Now this is a convention. We conventionally use sticks one meter long to measure things,
but that's not the only thing we could use. we could use sticks one inch long or a foot long or we could say how
many people long, or we could measure how long this Tarbosaurus is in wavelengths of a particular color of light.
So for example let's say I had a radio wave with a wavelength of 2 meters,
then the Tarbosaurus would be five wavelengths long. 1 2 3 4 5 on this little picture.
Or
to work it out mathematically,
we know that the Tarbosaurus is 10 meters long,
we know that one wavelength is 2 meters, which means it's 5 wavelengths of 2 meter light.
But there's no reason I have to use a 2 meter radio
wave, I could use a 600 nanometer light wave of the typical red laser pointer.
How many wavelengths long is a Tarbosaurus in this unit? Well 10 meters, and we know that one wavelength is
680x10e-9 meters, which gives us 1.471x10e7 wavelengths of
680 nanometer light.
This is the fundamental principal of optical path length, you're measuring distances using wavelengths of light instead of meter sticks.
Notice the formula for optical path length for which I will use this script L is the distance traveled X divided by the wavelength.
So what are the units of optical path length?
Well optical path length is a distance, that travelled by the light, divided by another distance, the wave length.
L is x over lambda.
Since both X and lambda are
distances and measured in meters optical path length will be unitless,
which should make sense optical path length is the number of wavelengths traveled by the light.
So I'm just counting wavelengths like we did in the previous slide and things that are counted should be unitless.
When I say that I have five apples, that's just five there's no meat unit on it.
So there is one quirk with using wavelengths to count as a measure of distance and that comes in with materials. As
you know from Unit 2 on geometric optics, the wavelength of light changes when it enters a material. The speed of light
goes from the speed, c,  3x10e8, down to v
whatever the v is in the material, where the index of refraction is this c over v.
When I combine this definition of n with the fundamental relationship for waves, V equals lambda F, and
recognizing that the frequency can't change,
then that tells me that the wavelength must change to the wavelength in the material, lambda must be lambda naught,
which is the wavelength in vacuum divided by the index of refraction.
The key point for optical path length is that you should always use the wavelength of the material that the light is currently in.
So here's an example with some materials. Imagine that we have a source which emits light of 500 nanometer wavelength.
The source is 3 millimeters from a glass slab with an index of refraction n=2.
Buried within the slab another three millimeters inside is a detector.
What is the optical path length between the source and the detector?
Well in this situation,
I have two different regions each with a different lambda outside the glass and inside the glass.
So I have to do this problem in two parts.
So first let's do the part outside the glass.
The optical path length before the glass is the distance over the wavelength we know the distance is 3 millimeters,
and we know the wavelength is 500 nanometers,
which when you divide that out gives us an optical path length before the glass of 6,000.
Now let's do the second part inside the glass. Well
the optical path length inside the glass is the distance the light travels divided by the wavelength of the light in the glass.
So to get the wavelength in the glass, we take the wavelength of light in vacuum, 500 nanometers, and divide it by 2 to get
250 nanometers.
Now we can proceed to calculate the optical path length.
3 millimeters, 3x10 e-3 meters,
divided by the wavelength of light in the glass,
250x10-9 meters gives us an optical path length in the glass of 12,000.
To get the total optical path length
we then just add these two values so the total optical path length is 6,000 plus 12,000 which is
18 thousand.
So in summary,
optical path length is essentially measuring by counting how many wavelength something is long instead of how many meter sticks,
which we will represent mathematically as a script L,
for the optical path length, which is going to be equal to the distance traveled X divided by the wavelength lambda.
Since different kinds of light have different wavelengths
the optical path length even for the same distance can vary depending upon the
wavelength of light being used. Think back to our
example with the dinosaur, it was five radio waves or
1.47x10e7
680 mm nanometer light.
It's also worth remembering that the optical path length is a unitless quantity and
finally you need to remember that you need to use the wavelength that is true
Where you happen to be. So you need to take into account any
indices of fraction. This is essentially returning to the very fundamental physics 131 idea of object egoism.
This concludes this video.
