# Preface! ✨

It’s your favorite material science & nanotechnology enthusiast! Today, I want to begin with the optics and some brief mathematics behind the transparency of aerogels!

If you’re stumped by those words, check out this article to understand what we will be discussing! 😁

For today, I want to begin with a term called optical depth and how it’s applied to aerogels.

# Optical Depth Explained! 💡

## Definitions! 🔑

The optical depth is the quantity of light that has been removed due to absorption, scattering, and reflection.

The extinction coefficient is a term for the volume of the aerogel which explains how easily the material can be penetrated by a beam of light, sound, particles, or other energy or matter. However, in this case, we want to focus on the light only. The extinction coefficient is a sum of the absorption coefficient and the scattering coefficient.

Isotropic means that a specific property of a medium/material is the same (or uniform) in all directions.

Diffuse Transmittance means that light passes through the aerogel and is scattered throughout the network of pores. The light will leave the aerogel at an angle rather than straight through the aerogel.

However, we want to know about the diffuse intensity of the light that passes through the aerogel in the form of diffuse transmittance.

Thanks to researchers from MIT using mathematics, we can now use the diffuse transmittance as a function of the optical depth!

Basically, all this means is that the specific intensity of the diffuse transmittance quantity/value is mathematically changing according to the optical depth.

## Opaque Or Transparent? 🔑

This equation, along with the optical depth calculation, allows us to describe the opaque and the transparent material.

The “β(e)” symbol is the extinction coefficient.

The “s” symbol stands for the distance along with two points in a material. “s(1)” means the starting point, and “s(2)” means the final point of calculating the optical depth. “ds” means the change of distance as the intensity changes as we travel between the points “s(1)” and “s(2)”.

The integral symbol “∫” is the optical depth when measured vertically as light passes through the material.

But Carlos, what about tau “τ”? 🤔

Tau is actually tau “τ”, which is the variable we will give the optical depth. The equation to calculate optical depth is:

This means that you have to divide the intensity of the light after it passed the material over the intensity of the light before it passed through the material.

This will result in the exponential function “e” (approx. 2.72) to the power of negative tau.

## Explanation! 🔑

For now, I only want you to think about e^-τ.

In this example, let’s give “τ” a number. “τ” will be 1. Now we have e^-1.

However, e^-1 can also be written as 1/e. When this happens, we can now be to calculate the optical depth.

The optical depth is the quantity of light that has been removed due to absorption, scattering, and reflection.

1/e has a transmissivity/transparency of about 37%.

So with this number, we can calculate that:

• If the optical depth is greater than 1/e or 1, the medium/material will be opaque, and photons will either be absorbed, reflected, or scatter many times before transmitting/leaving the aerogel. This means you can’t really see what’s on the other side of the material, and it can cause a haze because of the absorption or scattering of the incoming light.
• If the optical depth is less than 1/e or 1, the medium/material will appear transparent, and photons will either transmit directly through the material without being absorbed or scattered.

For example, if “τ” = 2.5, the transmissivity is approximately equal to 8%, which means the material is more opaque/blurry.

If “τ” = 0.5, the transmissivity is approximately equal to 60%, which means the material is more transparent/clear.

# Closing Thoughts! 💭

There we have it! The fundamentals of optical depth are complete! Now I can go more into depth the next time around!

See you tomorrow about The Mean Free Path of photons and the transparent aerogel particles! ✌🏽

# Vocabulary! 📓

Optical Depth → The quantity of light that has been removed due to absorption, scattering, and reflection; τ.

The Extinction Coefficient → A sum of the absorption coefficient and the scattering coefficient. The volume of the aerogel explains how easily the material can be penetrated by a beam of light, sound, particles, or other energy or matter; β(e).

IsotropicA specific property of a medium/material is the same (or uniform) in all directions.

Diffuse Transmittance Light passes through the aerogel and is scattered throughout the network of pores. The light will leave the aerogel at an angle rather than straight through the aerogel.

eEuler’s Number, an irrational number that can be used as an exponential number/constant; approximately 2.72.

Absorption The light is absorbed by the aerogel particles.

HazeIn aerogel, haze is defined as a lack of transparency, or that the aerogel looks somewhat cloudy rather than clear.

TransparencyAll the light will pass through a material, and it won’t be reflected, absorbed, or scatter in a material.