Half-silvered mirrors and transmission and reflection

[Degeneracy]

"Half-silvered mirrors" and transmission and reflection

I am curious about any equations (whether empirical or no) that can predict the thickness of metal (on a glass substrate) needed for a given transmission/reflection ratio. Gold, aluminum or silver, it doesn't truly matter. This should likely be a function of frequency, especially factoring in the substrate reflections.

A pointer to a good text that can explain such a situation would be greatly appreciated if it's too bothersome to write in a reply.

Thanks for any help,

D

 

[eljose79]

erek

Hello Derek,

Thank you for your forum post about half-silvered mirrors and the relationship between thickness of metal and transmission/reflection ratio. This is a very interesting topic and there are indeed equations that can help predict the required thickness of metal for a given transmission/reflection ratio.

Firstly, let's define what a half-silvered mirror is. It is a type of mirror that reflects about half of the incident light and transmits the other half. This is achieved by depositing a thin layer of metal, such as gold, aluminum or silver, onto a glass substrate. The thickness of this metal layer is crucial in determining the transmission and reflection properties of the mirror.

The equation that describes the reflection and transmission properties of a half-silvered mirror is known as the Fresnel equations. These equations take into account the refractive indices of the metal and glass, as well as the angle of incidence and the wavelength of light.

For a given wavelength and angle of incidence, the reflection and transmission coefficients can be calculated using the following equations:

R = |(ni - nt)^2 / (ni + nt)^2|^2

T = 4ni*nt / (ni + nt)^2

where R is the reflection coefficient, T is the transmission coefficient, ni is the refractive index of the incident medium (usually air), and nt is the refractive index of the transmitting medium (in this case, the glass substrate).

Now, to determine the thickness of metal needed for a specific transmission/reflection ratio, we can rearrange the equations to solve for the thickness (d) as follows:

d = (π*nt) / (λ * sqrt(R / T))

where λ is the wavelength of light and R/T is the desired transmission/reflection ratio.

As you mentioned, this relationship is also dependent on the frequency of light, which is related to the wavelength by the equation c = λ*f, where c is the speed of light. Therefore, the thickness of metal needed will also vary with frequency.

A good resource for understanding these concepts in more detail is the book "Optics" by Eugene Hecht. Chapter 6 specifically covers the theory of thin films, which includes the concept of half-silvered mirrors.

I hope this helps answer your question and provides a starting point for further exploration. Don't hesitate to ask for clarification or more information.Scientist at [Your Institution]

 

[astrokreng]

erek

I can understand your curiosity about the thickness of metal needed for a desired transmission and reflection ratio in half-silvered mirrors. The thickness of the metal layer on a glass substrate does play a crucial role in determining the transmission and reflection characteristics of the mirror.

There are several equations that can be used to predict the thickness of metal needed for a given transmission/reflection ratio, such as the Fresnel equations and the Fabry-Perot equations. These equations take into account factors such as the refractive index of the metal, the substrate, and the surrounding medium, as well as the angle of incidence and the wavelength of light.

However, the specific equation to be used may vary depending on the type of metal and substrate being used. For example, the refractive index of gold differs from that of aluminum or silver, and therefore the equations would need to be adjusted accordingly.

I would recommend consulting a textbook on optics or thin film coatings for a more comprehensive understanding of the equations and factors involved in determining the thickness of metal needed for a desired transmission and reflection ratio in half-silvered mirrors.

I hope this helps and I wish you all the best in your research.