Solving Light Refraction Boundary Conditions: Find T^-1

In summary, the question discusses the transmission of light with frequency \omega through multiple media with different refractive indices. The transmission coefficient T^{-1} is calculated using boundary conditions and the general Fresnel coefficients for reflection and transmission. The wave is reflected an infinite amount of times, resulting in a geometric series, and a phase is added while passing through one of the media. The final expression for T^{-1} includes a sine squared term.
  • #1
drcrabs
47
0
Can someone please help me with this question:

Light with frequency [tex]\omega[/tex] in media 1 ,with refractive index [tex]n_{1}[/tex] , is incident (normal) to an interface of media 2, with refractive index [tex]n_{2}[/tex], and then is incident on a second interface with refractive index [tex]n_{3}[/tex]. Using boundary conditions show that the transmission coefficient is:

[tex]T^{-1} = \frac{1}{4n_1n_3} ((n_1+n_3)^2 + \frac{(n_1^2-n_2^2)(n_3^2-n_2^2)}{n_2^2} Sin(\frac{n_2d\omega}{c}))[/tex]

So basically light starts in one media and passes though two different media and we get the above as the transmission coefficient
 
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  • #2
If I understand correctly, T is what comes out the n2 area, to n3 and infinity?
The point in this question is this:
1. find the genreal Fresnel Coefficients for reflection and transmition between two medias (you can find it around page 306 in Jackson 3re ed.).
2. Now look at your own wave - it is transmitted into n2, some passes into n3 but some reflects from the n2-n3 boundary back to n2, some passes back to n1 and some again is reflected to n3. The wave is reflected an infinite amount of times and there is an infinite sum involved of all the transmitted waves. Luckily, it's a geometric series.
P.S you must assume the media is non-magnetic to get this expression.
3. Don't forget the phase added to the wave while passing n2.
 
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  • #3
That problem is worked out on page 285 of Franklin "Classical Electromagnetism". The sin should be sin^2.
 

1. What is light refraction?

Light refraction is the bending of light as it passes through different mediums, such as air, water, or glass.

2. Why is solving light refraction boundary conditions important?

Solving light refraction boundary conditions allows us to understand how light behaves when passing through different mediums, which is crucial for many applications such as lens design, microscopy, and astronomy.

3. What are the boundary conditions for light refraction?

The boundary conditions for light refraction include the incident angle, the refractive indices of the two mediums, and the angle of refraction.

4. How is T^-1 calculated for light refraction boundary conditions?

T^-1, also known as the inverse of the transmission matrix, is calculated by using the Snell's law and the Fresnel equations to determine the coefficients for the reflection and transmission of light at the boundary between two mediums.

5. Can light refraction boundary conditions be solved analytically?

Yes, light refraction boundary conditions can be solved analytically using mathematical equations and principles such as Snell's law and the Fresnel equations. However, in some cases, numerical methods may be used for more complex scenarios.

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