Constructive/Destructive Interference of Two Glass Plates

[superspartan9]

Homework Statement

"Light is incident from above two plates of glass separated on both ends by small wires of diameter d = .6um. Considering only interference between light reflected from the bottom surface of the upper plate and light reflected from the upper surface of the lower plate, state whether the following wavelengths give constructive or destructive interference: λ = (a) 600 nm (b) 800 nm (c) 343 nm.

Homework Equations

Phase = (2m + 1)*pi for destructive interference, Phase = mλ for constructive interference, Phase constructive = 0 + 2N*pi, Phase destructive = pi + 2N*pi

The Attempt at a Solution

Looking at this problem, I can tell that when the light goes through the first plate into the air pocket between the two plates, there will be no phase shift because the index of refraction decreases between the mediums. When the light hits the second plate, there will be a phase shift of pi (if I understand reflection right, my professor is really bad at explaining these things). Beyond this, I'm stuck...

 

[ehild]

The red ray in the picture reflects directly from the first interface and the other (the blue) ray travels 2D distance in the air layer and changes pi phase at the air-glass interface. How much does the phase of a wave change when traveling some distance in a medium of refractive index N? What is the phase difference between the blue and red rays?


ehild

 

[superspartan9]

Phase doesn't change going through a single medium does it? It only shifts when it goes from a low N medium to a higher N medium, right? And I believe the phase difference would be Pi after reflecting off the second glass surface.

 

[ehild]

Phase doesn't change going through a single medium does it? It only shifts when it goes from a low N medium to a higher N medium, right? And I believe the phase difference would be Pi after reflecting off the second glass surface.

The phase is changing as a wave travels. For a sinusoidal light wave, (with angular frequency ω and wavelength λ) the electric field varies as E₀sin(2π*N*s/λ - ωt) along its path s in a medium of refractive index N. The red and blue waves meet at the same place and time, but then and there the blue one have traveled across the air layer twice. So the path difference, s, is twice the layer thickness, d: s=2d, and this means a phase change 4π*N*d/λ with respect to the red ray. This is added to the phase change at the interface. So the total phase difference is δ=4π*N*d/λ+π. If δ=2kπ the interference is constructive, the resultant of the waves is maximum. If δ=(2k+1)π the interference is destructive. Eliminating π, the condition for constructive interference is 4Nd=(2k-1)λ and that for the destructive one is 4Nd=(2k)λ, where k is an integer.
You need to find out if the given wavelengths correspond to constructive or destructive interference.

ehild

 

[superspartan9]

Ok, I think I'm starting to understand. The only thing I'm unsure about is the Δphase between the two pieces of glass, mainly how you know to multiply the Δpath = 2dn/λ by 2∏ to get the phase. Is the 2∏ a constant multiplier?

That aside, let me just clarify why things work like you've explained. The total path traveled is 2*d because the light travels through the medium, hits the opposite piece of glass (which reflects causing the ∏ shift seen in total phase change), and then travels back up through the top piece of glass to interfere with the wave.

The δ = 2∏k or 2∏k + ∏ comes from the fact that any odd multiplier of ∏ causes destructive and even ones cause constructive interference, correct?

Thank you so much for you help, by the way! :D

 

[ehild]

2π is the same constant that appears in the angular frequency, ω=2ωf where f is the frequency.
A wave is periodic both space and time. The time period is T=1/f, and the period in space is the wavelength, λ. The sine or cosine functions are periodic with 2π. sin(ωt) is a periodic function of time, with period T=2π/ω.
sin((2π/λ)*s) is periodic in length s, with period of λ, the wavelength. If s=λ or 2λ or k*λ , the sine function has the same value. The wave Asin((2π/λ)*s-(2π/T)*t)is periodic both in time and in space. If s changes with integer multiple of λ or/and t changes with integer multiple of T, the wave stays the same.

It looks to me that you have understood this interference thing.

ehild