- #1

Mr_Allod

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- 16

- Homework Statement
- Electromagnetic radiation is incident normally on a material whose index of refraction is ##n##. Show that the reflected wave can be eliminated by covering the material with a layer of a second material whose index of refraction is ##n^{\frac 1 2}## and whose thickness is one-quarter of a wavelength.

- Relevant Equations
- ##\tilde {\vec E} (z,t) = \tilde E_{0} e^{i(k_1z- \omega t)} \hat x##

##\tilde {\vec B} (z,t) = \frac 1 v \tilde E_{0} e^{i(k_1z- \omega t)} \hat y##

Boundary Conditions:

##\epsilon_1 E^{\perp}_1= \epsilon_2 E^{\perp}_2##

##B^{\perp}_1= B^{\perp}_2##

##\vec E^{\parallel}_1= \vec E^{\parallel}_2##

##\frac 1 {\mu} \vec B^{\parallel}_1=\frac 1 {\mu} \vec B^{\parallel}_2##

Hello there. I set up the problem like this, I have a wave incident from air on the anti-reflective coating consisting of:

##\tilde {\vec E_I} (z,t) = \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat x##

##\tilde {\vec B_I} (z,t) = \frac 1 v \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat y##

This wave gets both reflected and transmitted through the coating generating two new expressions:

##\tilde {\vec E_{R_1}} (z,t) = \tilde E_{0_{R_1}} e^{i(-k_1z- \omega t)} \hat x##

##\tilde {\vec E_{T_1}} (z,t) = \tilde E_{0_{T_1}} e^{i(k_2z- \omega t)} \hat x## (I will only write out the E-field expressions to keep things shorter)

I can now relate the expressions for air and coating with: ## \tilde {\vec E_{R_1}} + \tilde {\vec E_{T_1}} = \tilde {\vec E_I}## .

##\tilde {\vec E_{T_1}}## is then incident on the material of index n, once again both transmitting and reflecting:

##\tilde {\vec E_{R_2}} (z,t) = \tilde E_{0_{R_2}} e^{i(-k_2z- \omega t)} \hat x##

##\tilde {\vec E_{T_2}} (z,t) = \tilde E_{0_{T_2}} e^{i(k_3z- \omega t)} \hat x##

The expressions between coating and material can be related with: ## \tilde {\vec E_{R_2}} +\tilde {\vec E_{T_2}} =\tilde {\vec E_{T_1}}##

##\tilde {\vec E_{R_2}}## then travels back towards the interface between the anti-reflective coating and air, and its here that I start to get confused. Do I need to account for another transmission and reflection as the wave is leaving the anti-reflective coating? And if so how would this affect the expressions I already for the interfaces between other media? Because as I'm going now it seems I would be indefinitely accounting for reflections and transmissions. Eventually I believe I should be able to show that ##\tilde {\vec E_{R_1}}## and another wave leaving the reflective coating (##{\vec E_{T_3}}## perhaps?) are perfectly out of phase and with equal amplitudes therefore causing destructive interference. But I must be missing some key steps in between so I'd appreciate it if someone could give me a hand.

##\tilde {\vec E_I} (z,t) = \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat x##

##\tilde {\vec B_I} (z,t) = \frac 1 v \tilde E_{0_I} e^{i(k_1z- \omega t)} \hat y##

This wave gets both reflected and transmitted through the coating generating two new expressions:

##\tilde {\vec E_{R_1}} (z,t) = \tilde E_{0_{R_1}} e^{i(-k_1z- \omega t)} \hat x##

##\tilde {\vec E_{T_1}} (z,t) = \tilde E_{0_{T_1}} e^{i(k_2z- \omega t)} \hat x## (I will only write out the E-field expressions to keep things shorter)

I can now relate the expressions for air and coating with: ## \tilde {\vec E_{R_1}} + \tilde {\vec E_{T_1}} = \tilde {\vec E_I}## .

##\tilde {\vec E_{T_1}}## is then incident on the material of index n, once again both transmitting and reflecting:

##\tilde {\vec E_{R_2}} (z,t) = \tilde E_{0_{R_2}} e^{i(-k_2z- \omega t)} \hat x##

##\tilde {\vec E_{T_2}} (z,t) = \tilde E_{0_{T_2}} e^{i(k_3z- \omega t)} \hat x##

The expressions between coating and material can be related with: ## \tilde {\vec E_{R_2}} +\tilde {\vec E_{T_2}} =\tilde {\vec E_{T_1}}##

##\tilde {\vec E_{R_2}}## then travels back towards the interface between the anti-reflective coating and air, and its here that I start to get confused. Do I need to account for another transmission and reflection as the wave is leaving the anti-reflective coating? And if so how would this affect the expressions I already for the interfaces between other media? Because as I'm going now it seems I would be indefinitely accounting for reflections and transmissions. Eventually I believe I should be able to show that ##\tilde {\vec E_{R_1}}## and another wave leaving the reflective coating (##{\vec E_{T_3}}## perhaps?) are perfectly out of phase and with equal amplitudes therefore causing destructive interference. But I must be missing some key steps in between so I'd appreciate it if someone could give me a hand.

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