Total Internal Reflection (TIR)
Briefly, when light is coming from medium n1 onto interface with medium n2, where n1>n2, with angle of incidence larger than critical angle defined as asin(n2/n1), light is fully reflected back into medium n1. Small part though penetrate into medium n2, but carries no energy across the interface, and is referred as evanescent wave. Here I will cover analytical expressions for p- and s-polarized incident wave in TIR case. These expressions are suitable for COMSOL modeling in scattering formalism, when simulating scattering of evanescent waves.
For more information I recommend the following link, and related contest:
http://www.olympusmicro.com/primer/java/tirf/evaintensity/index.html
For more information I recommend the following link, and related contest:
http://www.olympusmicro.com/primer/java/tirf/evaintensity/index.html
TIR s-polarization
As stated above, n1/n2*sinθ1=sinθ2>1, thus if we remember that imaginary unit j=sqrt(-1),
we have cosθ2= ±j*sqrt((n1/n2*sinθ1)^2-1). Question is what expression we should choose, one with minus or one with plus.
To answer that, let´s remember from the page about Fresnel coefficients, that transmitted wave can be expressed as
E2=t*E0iy*exp(-j*k2*sinθ2*x)*exp(-j*k2*cosθ2*z). Since energy is not transmitted across the interface, E2 has to go to 0, when z goes to plus infinity. Thus physically meaningful solutions for E2 is when
cosθ2= -j*sqrt((n1/n2*sinθ1)^2-1) (1),
because j*j=-1.
Now we should modify expressions form Fresnel coefficient page using (1).
t=(2*n1*sqrt(1-(sinθ1)^2)))/(n1*sqrt(1-(sinθ1)^2))+n2*(-j)*sqrt((n1/n2*sinθ1)^2-1))),
r=(n1*sqrt(1-(sinθ1)^2))-n2*(-j)*sqrt((n1/n2*sinθ1)^2-1)))/((n1*sqrt(1-(sinθ1)^2))+n2*(-j)*sqrt((n1/n2*sinθ1)^2-1)))).
E=(E0x, E0y,E0z), where
E0x=0,
E0y=(z<0)*(exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2)*z))+r*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2)*z)))+(z=>0)*t*exp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt((n1/n2*sinθ1)^2-1)*z),
E0z=0.
we have cosθ2= ±j*sqrt((n1/n2*sinθ1)^2-1). Question is what expression we should choose, one with minus or one with plus.
To answer that, let´s remember from the page about Fresnel coefficients, that transmitted wave can be expressed as
E2=t*E0iy*exp(-j*k2*sinθ2*x)*exp(-j*k2*cosθ2*z). Since energy is not transmitted across the interface, E2 has to go to 0, when z goes to plus infinity. Thus physically meaningful solutions for E2 is when
cosθ2= -j*sqrt((n1/n2*sinθ1)^2-1) (1),
because j*j=-1.
Now we should modify expressions form Fresnel coefficient page using (1).
t=(2*n1*sqrt(1-(sinθ1)^2)))/(n1*sqrt(1-(sinθ1)^2))+n2*(-j)*sqrt((n1/n2*sinθ1)^2-1))),
r=(n1*sqrt(1-(sinθ1)^2))-n2*(-j)*sqrt((n1/n2*sinθ1)^2-1)))/((n1*sqrt(1-(sinθ1)^2))+n2*(-j)*sqrt((n1/n2*sinθ1)^2-1)))).
E=(E0x, E0y,E0z), where
E0x=0,
E0y=(z<0)*(exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2)*z))+r*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2)*z)))+(z=>0)*t*exp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt((n1/n2*sinθ1)^2-1)*z),
E0z=0.
TIR p-polarization
Analogously, for p-polarization same discussion stands with same conclusions, giving
t=(2*n1*sqrt(1-(sinθ1)^2)))/(n2*sqrt(1-(sinθ1)^2))+n1*(-j)*sqrt((n1/n2*sinθ1)^2-1))),
r=(n2*sqrt(1-(sinθ1)^2))-n1*(-j)*sqrt((n1/n2*sinθ1)^2-1)))/((n2*sqrt(1-(sinθ1)^2))+n1*(-j)*sqrt((n1/n2*sinθ1)^2-1)))).
E=(E0x, E0y,E0z), where
E0x=(z<0)*(-sqrt(1-(sinθ1)^2))*exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2))*z))+r*sqrt(1-(sinθ1)^2))*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2))*z)))+(z>0)*(t*j*sqrt((n1/n2*sinθ1)^2-1)êxp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt(1-(sinθ1)^2))*z)),
E0y=0,
E0z=(z<0)*(sinθ1*exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2))*z))+r*sinθ1*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2))*z)))+(z=>0)*(t*n1/n2*sinθ1*exp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt(1-(sinθ1)^2))*z)).
NOTE: If u find some typo errors, you should report.This expression can be directly copy/pasted to your COMSOL model.
t=(2*n1*sqrt(1-(sinθ1)^2)))/(n2*sqrt(1-(sinθ1)^2))+n1*(-j)*sqrt((n1/n2*sinθ1)^2-1))),
r=(n2*sqrt(1-(sinθ1)^2))-n1*(-j)*sqrt((n1/n2*sinθ1)^2-1)))/((n2*sqrt(1-(sinθ1)^2))+n1*(-j)*sqrt((n1/n2*sinθ1)^2-1)))).
E=(E0x, E0y,E0z), where
E0x=(z<0)*(-sqrt(1-(sinθ1)^2))*exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2))*z))+r*sqrt(1-(sinθ1)^2))*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2))*z)))+(z>0)*(t*j*sqrt((n1/n2*sinθ1)^2-1)êxp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt(1-(sinθ1)^2))*z)),
E0y=0,
E0z=(z<0)*(sinθ1*exp(-j*k0*n1*(sinθ1*x+sqrt(1-(sinθ1)^2))*z))+r*sinθ1*exp(-j*k0*n1*(sinθ1*x-sqrt(1-(sinθ1)^2))*z)))+(z=>0)*(t*n1/n2*sinθ1*exp(-j*k0*n1*sinθ1*x)*exp(-k0*n2*sqrt(1-(sinθ1)^2))*z)).
NOTE: If u find some typo errors, you should report.This expression can be directly copy/pasted to your COMSOL model.