Waveguide Modes, tan((pi*d/​lambda)-(m​*pi/2))=sq​rt((sin^2t​etac/sin^2​teta)-1)

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zeynep kahraman on 22 May 2018
Answered: John BG on 23 May 2018
tan((pi*d/lambda)-(m*pi/2))=sqrt((sin^2tetac/sin^2teta)-1)
John BG on 22 May 2018
Hi Zeynep
what type of wave-guide are you analyzing?
what is the frequency range?
is that c light speed in air? or is the wave-guide filled with dielectric?

zeynep kahraman on 23 May 2018
Hi JOHN; thetac=acos(n2/n1) (critical angle) lambda=lambdazero/n1 M=sin(thetac)/(t/(2*d)) number of modes and d,lambdazero,n1,n2 = input

John BG on 23 May 2018
Hi Zeynep Kahraman
1.
The equation itself can be approach in different ways, Symbolic or numeric.
I tend to go for numeric 1st, kind of habit.
d=.1
lambda0=1e-9
n1=1
n2=4
theta=[0:.01:2*pi];
f2= tan((pi*d*n1/lambda0)-(sin(acos(n2/n1))./(theta/(2*d)) *pi/2)-((sin(acos(n2/n1)))^2./(sin(theta)).^2-1).^.5)
subplot(2,2,1)
plot(theta,real(f2));title('real');grid on
subplot(2,2,3)
plot(theta,imag(f2));title('imag');grid on
subplot(2,2,2)
plot(theta,abs(f2));title('||');grid on
subplot(2,2,4)
plot(theta,angle(f2));title('phase');grid on
.
2.
That is the mathematical equation, I had to guess all the inputs, so if you supply the input parameters I will plot again with with d n1 n2 that you choose.
3.
Regarding the type of waveguide:
The equation resembles a dielectric interface for optical wave-guide.
Literature reference [RAMO]
Fields and Waves in Communications Electronics, 2nd ed
by Simon Ramo, John Whinnery, Theodore Duzer. Ed: JWiley&Sons
chapter 14
pg743
pg744
3.
Would you please give some details about the geometry of the interface the equation models? is it possible for you to attach a diagram to the question?
Zeynep Kahraman
If you find the above ok for you to proceed in your work then it means you find this answer useful. Then would you please be so kind to consider marking my answer as Accepted Answer?
Zeynep if you need further development, perhaps you would like to consider supplying values for d n1 n2 and lambda0.
thanks in advance for time and attention
John BG