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

 tan((pi*d/lambda)-(m*pi/2))=sqrt((sin^2tetac/sin^2teta)-1)

[Birdman]

Question?

[Walter Roberson]

My guess is

 syms c d lambda m teta real
 eqn = tan((pi*d./lambda)-(m*pi/2)) == sqrt(sin(teta*c).^2./sin(teta).^2-1);
 solve(eqn, teta)

[John BG]

Hi Zeynep

this is John BG <mailto:[e-mail] [e-mail]>

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?

Answer 1

[zeynep kahraman]

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

Answer 2

[John BG]

Hi Zeynep Kahraman

this is John BG <mailto:[e-mail] [e-mail]>

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?

To any other reader, if you find this answer useful please consider clicking on the thumbs-up vote link

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

<mailto:[e-mail] [e-mail]>