Which solver to use? I cannot get an answer, though I know there is one.

Greetings, I am looking to solve a nonlinear equation for a single variable. I have tried using MATLAB's 'solve' function, but it cannot find an answer. I know there is an answer, because I've plotted the left and right hand side, and there is an intersection point. Help? My code is as follows:

clear all;
clc;
close all;

micrometers = 1;
nanometers  = micrometers / 1000;
degrees = pi/180;

 eps1  = 2.5;                       % Permittivity, superstrate
 epsg  = 3;                         % Permittivity, grating
 eps3  = 2.5;                       % Permittivity, substrate
theta  = 0;                         % [deg.]
period = 1000 * nanometers;         % Grating period
gthick = 1000 * nanometers;         % Grating thickness
 ido   = 1;                         % diffractive order
 pol   = 'TE';
syms L
kwv = 2*pi/L;
 Bi = ((2*pi/L)*(sqrt(epsg)*sin(theta*degrees)-ido*L/period));
 ki = (epsg*kwv^2-Bi^2)^(1/2);
 gi = (Bi^2-eps1*kwv^2)^(1/2);
 di = (Bi^2-eps3*kwv^2)^(1/2);
lhs = tan(ki*gthick);
rhs = ((ki*(gi+di))/(ki^2-gi*di));
solve('lhs = rhs','L')

I get an answer of: Warning: Explicit solution could not be found. > In solve at 81 In Resonance_Calculator at 51

ans =

[ empty sym ]

If you plot LHS and RHS against each other, the intersection is around 1.670

Answer 1

[Matt Fig]

Use:

ROOT = solve(lhs-rhs,'L');

There are other roots. You might just want to use FZERO:

fzero(@(x) subs(lhs,x)-subs(rhs,x),1.68)
fzero(@(x) subs(lhs,x)-subs(rhs,x),1.65)

Answer 2

[Babak]

It seems that in yoru problem you ahve numerical value for all the variables and want to find L.

To do so, I don't recommend using the Symbolic Math Toolbox, but use the Optimization Toolbox.

Use fsolve() instead if you have MATLAB Optimization Toolbox License.

[Babak]

Type

help fsolve

in MATLAB Command window to see if you have it or not.