Yet, your solution is neat. Which makes me wonder whether q and p have, in fact, real values. I guess, this will be my next exploration step. But regardless of this, I will keep this way of function-analyzing in mind whenever Solve[] fails. Thanks. you've been very helpful.

Sometimes Solve, NSolve, Reduce, FindRoot, etc, etc can have great difficulty and take a long time looking for a zero of a complicated function. NMinimize can sometimes be faster, but you want it to find where your complicated function is near zero, whether it is positive, negative or even Complex, Norm will give a measure of the distance from zero for all those cases. If it finds the minimum of the norm is nonzero then that may indicate there is no zero but, as was shown, it is sometimes possible that it just did not search far enough to find a root.

Thanks for the reply. I am curious about the how using the Norm[] command would help finding the roots of the mentioned equation. It would be kind if you'd explain your method briefly. Regards.

Perfect! I am trying to calculate the propagation constant (Beta) for modes in an optical fiber according to a well known equation. The method, you used, works just fine. However, you assumed that p and q are both real. I was wondering what led to that assumption. I checked the reference, I am reading for the time being. It does not say anything about p and q being purely real. Of course, if they are complex, then I can't use the REDUCE[] command to find the areas, where the solutions exist. Any advice?

Thanks in advance

For any WorkingPrecision between 48 and 128 NMinimize converges on a beta about 153746 where the resulting minimum is significantly different from zero and does not get closer to zero with increasing Working precision.

In[7]:= a = 25 10^-6; n1 = 1458/1000 + 5 10^-3; n2 = 1458/1000; \[Lambda] = 15/10 10^-6;
k0 = (2 \[Pi])/\[Lambda]; p = Sqrt[n1^2 k0^2 - \[Beta]^2];q = Sqrt[\[Beta]^2 - n2^2 k0^2];
NMinimize[ Norm[((1/2 (BesselJ[0, p a] - BesselJ[2, p a]))/(p BesselJ[1, p a]) +
 (1/2 (-BesselK[0, q a] - BesselK[2, q a]))/(q BesselK[1, q a])) ((1/2 (BesselJ[0, p a] -
 BesselJ[2, p a]))/(p BesselJ[1, p a]) +n2^2/n1^2 (1/2 (-BesselK[0, q a] - BesselK[2, q a]))/
 (q BesselK[1, q a])) - (1)^2/a^2 (1/p^2 + 1/q^2) (1/p^2 + n2^2/n1^2 1/q^2)], \[Beta], 
 WorkingPrecision -> 128]

Out[9]= {2.6644908885382207535682631873522244614076638003775608946182982\
 185058485936623112823132496185424062412703162736336930371650049643*10^-14,
 {\[Beta] -> 153746.742756413486195499781367574310403493602174002003504819\
 78661459273051117012575097394837519002121150764177004071175454428111}}

Because of the way that minimum varies with WorkingPrecision I'm a little skeptical whether that is precisely correct or not.

Hi, I don't know why the code is corrupted. I copied the code as INLINE from Mathematica notebook then I paste it in the Sample Code field using "CTR k". However, I included another code snippet in this reply. I double checked it to make sure that it is not corrupted. Thanks

a = 25 10^-6; n1 = 1.458 + 5 10^-3; n2 = 1.458 ; \[Lambda] = 
 1.5 10^-6; k0 = (2 \[Pi])/\[Lambda]; p = Sqrt[n1^2 k0^2 - \[Beta]^2];
 q = Sqrt[\[Beta]^2 - n2^2 k0^2];


NSolve[((1/2 (BesselJ[0, p a] - BesselJ[2, p a]))/(
     p BesselJ[1, p a]) + (1/2 (-BesselK[0, q a] - BesselK[2, q a]))/(
     q BesselK[1, q a])) ((1/2 (BesselJ[0, p a] - BesselJ[2, p a]))/(
     p BesselJ[1, p a]) + 
     n2^2/n1^2 (1/2 (-BesselK[0, q a] - BesselK[2, q a]))/(
      q BesselK[1, q a])) == (1)^2/
   a^2 (1/p^2 + 1/q^2) (1/p^2 + n2^2/n1^2 1/q^2), \[Beta]]