bV diagram of TE modes guided by waveguides with a linearly tapered index profile

Question

I have an equation and I have tried solving it to find values of $b$ but it doesn't give me the same figure as in the book Foundations for guided-wave optics by Chin-Lin Chen.

The equation that I have tried to solve is Equation 3.14 in the book: $\newcommand{\Ai}{\operatorname{Ai}}$ $\newcommand{\Bi}{\operatorname{Bi}}$ $$ \frac{V^{1/3} \sqrt{b} \Ai(V^{2/3}b)+\Ai'(V^{2/3}b)}{V^{1/3} \sqrt{b} \Bi(V^{2/3}b)+\Bi'(V^{2/3}b)}= \frac{V^{1/3} \sqrt{a+b} \Ai[-V^{2/3}(1-b)]-\Ai'[-V^{2/3}(1-b)]}{V^{1/3} \sqrt{a+b} \Bi[-V^{2/3}(1-b)]-\Bi'[-V^{2/3}(1-b)]} $$

I have tried on it as follows:

a = 10; m = 1;
bValue = Table[
   FindRoot[((
      Subscript[V, i]^(1/3)*Sqrt[b]*AiryAi[Subscript[V, i]^(2/3)*b] + 
       AiryAiPrime[Subscript[V, i]^(2/3)*b])/(
      Subscript[V, i]^(1/3)*Sqrt[b]*AiryBi[Subscript[V, i]^(2/3)*b] + 
       AiryBiPrime[Subscript[V, i]^(2/3)*b])) - ((
      Subscript[V, i]^(1/3)*Sqrt[a + b]*
        AiryAi[-Subscript[V, i]^(2/3)*(1 - b)] - 
       AiryAiPrime[-Subscript[V, i]^(2/3)*(1 - b)])/(
      Subscript[V, i]^(1/3)*Sqrt[a + b]*
        AiryBi[-Subscript[V, i]^(2/3)*(1 - b)] - 
       AiryBiPrime[-Subscript[V, i]^(2/3)*(1 - b)])) - m Pi, {b, 
     0.037309}], {Subscript[V, i], 7.7, 10.2, 0.1}];
VV = Table[i, {i, 7.7, 10.2, 0.1}];
bValues = Re[b /. bValue]
bValuesList = Transpose[{Join[{b}, bValues]}]
bGrid = Grid[bValuesList, Frame -> All] 
bAgainstVdata = Partition[Riffle[VV, bValues], 2]

I have tried once add +m Pi and another trying add -m Pi where the m is the mode order and both of them did not give me the figure 3.4 in the book:

Answer 1

You've misunderstood the meaning of $m$ in the book. The following gives the desired plot:

ContourPlot[
 Table[(V^(1/3) Sqrt[b] AiryAi[V^(2/3) b] + AiryAiPrime[V^(2/3) b])/(
    V^(1/3) Sqrt[b] AiryBi[V^(2/3) b] + AiryBiPrime[V^(2/3) b]) == (
    V^(1/3) Sqrt[a + b] AiryAi[-V^(2/3) (1 - b)] - AiryAiPrime[-V^(2/3) (1 - b)])/(
    V^(1/3) Sqrt[a + b] AiryBi[-V^(2/3) (1 - b)] - 
     AiryBiPrime[-V^(2/3) (1 - b)]), {a, {10, 100}}] // Evaluate, {V, 0, 20}, {b, 0, 
  1}]