I am trying to plot a normalized polar plot for the following function with different values of $a$
$$\left\lvert \sum_{n=1}^\infty i^n (2n+1) \frac {P_n^1(cos(\theta))}{\sqrt{\frac{\pi k a}{2}}[-H_{n+\frac{3}{2}}^2 (ka) + \frac{n+1}{ka}H_{n+\frac{1}{2}}^2(ka)]} \right\rvert^2 $$
where $P_n^1(cos(\theta))$ is the associated Legendre polynomial and $H_{n+\frac{3}{2}}^2 (ka)$ and $H_{n+\frac{1}{2}}^2(ka)$ are Hankel function of 2nd kind. Here $k=2\pi$ and the value of $a$ vary as $a = [.25, .05, 2, 10, 20]$
I can get plots upto $a=2$ with $n=100$ but for $a=10, 20$ I am having difficulty plotting. I am running into issues where machine precision is lost and the norm for Hankel function becomes too big. This is my attempt below
k = 2 \[Pi] ;
Pr[a_, m_, \[Theta]_] :=
Abs[Sum[I^n (2 n + 1) LegendreP[n, 1, Cos[\[Theta]]]/(
Sqrt[(\[Pi] k a )/
2] (-HankelH2[n + 3/2, k a ] + (n + 1)/(k a)
HankelH2[n + 1/2, k a ])), {n, 1, m}]]^2;
ap05m = 2; normp05 = FindMaximum[Pr[.05, ap05m, \[Theta]], {\[Theta], 0, 2 \[Pi]}];
ap25m = 7 ; normp25 = FindMaximum[Pr[.25, ap25m, \[Theta]], {\[Theta], 0, 2 \[Pi]}];
a2m = 100; norm2 = FindMaximum[Pr[2.0, a2m, \[Theta]], {\[Theta], 0, 2 \[Pi]}];
a10m = 150; norm10 = FindMaximum[Pr[10.0, a10m, \[Theta]], {\[Theta], 0, 2 \[Pi]}];
a20m = 150; norm20 = FindMaximum[Pr[20.0, a20m, \[Theta]], {\[Theta], 0, 2 \[Pi]}];
PolarPlot[{Pr[.05, ap05m,\[Theta]]/normp05[[1]],
Pr[.25, ap25m, \[Theta]]/normp25[[1]],
Pr[2, a2m, \[Theta]]/norm2[[1]],
Pr[10, a10m, \[Theta]]/norm10[[1]],
Pr[20, a20m, \[Theta]]/norm20[[1]]}, {\[Theta], 0, 2 \[Pi]},
PolarAxes -> True, PlotRange -> Automatic,
PolarGridLines -> Automatic, PolarTicks -> {"Degrees", Automatic},
PolarAxesOrigin -> {0, 1}, PlotLegends -> "Expressions"]
The norm for $a=10$ and $a=20$ are
a10m = 150; norm10 = FindMaximum[Pr[10.0, a10m, \[Theta]], {\[Theta], 0, 2 \[Pi]}]
${7.24673*10^{27}, {\theta -> 0.121351}}$
a20m = 150; norm20 = FindMaximum[Pr[20.0, a20m, \[Theta]], {\[Theta], 0, 2 \[Pi]}]
${1.11063*10^{79}, {\theta -> 0.0809369}}$
These normalization factors are too big and as a result I don't see the graphs for $a=10$ and $a=20$
This is what the graph is supposed to look like given by the professor
This is what my current graph looks like
I have tried $Chop[]$ function, but it didn't work. I would appreciate any help in plotting the normalized graph for $a = 10, 20$ on the same graph with $a = .05, .25, 2$.