Clarify normalization of Zernike polynomials

openmc/filter_expansion.py

@@ -318,15 +318,15 @@ class ZernikeFilter(ExpansionFilter):
 
     This filter allows scores to be multiplied by Zernike polynomials of the
     particle's position normalized to a given unit circle, up to a
-    user-specified order. The Zernike polynomials are defined as
+    user-specified order. The Zernike polynomials follow the definition by `Noll
+    <https://doi.org/10.1364/JOSA.66.000207>`_ and are defined as
 
     .. math::
-        Z_n^m(\rho, \theta) = R_n^m(\rho) \cos (m\theta)
+        Z_n^m(\rho, \theta) = \sqrt{2n + 2} R_n^m(\rho) \cos (m\theta), \quad m > 0
 
-    and
+        Z_n^{m}(\rho, \theta) = \sqrt{2n + 2} R_n^{m}(\rho) \sin (m\theta), \quad m < 0
 
-    .. math::
-        Z_n^{-m}(\rho, \theta) = R_n^{-m}(\rho) \sin (m\theta)
+        Z_n^{m}(\rho, \theta) = \sqrt{n + 1} R_n^{m}(\rho), \quad m = 0
 
     where the radial polynomials are
 
@@ -334,6 +334,9 @@ class ZernikeFilter(ExpansionFilter):
         R_n^m(\rho) = \sum\limits_{k=0}^{(n-m)/2} \frac{(-1)^k (n-k)!}{k! (
         \frac{n+m}{2} - k)! (\frac{n-m}{2} - k)!} \rho^{n-2k}.
 
+    With this definition, the integral of :math:`(Z_n^m)^2` over the unit disk
+    is exactly :math:`\pi` for each polynomial.
+
     Specifying a filter with order N tallies moments for all :math:`n` from 0 to
     N and each value of :math:`m`. The ordering of the Zernike polynomial
     moments follows the ANSI Z80.28 standard, where the one-dimensional index

src/math.F90

@@ -575,8 +575,10 @@ contains
   end function calc_rn
 
 !===============================================================================
-! CALC_ZN calculates the n-th order Zernike polynomial moment for a given angle
-! (rho, theta) location in the unit disk.
+! CALC_ZN calculates the n-th order modified Zernike polynomial moment for a
+! given angle (rho, theta) location in the unit disk. The normlization of the
+! polynomials is such that the integral of Z_pq*Z_pq over the unit disk is
+! exactly pi
 !===============================================================================
 
   subroutine calc_zn(n, rho, phi, zn)