FFTW Array Order in Fortran - fortran

Question

I'd like to use FFTW to compute the Fourier Harmonics for a 2D dataset. And I think I'm getting the hang of it but I need to 'decipher' the complex output in terms of harmonics. First here's my toy code:
PROGRAM FFTW_TEST
USE, intrinsic :: iso_c_binding
IMPLICIT NONE
INTEGER :: i, j, n, m, l, k
REAL :: pi2, norm, norm2
REAL(C_DOUBLE), POINTER, DIMENSION(:) :: theta, func
COMPLEX(C_DOUBLE_COMPLEX), POINTER, DIMENSION(:) :: fmn_1d
REAL(C_DOUBLE), POINTER, DIMENSION(:,:) :: func_2d, r_2d, i_2d
COMPLEX(C_DOUBLE_COMPLEX), POINTER, DIMENSION(:,:) :: fmn_2d
INCLUDE 'fftw3.f03'
TYPE(C_PTR) :: plan
TYPE(C_PTR) :: real_1d, complex_1d
TYPE(C_PTR) :: real_2d, complex_2d
pi2 = 8.0 * ATAN(1.0)
!------ 1D Example
n = 64
real_1d = FFTW_ALLOC_REAL(INT(n,C_SIZE_T))
complex_1d = FFTW_ALLOC_COMPLEX(INT(n,C_SIZE_T))
CALL C_F_POINTER(real_1d, func, [n])
CALL C_F_POINTER(complex_1d, fmn_1d, [n])
ALLOCATE(theta(n))
FORALL(i=1:n) theta(i) = DBLE(i-1)/DBLE(n)
FORALL(i=1:n) func(i) = 1.0+2.0*cos(2*pi2*theta(i))-3.0*cos(3*pi2*theta(i)) ! Even Values
! FORALL(i=1:n) func(i) = 0.0+2.0*sin(4*pi2*theta(i))-3.0*sin(5*pi2*theta(i)) ! Odd Values
WRITE(401,*) func(1:n)
CALL FLUSH(401)
norm = n/2
norm2 = n
plan = FFTW_PLAN_DFT_R2C_1D(n, func, fmn_1d, FFTW_ESTIMATE)
CALL FFTW_EXECUTE_DFT_R2C(plan, func, fmn_1d)
CALL FFTW_DESTROY_PLAN(plan)
func = REAL(fmn_1d)/norm
func(1) = 2*func(1)
WRITE(402,*) func; CALL FLUSH(402)
func = -AIMAG(fmn_1d)/norm
func(1) = 2*func(1) ! note should be zero
WRITE(403,*) func; CALL FLUSH(403)
plan = FFTW_PLAN_DFT_C2R_1D(n, fmn_1d, func, FFTW_ESTIMATE)
CALL FFTW_EXECUTE_DFT_C2R(plan, fmn_1d, func)
CALL FFTW_DESTROY_PLAN(plan)
WRITE(404,*) func/norm2; CALL FLUSH(404)
CALL FFTW_FREE(real_1d)
CALL FFTW_FREE(complex_1d)
IF (ASSOCIATED(theta)) DEALLOCATE(theta)
!------- 2D Example
m=16
n=8
real_2d = FFTW_ALLOC_REAL(INT(m*n,C_SIZE_T))
complex_2d = FFTW_ALLOC_COMPLEX(INT(m*n,C_SIZE_T))
CALL C_F_POINTER(real_2d, func_2d, [m,n])
CALL C_F_POINTER(complex_2d, fmn_2d, [m,n])
PRINT *,LBOUND(func_2d,DIM=1)
ALLOCATE(fmn_1d(m),r_2d(m,n),i_2d(m,n))
func_2d = 0.5
DO i = 1, m
DO j = 1, n
DO l = 0, 8
DO k = -4,4
func_2d(i,j) = func_2d(i,j) + (l*100+k) * cos(l*pi2*(i-1)/m + k*pi2*(j-1)/n)
END DO
END DO
END DO
WRITE(501,*) func_2d(i,1:n); CALL FLUSH(501)
END DO
! Note in the m direction we read m=0,1,2,3,4...
! in the n direction we read n=0..nx/2,-nx/2+1...-1
norm = m*n/2
norm2 = norm*2
plan = FFTW_PLAN_DFT_R2C_2D(n, m, func_2d, fmn_2d, FFTW_ESTIMATE)
CALL FFTW_EXECUTE_DFT_R2C(plan, func_2d, fmn_2d)
CALL FFTW_DESTROY_PLAN(plan)
r_2d = REAL(fmn_2d)/norm
r_2d(1,1) = r_2d(1,1)/2
DO i = 1, m
WRITE(502,*) r_2d(i,1:n); CALL FLUSH(502)
END DO
i_2d = -AIMAG(fmn_2d)/norm
i_2d(1,1) = i_2d(1,1)/2
DO i = 1, m
WRITE(503,*) i_2d(i,1:n); CALL FLUSH(503)
END DO
DO i = 1, m
DO j = 1, n
IF (abs(r_2d(i,j))>1.0E-5 .or. ABS(i_2d(i,j))>1.0E-5) &
WRITE(6,*) i,j,r_2d(i,j),i_2d(i,j)
END DO
END DO
plan = FFTW_PLAN_DFT_C2R_2D(n, m, fmn_2d, func_2d, FFTW_ESTIMATE)
CALL FFTW_EXECUTE_DFT_C2R(plan, fmn_2d, func_2d)
CALL FFTW_DESTROY_PLAN(plan)
DO i = 1, m
WRITE(504,*) func_2d(i,1:n)/norm2; CALL FLUSH(504)
END DO
CALL FFTW_FREE(real_2d)
CALL FFTW_FREE(complex_2d)
IF (ASSOCIATED(fmn_1d)) DEALLOCATE(fmn_1d)
END PROGRAM FFTW_TEST
Now I'd like to extract the harmonics of a function which looks like:
$A_{mn} \cos(m\theta+n\phi)$, where $\theta$ and $\phi$ are the real-space grid. First, taking the real part of the complex array appears to be correct. The harmonics seem correct for simple functions. However, in the final test case it seems that the array has an odd ordering. Here's the ordering
m/n
0/0 1/0 2/0 3/0 4/0 5/0 6/0 7/0 8/0 0/1 1/1 2/1 3/1 4/1 5/1 6/1
7/1 8/1 0/2 1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 0/3 1/3 2/3 3/3 4/3
5/3 6/3 7/3 8/3 0/4 1/4 2/4 3/4 5/4 6/4 7/4 8/4 ?/? ?/? ?/? ?/?
For reference here's what I'd expect the ordering to look like:
m/n
0/0 1/0 2/0 3/0 4/0 5/0 6/0 7/0 8/0 -7/0 -6/0 -5/0 -4/0 -3/0 -2/0 -1/0
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 -7/1 -6/1 -5/1 -4/1 -3/1 -2/1 -1/1
0/2 1/2 2/2 3/2 4/2 5/2 6/2 7/2 8/2 -7/2 -6/2 -5/2 -4/2 -3/2 -2/2 -1/2
It seems like the code is just treating the array like a linear array. But what what I can read in the FFTW documentation this should not be the case. One dimension of the array should correspond to the transforms along one dimension and the other the other. Thus in some sense we should be able to read off m and n harmonics.
Can someone explain how the complex array should be ordered in terms of harmonics and the proper normalizations?