;+
; NAME:
;    sphere_coefficients
;
; PURPOSE:
;    Calculates the Mie scattering coefficients
;    for a multilayered sphere illuminated
;    by a coherent plane wave linearly polarized in the x direction.
;
; CATEGORY:
;    Holography, light scattering, microscopy
;
; CALLING SEQUENCE:
;    ab = sphere_coefficients(ap, np, nm, lambda)
;
; INPUTS:
;    ap: [nlayers] radii of layered sphere [micrometers]
;        NOTE: ap and np are reordered automatically so that
;        ap is in ascending order.
;
;    np: [nlayers] (complex) refractive indexes of sphere's layers
;
;    nm: (complex) refractive index of medium
;
;    lambda: wavelength of light [micrometers]
;
; OUTPUTS:
;    ab: [2,nc] complex a and b scattering coefficients.
;
; REFERENCES:
; 1. Adapted from Chapter 8 in
;    C. F. Bohren and D. R. Huffman,
;    Absorption and Scattering of Light by Small Particles,
;    (New York, Wiley, 1983).
;
; 2. W. Yang,
;    Improved recursive algorithm for
;    light scattering by a multilayered sphere,
;    Applied Optics 42, 1710--1720 (2003).
;
; 3. O. Pena, U. Pal,
;    Scattering of electromagnetic radiation by a multilayered sphere,
;    Computer Physics Communications 180, 2348--2354 (2009).
;    NB: Equation numbering follows this reference.
;
; 4. W. J. Wiscombe,
;    Improved Mie scattering algorithms,
;    Applied Optics 19, 1505-1509 (1980).
;
; EXAMPLE:
;  ab = sphere_coefficients([1.0, 1.5], [1.45, 1.56], 1.3326, 0.6328)
;
; MODIFICATION HISTORY:
; Replaces sphere_coefficients.pro, which calculated scattering
;    coefficients for a homogeneous sphere.  This earlier version
;    was written by Bo Sun and David G. Grier, and was based on
;    algorithms by W. J. Wiscombe (1980) and W. J. Lentz (1976).
;
; 10/31/2010 Written by F. C. Cheong, New York University
; 11/02/2010 David G. Grier (NYU) Formatting.  Explicit casts
;    to double precision.  Use complex functions.
; 04/10/2011 DGG Cleaned up Nstop code.  Use array math rather than
;    iteration where convenient.  Eliminate an and bn subroutines.
;    Simplify function names.  Use IDL 8 array notation.  Fixed
;    bug in bn coefficients for multilayer spheres.
; 05/31/2011 DGG Fixed corner condition in Nstop code.
;
; Copyright (c) 2010-2011, F. C. Cheong and D. G. Grier
;

;;;;;
;
; Nstop
;
; Number of terms to keep in the partial wave expansion
;
function Nstop, x, m

COMPILE_OPT IDL2, HIDDEN

;;; Wiscombe (1980)
xl = x[-1]
if xl lt 8. then $
   ns = floor(xl + 4. * xl^(1./3.) + 1.) $
else if xl lt 4200. then $
   ns = floor(xl + 4.05 * xl^(1./3.) + 2.) $
else if xl gt 4199. then $
   ns = floor(xl + 4. * xl^(1./3.) + 2.)

;;; Yang (2003) Eq. (30)
return, double(floor(max([ns, abs(x*m), abs(shift(x, -1)*m)])) + 15)

end

;;;;;
;
; sphere_coefficients
;
function sphere_coefficients, ap, np, nm, lambda

COMPILE_OPT IDL2

nlayers = n_elements(ap)

if n_elements(np) ne nlayers then $
   message, "Error Warning: ap and np must have the same number of elements"

; arrange shells in size order
if nlayers gt 1 then begin
   order = sort(ap)
   ap = ap[order]
   np = np[order]
endif

x = abs(2.d * !dpi * nm * ap / lambda) ; size parameter [array]
m = dcomplex(np/nm)                    ; relative refractive index [array]

nmax = Nstop(x, m)              ; number of terms in partial-wave expansion

ci = dcomplex(0, 1)             ; imaginary unit

; arrays for storing results
ab = dcomplexarr(2, nmax+1, /nozero)

D1     = dcomplexarr(nmax+2)
D1_a   = dcomplexarr(nlayers, nmax+2)
D1_am1 = dcomplexarr(nlayers, nmax+2)

D3     = dcomplexarr(nmax+1)
D3_a   = dcomplexarr(nlayers, nmax+1)
D3_am1 = dcomplexarr(nlayers, nmax+1)

Psi         = dcomplexarr(nmax+1)
Zeta        = dcomplexarr(nmax+1)
PsiZeta     = dcomplexarr(nmax+1)
PsiZeta_a   = dcomplexarr(nlayers, nmax+1)
PsiZeta_am1 = dcomplexarr(nlayers, nmax+1)

Q  = dcomplexarr(nlayers, nmax+1)
Ha = dcomplexarr(nlayers, nmax+1)
Hb = dcomplexarr(nlayers, nmax+1)

; Calculate D1, D3 and PsiZeta for Z1 in the first layer
z1 = x[0] * m[0]
; D1_a[0, nmax + 1] = dcomplex(0) ; Eq. (16a)
for n = nmax + 1.d, 1.d, -1.d do $ ; downward recurrence Eq. (16b)
   D1_a[0, n-1] = n/z1 - 1.d/(D1_a[0, n] + n/z1)

PsiZeta_a[0, 0] = 0.5d * (1.d - exp(2.d * ci * z1)) ; Eq. (18a)
D3_a[0, 0] = ci                                     ; Eq. (18a)
for n = 1.d, nmax do begin      ;upward recurrence Eq. (18b)
   PsiZeta_a[0, n] = PsiZeta_a[0, n-1] * $
                     (n/z1 - D1_a[0, n-1]) * (n/z1 - D3_a[0, n-1])
   D3_a[0, n] = D1_a[0, n] + ci/PsiZeta_a[0, n]
endfor

; Ha and Hb in the core
Ha[0, *] = D1_a[0, 0:-2]     ; Eq. (7a)
Hb[0, *] = D1_a[0, 0:-2]     ; Eq. (8a)

; Iterate from layer 2 to layer L
for ii = 1, nlayers - 1 do begin
   z1 = x[ii] * m[ii]
   z2 = x[ii-1] * m[ii]
   ; Downward recurrence for D1, Eqs. (16a) and (16b)
;   D1_a[ii, nmax+1]   = dcomplex(0)      ; Eq. (16a)
;   D1_am1[ii, nmax+1] = dcomplex(0)
   for n = nmax + 1.d, 1.d, -1.d do begin ; Eq. (16 b)
      D1_a[ii, n-1]   = n/z1 - 1.d/(D1_a[ii, n]   + n/z1)
      D1_am1[ii, n-1] = n/z2 - 1.d/(D1_am1[ii, n] + n/z2)
   endfor

   ; Upward recurrence for PsiZeta and D3, Eqs. (18a) and (18b)
   PsiZeta_a[ii, 0]   = 0.5d * (1.d - exp(2.d * ci * z1)) ; Eq. (18a)
   PsiZeta_am1[ii, 0] = 0.5d * (1.d - exp(2.d * ci * z2))
   D3_a[ii, 0]   = ci
   D3_am1[ii, 0] = ci
   for n = 1.d, nmax do begin   ; Eq. (18b)
      PsiZeta_a[ii, n]   = PsiZeta_a[ii, n-1] * $
                           (n/z1 -  D1_a[ii, n-1]) * $
                           (n/z1 -  D3_a[ii, n-1])
      PsiZeta_am1[ii, n] = PsiZeta_am1[ii, n-1] * $
                           (n/z2 - D1_am1[ii, n-1]) * $
                           (n/z2 - D3_am1[ii, n-1])
      D3_a[ii, n]   = D1_a[ii, n]   + ci/PsiZeta_a[ii, n]
      D3_am1[ii, n] = D1_am1[ii, n] + ci/PsiZeta_am1[ii, n]
   endfor

   ; Upward recurrence for Q
   Q[ii, 0] = (exp(-2.d * ci * z2) - 1.d) / (exp(-2.d * ci * z1) - 1.d)
   for n = 1.d, nmax do begin
      Num = (z1 * D1_a[ii, n]   + n) * (n - z1 * D3_a[ii, n-1])
      Den = (z2 * D1_am1[ii, n] + n) * (n - z2 * D3_am1[ii, n-1])
      Q[ii, n] = (x[ii-1]/x[ii])^2 * Q[ii, n-1] * Num/Den
   endfor

   ; Upward recurrence for Ha and Hb, Eqs. (7b), (8b) and (12) - (15)
   for n = 1.d, nmax do begin
      G1 = m[ii] * Ha[ii-1, n] - m[ii-1] * D1_am1[ii, n]
      G2 = m[ii] * Ha[ii-1, n] - m[ii-1] * D3_am1[ii, n]
      Temp = Q[ii, n] * G1
      Num = G2 * D1_a[ii, n] - Temp * D3_a[ii, n]
      Den = G2 - Temp
      Ha[ii, n] = Num/Den

      G1 = m[ii-1] * Hb[ii-1, n] - m[ii] * D1_am1[ii, n]
      G2 = m[ii-1] * Hb[ii-1, n] - m[ii] * D3_am1[ii, n]
      Temp = Q[ii, n] * G1
      Num = G2 * D1_a[ii, n] - Temp * D3_a[ii, n]
      Den = G2 - Temp
      Hb[ii, n] = Num/Den
   endfor
endfor                          ;ii (layers)

z1 = dcomplex(x[-1])
; Downward recurrence for D1, Eqs. (16a) and (16b)
; D1[nmax+1] = dcomplex(0)   ; Eq. (16a)
for n = nmax, 1.d, -1.d do $ ; Eq. (16b)
   D1[n-1] = n/z1 - (1.d/(D1[n] + n/z1))

; Upward recurrence for Psi, Zeta, PsiZeta and D3, Eqs. (18a) and (18b)
Psi[0]     = sin(z1)       ; Eq. (18a)
Zeta[0]    = -ci * exp(ci * z1)
PsiZeta[0] = 0.5d * (1.d - exp(2.d * ci * z1))
D3[0] = ci
for n = 1.d, nmax do begin ; Eq. (18b)
   Psi[n]  = Psi[n-1]  * (n/z1 - D1[n-1])
   Zeta[n] = Zeta[n-1] * (n/z1 - D3[n-1])
   PsiZeta[n] = PsiZeta[n-1] * (n/z1 -D1[n-1]) * (n/z1 - D3[n-1])
   D3[n] = D1[n] + ci/PsiZeta[n]
endfor

; Scattering coefficients, Eqs. (5) and (6)
n = dindgen(nmax + 1)
ab[0, *]  = (Ha[-1, *]/m[-1] + n/x[-1]) * Psi  - shift(Psi,  1) ; Eq. (5)
ab[0, *] /= (Ha[-1, *]/m[-1] + n/x[-1]) * Zeta - shift(Zeta, 1)
ab[1, *]  = (Hb[-1, *]*m[-1] + n/x[-1]) * Psi  - shift(Psi,  1) ; Eq. (6)
ab[1, *] /= (Hb[-1, *]*m[-1] + n/x[-1]) * Zeta - shift(Zeta, 1)
ab[*, 0]  = dcomplex(0)

return, ab
end