/*
In: tau (lattice parameter)
Out: g2 -> g[0]
g3 -> g[1]
*/
void compute_invariants(gsl_complex tau, gsl_complex *g)
{
gsl_complex q, q14;
gsl_complex t2,t3,t24,t34;
gsl_complex g3_term1, g3_term2;
gsl_complex g2, g3;
q = gsl_complex_exp(gsl_complex_mul_imag(tau,M_PI));
q14 = gsl_complex_exp(gsl_complex_mul_imag(tau,M_PI_4));
t2=theta20(q,q14);
t3=theta30(q);
t24 = pow4(t2);
t34 = pow4(t3);
g2 = gsl_complex_mul_real(gsl_complex_sub(gsl_complex_add(gsl_complex_mul(t24,t24),gsl_complex_mul(t34,t34)),gsl_complex_mul(t24,t34)),_CONST_43PI4);
g3_term1 = gsl_complex_add(gsl_complex_mul(t24,gsl_complex_mul(t24,t24)),gsl_complex_mul(t34,gsl_complex_mul(t34,t34)));
g3_term2 = gsl_complex_mul(gsl_complex_add(t24,t34),gsl_complex_mul(t24,t34));
g3 = gsl_complex_sub( gsl_complex_mul_real(g3_term1, _CONST_827PI6),
gsl_complex_mul_real(g3_term2, _CONST_49PI6) );
g[0] = g2;
g[1] = g3;
}
/* ------------------------------------------------------ */
gsl_complex gsl_complex_arctan2 (gsl_complex a, gsl_complex b)
{
gsl_complex z, p;
if(GSL_REAL(b) != 0.0)
{
z = gsl_complex_arctan(gsl_complex_div(a, b));
if(GSL_REAL(b) < 0.0){
GSL_SET_COMPLEX (&p, M_PI, 0);
if(GSL_REAL(a) >= 0.0)
z = gsl_complex_add(z, p);
else
z = gsl_complex_sub(z, p);
}
}
else
{
if(GSL_REAL(a) >= 0.0)
{
GSL_SET_COMPLEX (&z, M_PI/2.0, 0.0);
}
else
{
GSL_SET_COMPLEX (&z, -M_PI/2.0, 0.0);
}
}
return z;
}
/* NOTE: Assumes z is in fundamental parallelogram */
void wP_and_prime(gsl_complex z, gsl_complex tau, const gsl_complex *g, gsl_complex *p, gsl_complex *pp)
{
int N = 6; /* Enough iterations for good P, not so good P' */
int i;
gsl_complex z0;
gsl_complex z02;
gsl_complex pout, ppout;
gsl_complex ppsolve;
z = near_origin(z,tau);
z0 = gsl_complex_div_real(z,(double)(1 << N));
z02 = gsl_complex_mul(z0,z0);
/* Laurent expansion: P \approx 1/z^2 + (g2/20)z^2 + (g3/28) z^4 */
pout = gsl_complex_add(gsl_complex_inverse(z02),
gsl_complex_add(gsl_complex_mul(z02,gsl_complex_mul_real(g[0],0.05)),
gsl_complex_mul(gsl_complex_mul(z02,z02),gsl_complex_mul_real(g[1],_CONST_1_28))));
/* Laurent expansion: P' \approx -2/z^3 + g2/10z + g3/7 z^3 */
ppout = gsl_complex_add(gsl_complex_mul_real(gsl_complex_inverse(gsl_complex_mul(z0,z02)),-2.0),
gsl_complex_add(gsl_complex_mul(z0,gsl_complex_mul_real(g[0],0.1)),
gsl_complex_mul(gsl_complex_mul(z0,z02),gsl_complex_mul_real(g[1],_CONST_1_7))));
for (i=0;i<N;i++) {
P_and_Pprime_doubler(&pout, &ppout, g);
}
/* At this point ppout is a decent but not great approximation of P'(z) */
/* Instead of using it directly, we use it as a guide for which square root of */
/* (4P^3 - g2 P - g3) should be selected. */
ppsolve = gsl_complex_sqrt(
gsl_complex_sub(
gsl_complex_mul_real(gsl_complex_mul(pout,gsl_complex_mul(pout,pout)),4.0),
gsl_complex_add(gsl_complex_mul(g[0],pout),g[1])
)
);
*p = pout;
if (gsl_complex_abs(gsl_complex_sub(ppsolve,ppout)) < gsl_complex_abs(gsl_complex_add(ppsolve,ppout)))
*pp = ppsolve;
else
*pp = gsl_complex_negative(ppsolve);
}
/* Assuming z is in the (1,tau) parallelogram, return the point
closest to the origin among all translates of z by the lattice. */
gsl_complex near_origin(gsl_complex z, gsl_complex tau)
{
gsl_complex znew;
znew = gsl_complex_sub_real(z,1.0);
if (gsl_complex_abs(z) > gsl_complex_abs(znew))
z = znew;
znew = gsl_complex_sub(z,tau);
if (gsl_complex_abs(z) > gsl_complex_abs(znew))
z = znew;
znew = gsl_complex_sub(z,gsl_complex_add_real(tau,1.0));
if (gsl_complex_abs(z) > gsl_complex_abs(znew))
z = znew;
return z;
}
/* The extended Lattes map (rational function doubling on the elliptic curve) */
void P_and_Pprime_doubler(gsl_complex *p, gsl_complex *pp, const gsl_complex *g)
{
gsl_complex pp3;
gsl_complex ppp, ppp3;
/* p'' */
ppp = gsl_complex_sub(gsl_complex_mul_real(gsl_complex_mul(*p,*p),6.0),
gsl_complex_mul_real(g[0],0.5));
ppp3 = gsl_complex_mul(ppp,gsl_complex_mul(ppp,ppp));
pp3 = gsl_complex_mul(*pp,gsl_complex_mul(*pp,*pp));
*pp = gsl_complex_sub(gsl_complex_add(gsl_complex_mul_real(gsl_complex_div(gsl_complex_mul(*p,ppp),*pp),3.0),
gsl_complex_mul_real(gsl_complex_div(ppp3,pp3),-0.25)),
*pp);
*p = P_doubler(*p,g);
}
/* psi(z) for complex z in the right half-plane */
static int
psi_complex_rhp(
gsl_complex z,
gsl_sf_result * result_re,
gsl_sf_result * result_im
)
{
int n_recurse = 0;
int i;
gsl_complex a;
if(GSL_REAL(z) == 0.0 && GSL_IMAG(z) == 0.0)
{
result_re->val = 0.0;
result_im->val = 0.0;
result_re->err = 0.0;
result_im->err = 0.0;
return GSL_EDOM;
}
/* compute the number of recurrences to apply */
if(GSL_REAL(z) < 20.0 && fabs(GSL_IMAG(z)) < 20.0)
{
const double sp = sqrt(20.0 + GSL_IMAG(z));
const double sn = sqrt(20.0 - GSL_IMAG(z));
const double rhs = sp*sn - GSL_REAL(z);
if(rhs > 0.0) n_recurse = ceil(rhs);
}
/* compute asymptotic at the large value z + n_recurse */
a = psi_complex_asymp(gsl_complex_add_real(z, n_recurse));
result_re->err = 2.0 * GSL_DBL_EPSILON * fabs(GSL_REAL(a));
result_im->err = 2.0 * GSL_DBL_EPSILON * fabs(GSL_IMAG(a));
/* descend recursively, if necessary */
for(i = n_recurse; i >= 1; --i)
{
gsl_complex zn = gsl_complex_add_real(z, i - 1.0);
gsl_complex zn_inverse = gsl_complex_inverse(zn);
a = gsl_complex_sub(a, zn_inverse);
/* accumulate the error, to catch cancellations */
result_re->err += 2.0 * GSL_DBL_EPSILON * fabs(GSL_REAL(zn_inverse));
result_im->err += 2.0 * GSL_DBL_EPSILON * fabs(GSL_IMAG(zn_inverse));
}
result_re->val = GSL_REAL(a);
result_im->val = GSL_IMAG(a);
result_re->err += 2.0 * GSL_DBL_EPSILON * fabs(result_re->val);
result_im->err += 2.0 * GSL_DBL_EPSILON * fabs(result_im->val);
return GSL_SUCCESS;
}
double rpp(gsl_matrix_complex * M)
{
gsl_complex t11, t43, t41, t13, t33, t31, mul1, mul2, mul3, mul4, sub1, sub2, div1;
t11 = gsl_matrix_complex_get(M,0,0);
t43 = gsl_matrix_complex_get(M,3,2);
t41 = gsl_matrix_complex_get(M,3,0);
t13 = gsl_matrix_complex_get(M,0,2);
t33 = gsl_matrix_complex_get(M,2,2);
t31 = gsl_matrix_complex_get(M,2,0);
mul1 = gsl_complex_mul(t11,t43);
mul2 = gsl_complex_mul(t41,t13);
mul3 = gsl_complex_mul(t11,t33);
mul4 = gsl_complex_mul(t13,t31);
sub1 = gsl_complex_sub(mul1,mul2);
sub2 = gsl_complex_sub(mul3,mul4);
div1 = gsl_complex_div(sub1,sub2);
return gsl_complex_abs2(div1);
}
void gtmiso(gsl_matrix_complex * Tiso, gsl_complex eiso, double k0, double eta, double diso)
{
double eta2 = pow(eta,2);
gsl_complex rb = gsl_complex_rect(eta2,0);
gsl_complex za = gsl_complex_sub_real(eiso,eta2);
gsl_complex qiso = gsl_complex_sqrt(za);
gsl_complex zb = gsl_complex_mul_real(qiso,k0);
gsl_complex zc = gsl_complex_mul_real(zb,diso);
gsl_complex zd = gsl_complex_div(rb,eiso);
gsl_complex carg = gsl_complex_cos(zc);
gsl_complex sarg = gsl_complex_sin(zc);
gsl_complex i = gsl_complex_rect(0,1);
gsl_complex one = gsl_complex_rect(1,0);
gsl_matrix_complex_set_zero(Tiso);
gsl_matrix_complex_set(Tiso,0,0,carg);
gsl_matrix_complex_set(Tiso,1,1,carg);
gsl_matrix_complex_set(Tiso,2,2,carg);
gsl_matrix_complex_set(Tiso,3,3,carg);
gsl_complex zd1 = gsl_complex_sub(one,zd);
gsl_complex zd2 = eiso;
gsl_complex zd3 = one;
gsl_complex zd4 = gsl_complex_sub_real(eiso,eta2);
gsl_complex zmult1 = gsl_complex_mul(zd1,i);
gsl_complex zmult2 = gsl_complex_mul(zd2,i);
gsl_complex zmult3 = gsl_complex_mul(zd3,i);
gsl_complex zmult4 = gsl_complex_mul(zd4,i);
gsl_complex zdiv1 = gsl_complex_div(zmult1,qiso);
gsl_complex zdiv2 = gsl_complex_div(zmult2,qiso);
gsl_complex zdiv3 = gsl_complex_div(zmult3,qiso);
gsl_complex zdiv4 = gsl_complex_div(zmult4,qiso);
gsl_complex zmult5 = gsl_complex_mul(zdiv1,sarg);
gsl_complex zmult6 = gsl_complex_mul(zdiv2,sarg);
gsl_complex zmult7 = gsl_complex_mul(zdiv3,sarg);
gsl_complex zmult8 = gsl_complex_mul(zdiv4,sarg);
gsl_matrix_complex_set(Tiso,0,1,zmult5);
gsl_matrix_complex_set(Tiso,1,0,zmult6);
gsl_matrix_complex_set(Tiso,2,3,zmult7);
gsl_matrix_complex_set(Tiso,3,2,zmult8);
}
int main(int argc, char** argv)
{
gsl_complex a,b;
GSL_SET_COMPLEX(&a,3,4);//a=3+4i
GSL_SET_COMPLEX(&b,6,8);//b=6+8i
gsl_complex c = gsl_complex_add(a,b);
printf("a+b\treal : %f image : %f\n",c.dat[0],c.dat[1]);
c = gsl_complex_sub(a,b);
printf("a-b\treal : %f image : %f\n",c.dat[0],c.dat[1]);
c = gsl_complex_mul(a,b);
printf("a*b\treal : %f image : %f\n",c.dat[0],c.dat[1]);
c = gsl_complex_div(a,b);
printf("a/b\treal : %f image : %f\n",c.dat[0],c.dat[1]);
// system("PAUSE");
return 0;
}
/* The Lattes map */
gsl_complex P_doubler(gsl_complex p, const gsl_complex *g)
{
gsl_complex p2, p3;
gsl_complex num;
gsl_complex denom;
gsl_complex term;
p2 = gsl_complex_mul(p,p);
p3 = gsl_complex_mul(p2,p);
/* denom = 4p^3 - g2p - g3 */
denom = gsl_complex_sub(gsl_complex_mul_real(p3,4.0),
gsl_complex_add(gsl_complex_mul(p,g[0]),g[1]));
/* num = (p^2 + g2/4)^2 + 2g3p */
term = gsl_complex_add(p2,gsl_complex_mul_real(g[0],0.25));
num = gsl_complex_add(gsl_complex_mul(p,gsl_complex_mul_real(g[1],2.0)),
gsl_complex_mul(term,term));
return gsl_complex_div(num,denom);
}
void tabulate(FILE *os,
ComplexFunction fun,
void *params,
gsl_complex z0,
gsl_complex z1,
int n)
{
gsl_complex k = gsl_complex_sub(z1, z0);
for (int i = 0; i < n; ++i)
{
double t = (double)i / (n - 1);
gsl_complex z = gsl_complex_add(z0, gsl_complex_mul_real(k, t));
gsl_complex f = fun(z, params);
double abs = gsl_complex_abs(f);
fprintf(os, "%g %g %g %g %g %g %g\n",
t,
GSL_REAL(z), GSL_IMAG(z),
GSL_REAL(f), GSL_IMAG(f), abs, -abs);
}
}
/******************************************************************************
* calc_X()
* This function calcs the X's and g's for a given theta.
*
* Parameters
* ---------
*
* Returns
* -------
*
* Notes
* -----
* This function computes
* X[j] = Ar[j]/Ai[j]
* where Ar[j] and Ai[j] are the field amplitdues within layer j at
* the j/j-1 interface. For the base layer X[0] = 0,
* ie there is no reflected field in the substrate. While
* X[nlayers-1] is the field intensity at the top of the multilayer,
* therefore |X[nlayers-1]|^2 = R
*
* The X's are calculated using the optical recursion formlua, starting at the bottom
* where X[0] = 0 and calc X[1], and proceding to calc of X[nlayer-1]:
*
* X[j] = ( r[j] + X[j-1]*phase) / (1 + r[j]*X[j-1]*phase)
*
* The r[j] values are the reflection coefficients for the j/j-1 interface,
* and calculated from:
*
* r[j] = (g[j]-g[j-1])/(g[j]+g[j-1])
*
* This function also includes an interfacial roughness term mult into
* the reflection coefficient for the j/j-1 interface:
*
* r[j] = r[j]*exp(-1.0(q*sig[j])^2)
*
* The phase term accounts for the phase shift and attentuation of the field
* traversing the j-1 layer (ie from the j-1/j-2 interface to the j/j-1 interface)
*
* phase = exp( -i* 2*k*d[j-1] * g[j-1])
*
* Note if j == 1 then the phase for layer j-1 is 0.0,
* ie infinitley thick base layer
*
* Note: g[j] = sqrt( n[j]^2 - cos(theta)^2 )
* where n[j] is the layers index of refraction: n[j] = 1-del[j]-i*bet[j]
* Hence, each layer has a unique g value. We assume here
* that the del[j] and bet[j] have already been calculated
* (see calc_del_bet(), this should be called before this function).
*
*
* Note: double check how doing the phase calc below, is that the best way??
* e.g. could we use:
* phase = gsl_complex_exp( gsl_complex_mul_real(g1,-1.0*k*d[j]));
* instead of
* phase = gsl_complex_exp( gsl_complex_mul(phase,g1));
*
* Note we should move the g and r calc to a new function, possibly calc the
* t values as well and store these so we dont have to recalc in calc_A() (???)
*
******************************************************************************/
int calc_X(double theta, ref_model *ref, angle_calc *ang_c, layer_calc *lay_c){
int i, j;
double cos_sqr_thet, k, q, dw;
double *d, *sig;
gsl_complex g1, g2, num, den, r, X2, X1, n1, n2, phase;
// convenience vars
cos_sqr_thet = square(cos(theta*M_PI/180.));
k = ref->k;
q = 2.0*k*sin((theta*M_PI/180.));
d = ref->d;
sig = ref->sigma;
// start at bottom
// ie interface 0
ang_c->Re_X[0] = GSL_REAL(X1) = 0.0;
ang_c->Im_X[0] = GSL_IMAG(X1) = 0.0;
// calc g for base layer (g[0])
// n = 1 - del -i*bet
// g = sqrt( n^2 - cos(theta)^2 )
GSL_REAL(n1) = 1.0 - lay_c->del[0];
GSL_IMAG(n1) = -1.0*lay_c->bet[0];
g1 = gsl_complex_sqrt( gsl_complex_sub_real(gsl_complex_mul(n1, n1), cos_sqr_thet) );
// store g for base layer
ang_c->Re_g[0] = GSL_REAL(g1);
ang_c->Im_g[0] = GSL_IMAG(g1);
// loop over all layers
// note j is upper layer and i is the interface number
// the loop starts considering the first interface (i=0)
// which is the interface between the base layer (j=0)
// and the first layer (j=1) ie the 1/0 interface
// note there are nlayer-1 interfaces...
i = 0;
for ( j = 1; j < ref->nlayer; j++){
// calc g for upper layer
// n = 1 - del -i*bet
// g2 = sqrt( n^2 - cos(theta)^2 )
GSL_REAL(n2) = 1.0 - lay_c->del[j];
GSL_IMAG(n2) = -1.0 * lay_c->bet[j];
g2 = gsl_complex_sqrt( gsl_complex_sub_real(gsl_complex_mul(n2, n2), cos_sqr_thet) );
// store g for layer j
ang_c->Re_g[j] = GSL_REAL(g2);
ang_c->Im_g[j] = GSL_IMAG(g2);
// calculate r for the j/j-1 interface
// r_j = (g2-g1)/(g2+g1)
//GSL_SET_COMPLEX(&num,0.0,0.0);
//GSL_SET_COMPLEX(&den,0.0,0.0);
//GSL_SET_COMPLEX(&r,0.0,0.0);
num = gsl_complex_sub(g2,g1);
den = gsl_complex_add(g2,g1);
r = gsl_complex_div(num,den);
// calc r including roughness
// simple dw roughness model
// r = r*exp(-1.0(q*sig)^2)
// sigma[i]
if (ref->sigma[j] > 0.0){
//dw = exp(-1.0*square(q*ref->sigma[j]));
dw = exp(-1.0*square(q*ref->sigma[i]));
r = gsl_complex_mul_real(r,dw);
}
//calc phase shift and attentuation for
//field traversing the j-1 layer
//phase = exp( -i* 2*k*d[j-1] * g1)
// if j == 1 then the phase for layer j-1
// is 0.0, ie infinitley thick base layer
if (j == 1){
GSL_REAL(phase) = 0.0;
GSL_IMAG(phase) = 0.0;
} else {
// check here
GSL_REAL(phase) = 0.0;
GSL_IMAG(phase) = -2.0*k*d[j-1];
phase = gsl_complex_exp( gsl_complex_mul(phase,g1));
}
// calc Xj
// X2 = ( r + X1*phase) / (1 + r*X1*phase)
//GSL_SET_COMPLEX(&num,0.0,0.0);
//GSL_SET_COMPLEX(&den,0.0,0.0);
num = gsl_complex_add(r,gsl_complex_mul(X1,phase));
den = gsl_complex_add_real(gsl_complex_mul(r,gsl_complex_mul(X1,phase)), 1.0);
X2 = gsl_complex_div(num,den);
if (fabs(GSL_REAL(X2)) < 1e-10){
GSL_REAL(X2) = 0.0;
}
if (fabs(GSL_IMAG(X2)) < 1e-10){
GSL_IMAG(X2) = 0.0;
}
// store values of Xj
ang_c->Re_X[j] = GSL_REAL(X2);
ang_c->Im_X[j] = GSL_IMAG(X2);
// use top layer as bottom for next time through
GSL_REAL(X1) = GSL_REAL(X2);
GSL_IMAG(X1) = GSL_IMAG(X2);
// do the same for g's
GSL_REAL(g1) = GSL_REAL(g2);
GSL_IMAG(g1) = GSL_IMAG(g2);
//increment the interface number
i++;
}
return (SUCCESS);
}
int
gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A, gsl_permutation * p, int *signum)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("LU decomposition requires square matrix", GSL_ENOTSQR);
}
else if (p->size != A->size1)
{
GSL_ERROR ("permutation length must match matrix size", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
size_t i, j, k;
*signum = 1;
gsl_permutation_init (p);
for (j = 0; j < N - 1; j++)
{
/* Find maximum in the j-th column */
gsl_complex ajj = gsl_matrix_complex_get (A, j, j);
double max = gsl_complex_abs (ajj);
size_t i_pivot = j;
for (i = j + 1; i < N; i++)
{
gsl_complex aij = gsl_matrix_complex_get (A, i, j);
double ai = gsl_complex_abs (aij);
if (ai > max)
{
max = ai;
i_pivot = i;
}
}
if (i_pivot != j)
{
gsl_matrix_complex_swap_rows (A, j, i_pivot);
gsl_permutation_swap (p, j, i_pivot);
*signum = -(*signum);
}
ajj = gsl_matrix_complex_get (A, j, j);
if (!(GSL_REAL(ajj) == 0.0 && GSL_IMAG(ajj) == 0.0))
{
for (i = j + 1; i < N; i++)
{
gsl_complex aij_orig = gsl_matrix_complex_get (A, i, j);
gsl_complex aij = gsl_complex_div (aij_orig, ajj);
gsl_matrix_complex_set (A, i, j, aij);
for (k = j + 1; k < N; k++)
{
gsl_complex aik = gsl_matrix_complex_get (A, i, k);
gsl_complex ajk = gsl_matrix_complex_get (A, j, k);
/* aik = aik - aij * ajk */
gsl_complex aijajk = gsl_complex_mul (aij, ajk);
gsl_complex aik_new = gsl_complex_sub (aik, aijajk);
gsl_matrix_complex_set (A, i, k, aik_new);
}
}
}
}
return GSL_SUCCESS;
}
}
void SoftDemapper::Run() {
unsigned int nbits,nsymbs,count;
/// fetch data objects
gsl_vector_complex_class input = vin1.GetDataObj();
// number of input symbs
nsymbs = input.vec->size;
// cout << "received " << nsymbs << " elements in vector." << endl;
// number of output symbols
nbits = nsymbs * Nb();
gsl_vector *llr=gsl_vector_alloc(nbits);
double *den = (double *)calloc( Nb(), sizeof(double) );
double *num = (double *)calloc( Nb(), sizeof(double) );
// determine symb_likelyhood
for (int i=0;i<nsymbs;i++) { // cycle through received symbols
for (int k=0;k<Nb();k++) {
num[k] = GSL_NEGINF;
den[k] = GSL_NEGINF;
}
// the received symbol
gsl_complex recsym = gsl_vector_complex_get(input.vec,i);
// cout << "received symbol = ("
// << GSL_REAL(recsym)
// << ","
// << GSL_IMAG(recsym)
// << ") " << endl;
for (int j=0;j<Ns;j++) { // cycle through postulated symbol
// gray encoded symbol id
unsigned int symbol_id = gsl_vector_uint_get(gray_encoding,j);
// complex symbol
gsl_complex refsym = gsl_complex_polar(1.0,symbol_arg * symbol_id );
// likelyhood metric
double metric = gsl_complex_abs( gsl_complex_sub(refsym,recsym) );
metric = -EsNo()*metric*metric;
//gsl_matrix_complex_set(symb_likelihoods,j,i);
//
// HERE is available a metric for symb i and refsymb j
//
int mask = 1 << Nb() - 1;
for (int k=0;k<Nb();k++) { /* loop over bits */
if (mask&j) {
/* this bit is a one */
num[k] = ( *max_star[LMAP()] )( num[k], metric );
} else {
/* this bit is a zero */
den[k] = ( *max_star[LMAP()] )( den[k], metric );
}
mask = mask >> 1;
} //bits
} // alphabet
for (int k=0;k<Nb();k++) {
gsl_vector_set(llr,Nb()*i+k,num[k] - den[k]);
}
} // symbols
gsl_vector_class outv(llr);
// outv.show();
// output bitwise LLR
vout1.DeliverDataObj(outv);
gsl_vector_free(llr);
// free dynamic structures
free(num);
free(den);
}
complex& complex::operator-=(const complex& z1)
{
_complex = gsl_complex_sub(_complex, z1._complex);
return *this;
}
complex complex::operator-(const complex& z1) const
{
gsl_complex rl = gsl_complex_sub(_complex, z1._complex);
return complex(rl);
}
/******************************************************************************
* calc_A()
* This function calcs the A's for a given theta.
*
* Parameters
* ---------
*
* Returns
* -------
*
* Notes
* -----
* This function assumes that the X[j] and g[j] values have already
* been calculated (see calc_X())
*
* Note: X[j] = Ar[j]/Ai[j]
* And we can derive:
*
* Ai[j] = phase*Ai[j+1]*t[j+1] / ( 1 + phase^2 * X[j] * r[j+1])
* Ar[j] = Ai[j]*X[j]
*
* where Ar[j] and Ai[j] are the field amplitudes within layer j at
* the j/j-1 interface.
*
* For the top layer Ai[nlayer-1] = 1. ie everything is referenced to
* unit intensity (Io=|Ai|^2) within the top layer. Therefore Ar[0] = X[0].
*
* Given the known values for the top layer we can procede to calculate each Ai and Ar
* value starting at the top.
*
* For the base layer X[0] = 0, ie Ar[0] = 0
*
* The values of r[j+1] and t[j+1] are the reflection and transmission coefficients
* for the j+1/j interface, respectively, and calculated from:
*
* r[j+1] = (g[j+1]-g[j])/(g[j+1]+g[j])
* t[j+1] = 2*g[j+1]/(g[j+1] + g[j]) = 1 + r[j+1]
*
* This function also includes an interfacial roughness term mult into
* the reflection coefficient for the j+1/j interface:
*
* r[j+1] = r[j+1]*exp(-1.0(q*sig[j+1])^2)
*
* Note we include roughness in the t calc using the below:
* t[j+1] = 1 + r[j+1]*exp(-1.0(q*sig[j+1])^2)
*
* Therefore as r-> 0, t-> 1. This is only done if calc_params[10] > 0.
* Otherwise t is calc explicitley from the g's
*
* The phase term accounts for the phase shift and attentuation of the field
* traversing the j layer (ie from the j+1/j interface to the j/j-1 interface),
* ie the Ai[j] and Ar[j] are the field amplitudes at the j/j-1 interface:
*
* phase = exp( -i*k*d[j] * g[j])
*
* Note if j == 0 then the phase for layer j is 0.0, and X[0] = 0
* ie in the infinitley thick base layer we assumer the is no reflected
* field (Ar[0] = 0.0) and we are calculating the the field amplitude at the
* top of layer 0 rather than the bottom of layer 0.
*
*
* Note: double check how doing the phase calc below, is that the best way??
* e.g. could we use:
* phase = gsl_complex_exp( gsl_complex_mul_real(g1,-1.0*k*d[j]));
* instead of
* phase = gsl_complex_exp( gsl_complex_mul(phase,g1));
*
*
******************************************************************************/
int calc_A(double theta, ref_model *ref, angle_calc *ang_c, layer_calc *lay_c){
int i, j, idx;
double cos_sqr_thet, k, q, dw;
double *d, *sig;
gsl_complex g1, g2, num, den, r, t, Ai_2, Ar_2, Ai_1, Ar_1, X1, phase, phase_sqr;
//gsl_complex n1, n2;
// convenience vars
cos_sqr_thet = square(cos(theta*M_PI/180.));
k = ref->k;
q = 2.0*k*sin((theta*M_PI/180.));
d = ref->d;
sig = ref->sigma;
// start at top
idx = ref->nlayer - 1;
// get g for upper layer, ie air/vaccum slab
// ie del ~ bet ~ 0.
// n = 1 - del -i*bet ~ 1
// g2 = sqrt( n^2 - cos(theta)^2 )
//GSL_REAL(n2) = 1.0 - lay_c->del[idx];
//GSL_IMAG(n2) = -1.0*lay_c->bet[idx];
//g2 = gsl_complex_sqrt( gsl_complex_sub_real(gsl_complex_mul(n2, n2), cos_sqr_thet) );
GSL_SET_COMPLEX(&g2,ang_c->Re_g[idx],ang_c->Im_g[idx]);
// for the top (j=nlayer-1) layer, Ai =1.0
// therefore Ar = Ai*X = X
// store these in A arrays
ang_c->Re_Ai[idx] = GSL_REAL(Ai_2) = 1.0;
ang_c->Im_Ai[idx] = GSL_IMAG(Ai_2) = 0.0;
ang_c->Re_Ar[idx] = GSL_REAL(Ar_2) = ang_c->Re_X[idx];
ang_c->Im_Ar[idx] = GSL_IMAG(Ar_2) = ang_c->Im_X[idx];
// loop over all layers, start at top - 1
// ie loop from j = nlayer-2 --> j = 0
// note j is lower layer in the r and t calcs
// therefore we are considering the j+1/j interface
// i is the interface number, ie which interface
// are the r's and t's calc for
i = idx - 1;
for ( j = idx - 1; j > -1 ; j--){
// calc g and phase for lower layer
// n = 1 - del -i*bet
// g1 = sqrt( n^2 - cos(theta)^2 )
//GSL_REAL(n1) = 1.0 - xpar->del[j];
//GSL_IMAG(n1) = -1.0 * xpar->bet[j];
//g1 = gsl_complex_sqrt( gsl_complex_sub_real(gsl_complex_mul(n1, n1), cos_sqr_thet) );
GSL_SET_COMPLEX(&g1,ang_c->Re_g[j],ang_c->Im_g[j]);
//calc phase and attenuation
// phase = exp( -i*k*d[j] * g1)
// phase_sqr = (phase)^2
if (j == 0){
GSL_REAL(phase) = 1.0;
GSL_IMAG(phase) = 0.0;
GSL_REAL(phase_sqr) = 1.0;
GSL_IMAG(phase_sqr) = 0.0;
} else {
GSL_REAL(phase) = 0.0;
GSL_IMAG(phase) = -1.0*k*d[j];
phase = gsl_complex_exp( gsl_complex_mul(phase,g1));
GSL_REAL(phase_sqr) = 0.0;
GSL_IMAG(phase_sqr) = -2.0*k*d[j];
phase_sqr = gsl_complex_exp( gsl_complex_mul(phase_sqr,g1));
}
// calc r for upper layer
// r = (g2-g1)/(g2+g1)
GSL_SET_COMPLEX(&num,0.0,0.0);
GSL_SET_COMPLEX(&den,0.0,0.0);
num = gsl_complex_sub(g2,g1);
den = gsl_complex_add(g2,g1);
r = gsl_complex_div(num,den);
// include simple dw type roughness
// r = r*exp(-1.0(q*sig)^2)
if (ref->sigma[j+1] > 0.0){
//dw = exp(-1.0*square(q*ref->sigma[j+1]));
dw = exp(-1.0*square(q*ref->sigma[i]));
r = gsl_complex_mul_real(r,dw);
}
// calc t for upper layer
// t = 2*g2/(g2 + g1)
if (ref->calc_params[10] == 0.0){
GSL_SET_COMPLEX(&num,0.0,0.0);
num = gsl_complex_mul_real(g2, 2.0);
t = gsl_complex_div(num,den);
} else {
// note as as r->0, t->1
// ie t = 1+r, so we could just calc t from r?
t = gsl_complex_add_real(r,1.0);
}
// calc Ai_1 and Ar_1
// these are field mag in the lower slab (layer j)
// at the bottom of the slab, relative to Ai in the top layer (j=nlayer-1)
// Ai_1 = phase*Ai_2*t / ( 1 + phase^2 * X1 * r)
// Ar_1 = Ai_1*X1
GSL_REAL(X1) = ang_c->Re_X[j];
GSL_IMAG(X1) = ang_c->Im_X[j];
GSL_SET_COMPLEX(&num,0.0,0.0);
GSL_SET_COMPLEX(&den,0.0,0.0);
num = gsl_complex_mul(t,gsl_complex_mul(Ai_2,phase));
den = gsl_complex_add_real(gsl_complex_mul(phase_sqr,gsl_complex_mul(X1,r)),1.0);
Ai_1 = gsl_complex_div(num,den);
if (fabs(GSL_REAL(Ai_1)) < 1.0e-12){
GSL_REAL(Ai_1) = 0.0;
}
if (fabs(GSL_IMAG(Ai_1)) < 1.0e-12){
GSL_IMAG(Ai_1) = 0.0;
}
Ar_1 = gsl_complex_mul(Ai_1,X1);
//store results
ang_c->Re_Ai[j] = GSL_REAL(Ai_1);
ang_c->Im_Ai[j] = GSL_IMAG(Ai_1);
ang_c->Re_Ar[j] = GSL_REAL(Ar_1);
ang_c->Im_Ar[j] = GSL_IMAG(Ar_1);
// this is the field intensity at the bottom of layer j
// except for layer j = 0, this is the intensity at the top
// of the base layer (bc of how phase is calc above)
// use bottom layer as top for next time through
GSL_REAL(Ai_2) = GSL_REAL(Ai_1);
GSL_IMAG(Ai_2) = GSL_IMAG(Ai_1);
GSL_REAL(Ar_2) = GSL_REAL(Ar_1);
GSL_IMAG(Ar_2) = GSL_IMAG(Ar_1);
// do the same for g's
GSL_REAL(g2) = GSL_REAL(g1);
GSL_IMAG(g2) = GSL_IMAG(g1);
// increment interface number
i--;
}
return (SUCCESS);
}
gsl_complex integrate_line_segment(Params *params,
gsl_complex p0,
gsl_complex p1)
{
gsl_complex k = gsl_complex_sub(p1, p0);
const double r = params->r;
const double a = 0.0; // parameter interval start
const double b = 1.0; // parameter interval end
const double L = b - a; // length of parameter interval
const size_t table_size = 1000;
// calculate frequency of oscillatory part
double omega = GSL_REAL(k) * r;
gsl_integration_workspace *ws = gsl_integration_workspace_alloc(table_size);
// prepare sine/cosine tables for integration
gsl_integration_qawo_table *table_cos = gsl_integration_qawo_table_alloc(omega, L, GSL_INTEG_COSINE, table_size);
gsl_integration_qawo_table *table_sin = gsl_integration_qawo_table_alloc(omega, L, GSL_INTEG_SINE, table_size);
LineSegmentParams lsp;
lsp.p0 = p0;
lsp.k = k;
lsp.damping = params->r * GSL_IMAG(k);
lsp.params = params;
//fprintf(stderr, "p0 = %g %g, p1 = %g %g, k = %g %g, r = %g, omega = %g, damping = %g\n",
// GSL_REAL(p0), GSL_IMAG(p0), GSL_REAL(p1), GSL_IMAG(p1),
// GSL_REAL(k), GSL_IMAG(k), r, omega, lsp.damping);
gsl_function F;
F.function = &line_segment_integrand_wrapper;
F.params = &lsp;
double result_real_cos, abserr_real_cos;
double result_real_sin, abserr_real_sin;
double result_imag_cos, abserr_imag_cos;
double result_imag_sin, abserr_imag_sin;
double epsabs = 1e-9;
double epsrel = 1e-9;
lsp.part = REAL;
gsl_integration_qawo(&F, a, epsabs, epsrel, table_size, ws, table_cos, &result_real_cos, &abserr_real_cos);
gsl_integration_qawo(&F, a, epsabs, epsrel, table_size, ws, table_sin, &result_real_sin, &abserr_real_sin);
lsp.part = IMAG;
gsl_integration_qawo(&F, a, epsabs, epsrel, table_size, ws, table_cos, &result_imag_cos, &abserr_imag_cos);
gsl_integration_qawo(&F, a, epsabs, epsrel, table_size, ws, table_sin, &result_imag_sin, &abserr_imag_sin);
//fprintf(stderr, " cos: %g (+- %g) %g (+- %g) sin: %g (+- %g) %g (+- %g)\n",
// result_real_cos, abserr_real_cos, result_imag_cos, abserr_imag_cos,
// result_real_sin, abserr_real_sin, result_imag_sin, abserr_imag_sin);
gsl_complex cos_part = gsl_complex_rect(result_real_cos, result_imag_cos);
gsl_complex sin_part = gsl_complex_rect(-result_imag_sin, result_real_sin);
gsl_complex sum = gsl_complex_add(cos_part, sin_part);
gsl_complex result = gsl_complex_mul(
k,
gsl_complex_mul(sum, gsl_complex_exp(gsl_complex_mul_imag(p0, params->r))));
gsl_integration_qawo_table_free(table_sin);
gsl_integration_qawo_table_free(table_cos);
gsl_integration_workspace_free(ws);
return result;
}
