planewave.h 10.2 KB
``````#ifndef STIM_PLANEWAVE_H
#define STIM_PLANEWAVE_H

#include <string>
#include <sstream>
#include <cmath>

#include "../math/vector.h"
#include "../math/quaternion.h"
#include "../math/constants.h"
#include "../math/plane.h"
#include "../math/complex.h"

namespace stim{
namespace optics{

/// evaluate the scalar field produced by a plane wave at a point (x, y, z)

/// @param x is the x-coordinate of the point
/// @param y is the y-coordinate of the point
/// @param z is the z-coordinate of the point
/// @param A is the amplitude of the plane wave, specifically the field at (0, 0, 0)
/// @param kx is the k-vector component in the x direction
/// @param ky is the k-vector component in the y direction
/// @param kz is the k-vector component in the z direction
template<typename T>
stim::complex<T> planewave_scalar(T x, T y, T z, stim::complex<T> A, T kx, T ky, T kz){
T d = x * kx + y * ky + z * kz;						//calculate the dot product between k and p = (x, y, z) to find the distance p is along the propagation direction
stim::complex<T> di = stim::complex<T>(0, d);		//calculate the phase shift that will have to be applied to propagate the wave distance d
return A * exp(di);									//multiply the phase term by the amplitude at (0, 0, 0) to propagate the wave to p
}

/// evaluate the scalar field produced by a plane wave at several positions

/// @param field is a pre-allocated block of memory that will store the complex field at all points
/// @param N is the number of field values to be evaluated
/// @param x is a set of x coordinates defining positions within the field (NULL implies that all values are zero)
/// @param y is a set of y coordinates defining positions within the field (NULL implies that all values are zero)
/// @param z is a set of z coordinates defining positions within the field (NULL implies that all values are zero)
/// @param A is the amplitude of the plane wave, specifically the field at (0, 0, 0)
/// @param kx is the k-vector component in the x direction
/// @param ky is the k-vector component in the y direction
/// @param kz is the k-vector component in the z direction
template<typename T>
void cpu_planewave_scalar(stim::complex<T>* field, size_t N, T* x, T* y = NULL, T* z = NULL, stim::complex<T> A = 1.0, T kx = 0.0, T ky = 0.0, T kz = 0.0){
T px, py, pz;
for(size_t i = 0; i < N; i++){										// for each element in the array
(x == NULL) ? px = 0 : px = x[i];								// test for NULL values
(y == NULL) ? py = 0 : py = y[i];
(z == NULL) ? pz = 0 : pz = z[i];

field[i] = planewave_scalar(px, py, pz, A, kx, ky, kz);			// call the single-value plane wave function
}
}

template<typename T>
class planewave{

protected:

stim::vec<T> k;							//k-vector, pointed in propagation direction with magnitude |k| = tau / lambda = 2pi / lambda
stim::vec< stim::complex<T> > E0;		//amplitude (for a scalar plane wave, only E0[0] is used)

/// Bend a plane wave via refraction, given that the new propagation direction is known
CUDA_CALLABLE planewave<T> bend(stim::vec<T> kn) const{

stim::vec<T> kn_hat = kn.norm();				//normalize the new k
stim::vec<T> k_hat = k.norm();					//normalize the current k

planewave<T> new_p;								//create a new plane wave

T k_dot_kn = k_hat.dot(kn_hat);					//if kn is equal to k or -k, handle the degenerate case

//if k . n < 0, then the bend is a reflection
if(k_dot_kn < 0) k_hat = -k_hat;				//flip k_hat

if(k_dot_kn == -1){
new_p.k = -k;
new_p.E0 = E0;
return new_p;
}
else if(k_dot_kn == 1){
new_p.k = k;
new_p.E0 = E0;
return new_p;
}

vec<T> r = k_hat.cross(kn_hat);					//compute the rotation vector
T theta = asin(r.len());						//compute the angle of the rotation about r
quaternion<T> q;								//create a quaternion to capture the rotation
q.CreateRotation(theta, r.norm());
vec< complex<T> > E0n = q.toMatrix3() * E0;		//apply the rotation to E0
new_p.k = kn_hat * kmag();
new_p.E0 = E0n;

return new_p;
}

public:

///constructor: create a plane wave propagating along k
CUDA_CALLABLE planewave(vec<T> kvec = stim::vec<T>(0, 0, stim::TAU),
vec< complex<T> > E = stim::vec<T>(1, 0, 0))
{
//phi = phase;

k = kvec;
vec< complex<T> > k_hat = k.norm();

if(E.len() == 0)			//if the plane wave has an amplitude of 0
E0 = vec<T>(0);			//just return it
else{
vec< complex<T> > s = (k_hat.cross(E)).norm();		//compute an orthogonal side vector
vec< complex<T> > E_hat = (s.cross(k)).norm();	//compute a normalized E0 direction vector
E0 = E_hat;// * E_hat.dot(E);					//compute the projection of _E0 onto E0_hat
}

E0 = E0 * exp( complex<T>(0, phase) );
}

///multiplication operator: scale E0
CUDA_CALLABLE planewave<T> & operator* (const T & rhs){
E0 = E0 * rhs;
return *this;
}

CUDA_CALLABLE T lambda() const{
return stim::TAU / k.len();
}

CUDA_CALLABLE T kmag() const{
return k.len();
}

CUDA_CALLABLE vec< complex<T> > E(){
return E0;
}

CUDA_CALLABLE vec<T> kvec(){
return k;
}

/// calculate the value of the field produced by the plane wave given a three-dimensional position
CUDA_CALLABLE vec< complex<T> > pos(T x, T y, T z){
return pos( stim::vec<T>(x, y, z) );
}

CUDA_CALLABLE vec< complex<T> > pos(vec<T> p = vec<T>(0, 0, 0)){
vec< complex<T> > result;

T kdp = k.dot(p);
complex<T> x = complex<T>(0, kdp);
complex<T> expx = exp(x);

result[0] = E0[0] * expx;
result[1] = E0[1] * expx;
result[2] = E0[2] * expx;

return result;
}

//scales k based on a transition from material ni to material nt
CUDA_CALLABLE planewave<T> n(T ni, T nt){
return planewave<T>(k * (nt / ni), E0);
}

CUDA_CALLABLE planewave<T> refract(stim::vec<T> kn) const{
return bend(kn);
}

/// Calculate the result of a plane wave hitting an interface between two refractive indices

/// @param P is a plane representing the position and orientation of the surface
/// @param n0 is the refractive index outside of the surface (in the direction of the normal)
/// @param n1 is the refractive index inside the surface (in the direction away from the normal)
/// @param r is the reflected component of the plane wave
/// @param t is the transmitted component of the plane wave
void scatter(stim::plane<T> P, T n0, T n1, planewave<T> &r, planewave<T> &t){
scatter(P, n1/n0, r, t);
}

/// Calculate the scattering result when nr = n1/n0

/// @param P is a plane representing the position and orientation of the surface
/// @param r is the ration n1/n0
/// @param n1 is the refractive index inside the surface (in the direction away from the normal)
/// @param r is the reflected component of the plane wave
/// @param t is the transmitted component of the plane wave
void scatter(stim::plane<T> P, T nr, planewave<T> &r, planewave<T> &t){

int facing = P.face(k);		//determine which direction the plane wave is coming in

if(facing == -1){		//if the wave hits the back of the plane, invert the plane and nr
P = P.flip();			//flip the plane
nr = 1/nr;				//invert the refractive index (now nr = n0/n1)
}

//use Snell's Law to calculate the transmitted angle
T cos_theta_i = k.norm().dot(-P.norm());				//compute the cosine of theta_i
T theta_i = acos(cos_theta_i);							//compute theta_i
T sin_theta_t = (1/nr) * sin(theta_i);						//compute the sine of theta_t using Snell's law
T theta_t = asin(sin_theta_t);							//compute the cosine of theta_t

bool tir = false;						//flag for total internal reflection
if(theta_t != theta_t){
tir = true;
theta_t = stim::PI / (T)2;
}

//handle the degenerate case where theta_i is 0 (the plane wave hits head-on)
if(theta_i == 0){
T rp = (1 - nr) / (1 + nr);		//compute the Fresnel coefficients
T tp = 2 / (1 + nr);
vec<T> kr = -k;
vec<T> kt = k * nr;			//set the k vectors for theta_i = 0
vec< complex<T> > Er = E0 * rp;		//compute the E vectors
vec< complex<T> > Et = E0 * tp;
T phase_t = P.p().dot(k - kt);	//compute the phase offset
T phase_r = P.p().dot(k - kr);

//create the plane waves
r = planewave<T>(kr, Er, phase_r);
t = planewave<T>(kt, Et, phase_t);
return;
}

//compute the Fresnel coefficients
T rp, rs, tp, ts;
rp = tan(theta_t - theta_i) / tan(theta_t + theta_i);
rs = sin(theta_t - theta_i) / sin(theta_t + theta_i);

if(tir){
tp = ts = 0;
}
else{
tp = ( 2 * sin(theta_t) * cos(theta_i) ) / ( sin(theta_t + theta_i) * cos(theta_t - theta_i) );
ts = ( 2 * sin(theta_t) * cos(theta_i) ) / sin(theta_t + theta_i);
}

//compute the coordinate space for the plane of incidence
vec<T> z_hat = -P.norm();
vec<T> y_hat = P.parallel(k).norm();
vec<T> x_hat = y_hat.cross(z_hat).norm();

//compute the k vectors for r and t
vec<T> kr, kt;
kr = ( y_hat * sin(theta_i) - z_hat * cos(theta_i) ) * kmag();
kt = ( y_hat * sin(theta_t) + z_hat * cos(theta_t) ) * kmag() * nr;

//compute the magnitude of the p- and s-polarized components of the incident E vector
complex<T> Ei_s = E0.dot(x_hat);
int sgn = E0.dot(y_hat).sgn();
vec< complex<T> > cx_hat = x_hat;
complex<T> Ei_p = ( E0 - cx_hat * Ei_s ).len() * sgn;
//compute the magnitude of the p- and s-polarized components of the reflected E vector
complex<T> Er_s = Ei_s * rs;
complex<T> Er_p = Ei_p * rp;
//compute the magnitude of the p- and s-polarized components of the transmitted E vector
complex<T> Et_s = Ei_s * ts;
complex<T> Et_p = Ei_p * tp;

//compute the reflected E vector
vec< complex<T> > Er = vec< complex<T> >(y_hat * cos(theta_i) + z_hat * sin(theta_i)) * Er_p + cx_hat * Er_s;
//compute the transmitted E vector
vec< complex<T> > Et = vec< complex<T> >(y_hat * cos(theta_t) - z_hat * sin(theta_t)) * Et_p + cx_hat * Et_s;

T phase_t = P.p().dot(k - kt);
T phase_r = P.p().dot(k - kr);

//create the plane waves
r.k = kr;
r.E0 = Er * exp( complex<T>(0, phase_r) );

t.k = kt;
t.E0 = Et * exp( complex<T>(0, phase_t) );
}

std::string str()
{
std::stringstream ss;
ss<<"Plane Wave:"<<std::endl;
ss<<"	"<<E0<<" e^i ( "<<k<<" . r )";
return ss.str();
}
};					//end planewave class
}					//end namespace optics
}					//end namespace stim

template <typename T>
std::ostream& operator<<(std::ostream& os, stim::optics::planewave<T> p)
{
os<<p.str();
return os;
}

#endif``````