-
Notifications
You must be signed in to change notification settings - Fork 2
/
OAM_Fourier_Optics.m
295 lines (241 loc) · 6.7 KB
/
OAM_Fourier_Optics.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
clc;clear
%% configuration
wavelength=1; % ***wavelength
k0=2*pi/wavelength; % wavenumber
Z=120*pi; % wave impedance
Volume=1;
%an=(Volume/(pi*4/3)).^(1/3);
TL=2; % Topological charge ***
%% number and position of source points
N=12; % *** number of point sources
% radius and angle
radius= 0.5; % *** radius
theta=linspace(0,2*pi-2*pi/(N),N);
[x00,y00]=pol2cart(theta,radius);
%% sampling the observation area xoy plane
% z direction slices
delta=0.4; % step ***
size=30; % xoy range ***
% xoy plane
xo=-size:delta:size-delta;
yo=-size:delta:size-delta;
zo=1; % observation z ***
% size
Nx=length(xo);
Ny=length(yo);
% position of observation point
index=0;
for m=1:Nx; % X
for n=1:Ny; % Y
index=index+1;
posx(index)=xo(m);
posy(index)=yo(n);
posz(index)=zo(1);
end
end
N_p=index;
figure(1);
plot(x00,y00,'r.')
hold on
plot([x00,0.5],[y00,0],'b--')
xlabel('x (a.u.)')
ylabel('y (a.u.)')
axis equal
% N_p=index;
title('Location of point sources')
%% dyadic Green function
Greenxx=zeros(N_p,N);
Greenyy=zeros(N_p,N);
Greenzz=zeros(N_p,N);
Greenxy=zeros(N_p,N);
Greenxz=zeros(N_p,N);
Greenyz=zeros(N_p,N);
for index_f=1:N_p; % field point number
% counting
%N_p-index_f
% position of observation point
x=posx(index_f);
y=posy(index_f);
z=posz(index_f);
for index_s=1:N;
% position of source point
xx=x00(index_s);
yy=y00(index_s);
zz=0;
% distance
R=sqrt((xx-x)^2+(yy-y)^2+(zz-z)^2);
alpha=k0*R;
% direction
cosx=(x-xx)/R;
cosy=(y-yy)/R;
cosz=(z-zz)/R;
% costant
const1=-j*k0*Z*k0*Volume*exp(-j*alpha)/(4*pi*alpha^3);
const2=3-alpha^2+3*j*alpha;
const3=(alpha)^2-1-j*alpha;
Greenxx(index_f,index_s)=const1*(const3+cosx*cosx*const2);
Greenyy(index_f,index_s)=const1*(const3+cosy*cosy*const2);
Greenzz(index_f,index_s)=const1*(const3+cosz*cosz*const2);
Greenxy(index_f,index_s)=const1*cosx*cosy*const2;
Greenxz(index_f,index_s)=const1*cosx*cosz*const2;
Greenyz(index_f,index_s)=const1*cosy*cosz*const2;
end
end
% dyadic Green function
G=[Greenxx Greenxy Greenxz;
Greenxy Greenyy Greenyz;
Greenxz Greenyz Greenzz];
% decomposition of source point (x polaried OAM source)
for index_s=1:N;
Jx(index_s)=exp(i*TL*theta(index_s)); % helical wavefront
Jy(index_s)=0;
Jz(index_s)=0;
end
J=[Jx Jy Jz].';
% radiated field
Etot=G*J;
Ex=Etot([1:N_p],1);
Ey=Etot([N_p+1:2*N_p],1);
Ez=Etot([2*N_p+1:3*N_p],1);
% transverse component
[x,y]=meshgrid(xo,yo);
[phi, rho]=cart2pol(x,y);
Ex_re=reshape(Ex,Ny,Nx);
%Ex_re=Ex_re./norm(Ex_re,'fro');
figure(2)
subplot(2,1,1)
pcolor(x,y,real(Ex_re));
shading interp;
colorbar;
axis equal
axis([- size size-delta -size size-delta])
title('Re(E_x) at an initial plane')
subplot(2,1,2)
pcolor(x,y,abs(Ex_re));
shading interp;
colorbar;
axis equal
axis([- size size-delta -size size-delta])
title('|E_x| at an initial plane')
%%
% initial parameters for Fourier optics
%lambda = 632.8e-9; % wavelength
%k = 2*pi / lambda; % wavenumber
Scale=5; % scaling factor for zero padding
L = Scale*2*size; % total length
M = Scale*Nx; % total points
z = 5; % propagating distance ***
% 网格定义
dx = L / M;
% x = -L/2 : dx : L/2 - dx;
% [X, Y] = meshgrid(x, x);
% initial vortex beam (in real space)
U0 = Ex_re;
Upad=padarray(U0,[(M-Nx)/2,(M-Nx)/2],0,'both'); % zero padding
% angular spectrum representation of vortex beam
F_U0 = fftshift(fft2((Upad)));
% angular frequency coordinates
df = 1 / (L);
fX = -1/(2*dx) : df : 1/(2*dx) - df;
[ffX, ffY] = meshgrid(fX, fX);
% angular propagation
flag=(1 - wavelength^2 * (ffX.^2 + ffY.^2))>=0;
%sum(sum(flag))
H = exp(-j * k0* z * conj(sqrt(1 - wavelength^2 * (ffX.^2 + ffY.^2))));
F_Uz = H .* F_U0;
% inverse Fourier transform
Uz = (ifft2(fftshift(F_Uz)));
Uz = Uz((M-Nx)/2+1:(M+Nx)/2,(M-Nx)/2+1:(M+Nx)/2);
%% Solution by Green's function method
% z direction slices
%delta=0.2; % step ***
%size=8; % xoy range ***
% xoy plane
xo=-size:delta:size-delta;
yo=-size:delta:size-delta;
zo=zo+z; % observation z
% size
Nx=length(xo);
Ny=length(yo);
% position of observation point
index=0;
for m=1:Nx; % X
for n=1:Ny; % Y
index=index+1;
posx(index)=xo(m);
posy(index)=yo(n);
posz(index)=zo(1);
end
end
N_p=index;
%% dyadic Green function
for index_f=1:N_p; % field point number
% counting
%N_p-index_f
% position of observation point
x=posx(index_f);
y=posy(index_f);
z=posz(index_f);
for index_s=1:N;
% position of source point
xx=x00(index_s);
yy=y00(index_s);
zz=0;
% distance
R=sqrt((xx-x)^2+(yy-y)^2+(zz-z)^2);
alpha=k0*R;
% direction
cosx=(x-xx)/R;
cosy=(y-yy)/R;
cosz=(z-zz)/R;
% costant
const1=-j*k0*Z*k0*Volume*exp(-j*alpha)/(4*pi*alpha^3);
const2=3-alpha^2+3*j*alpha;
const3=(alpha)^2-1-j*alpha;
Greenxx(index_f,index_s)=const1*(const3+cosx*cosx*const2);
Greenyy(index_f,index_s)=const1*(const3+cosy*cosy*const2);
Greenzz(index_f,index_s)=const1*(const3+cosz*cosz*const2);
Greenxy(index_f,index_s)=const1*cosx*cosy*const2;
Greenxz(index_f,index_s)=const1*cosx*cosz*const2;
Greenyz(index_f,index_s)=const1*cosy*cosz*const2;
end
end
% dyadic green function
G=[Greenxx Greenxy Greenxz;
Greenxy Greenyy Greenyz;
Greenxz Greenyz Greenzz];
% decomposition of source point (x polaried OAM source)
for index_s=1:N;
Jx(index_s)=exp(i*TL*theta(index_s));
Jy(index_s)=0;
Jz(index_s)=0;
end
J=[Jx Jy Jz].';
% radiated field
Etot=G*J;
Ex=Etot([1:N_p],1);
Ey=Etot([N_p+1:2*N_p],1);
Ez=Etot([2*N_p+1:3*N_p],1);
% transverse component ***
[x,y]=meshgrid(xo,yo);
[phi, rho]=cart2pol(x,y);
Ex_re=reshape(Ex,Ny,Nx);
%Ex_re=Ex_re./norm(Ex_re,'fro');
figure(3)
subplot(2,1,1)
pcolor(x,y,abs(Ex_re));
shading interp;
colorbar;
axis equal
axis([- size size-delta -size size-delta])
title('|E_x| by Green fuction method')
subplot(2,1,2)
pcolor(x,y,abs(Uz));
%rpcolor(abs(Uz));
shading interp;
colorbar;
axis equal
axis([- size size-delta -size size-delta])
title('|E_x| by Fourier optics')
% calculation error
norm(Uz-Ex_re,'fro')/norm(Ex_re,'fro')