-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathpropagation_through_axicon_pair_to_Julien.m
320 lines (253 loc) · 13.5 KB
/
propagation_through_axicon_pair_to_Julien.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
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
% Clear the memory, close all figures and clear the screen
clear; close; clc;
%Input Coordinate System
x_range = 10000 * 10 ^ (-6);
y_range = 10000 * 10 ^ (-6);
N = 4000;
period = x_range/N;
lambda = 828 * 10 ^ (-9);
nPixel = (-N/2:(N/2-1)); % create an array of the sample number, with zero in the centre
x = nPixel/N*x_range; % calculate the spatial coordinates
y = nPixel/N*y_range; % calculate the spatial coordinates
[X,Y] = meshgrid(x,y);
%Second Axicon Coordinate System
dx5 = period; %spatial period x-direction
dy5 = period; %spatial period y-direction
fx5=-1/(2*dx5):1/x_range:1/(2*dx5)-1/x_range; % calculate the FT coordinates
fy5=-1/(2*dy5):1/y_range:1/(2*dy5)-1/y_range; % calculate the FT coordinates
FT5_period = 1/y_range;
FT5_full_range = N/y_range;
[FX5,FY5] = meshgrid(fx5,fy5);
z4 = 120 * 10 ^(-3); % separation between two axicons
x_prime5 = fx5.*lambda.*z4; % calculate the FT spatial coordinates(Scaled)
y_prime5 = fy5.*lambda.*z4; % calculate the FT spatial coordinates(Scaled)
[X_prime5,Y_prime5] = meshgrid(x_prime5,y_prime5);
xprime5_period = FT5_period.*lambda.*z4;
xprime5_full_range = FT5_full_range.*lambda.*z4;
%OBSERV plane Coordinate System
dx6 = xprime5_period; %spatial period x-direction
dy6 = xprime5_period; %spatial period y-direction
fx6=-1/(2*dx6):1/xprime5_full_range:1/(2*dx6)-1/xprime5_full_range; % calculate the FT coordinates
fy6=-1/(2*dy6):1/xprime5_full_range:1/(2*dy6)-1/xprime5_full_range; % calculate the FT coordinates
FT6_period = 1/xprime5_full_range;
FT6_full_range = N/xprime5_full_range;
[FX6,FY6] = meshgrid(fx6,fy6);
z5 = 0.2; % position of the OBSERV plane
x_prime6 = fx6.*lambda.*z5; % calculate the FT spatial coordinates(Scaled)
y_prime6 = fy6.*lambda.*z5; % calculate the FT spatial coordinates(Scaled)
[X_prime6,Y_prime6] = meshgrid(x_prime6,y_prime6);
xprime6_period = FT6_period.*lambda.*z5;
xprime6_full_range = FT6_full_range.*lambda.*z5;
%Gaussian Input Beam
k = 2*pi/lambda;
A = 1;
w0 = 2700/2 * 10 ^ (-6);
zR = pi * w0 ^ 2/lambda; %Rayleigh range
z = 0; %z_bef, position(in m) of the GB beam waist comparing with the position of the lens
t = 0;
rho = sqrt(X.^2 + Y.^2);
if z == 0
Rz = Inf; %at beam waist plane wave
else
Rz = z + zR^2/z;%at z =! 0
end
[E_GB, I_GB, power] = getGB(A,z,lambda,w0,zR, t, rho, Rz);%Obtain GB profile and its total power
%plot GB input
figure(1);
pcolor(X, Y, I_GB);shading interp;colorbar;axis equal;
title('Input GB Intensity Profile, w0 = 2.7mm/2, zbef = 0m');
xlabel('x(m)')
ylabel('y(m)')
zlabel('Amplitude')
%%%Axicon parameters%%%%%%
r_axi = 25.4 /2* 10 ^ (-3); %radius of axicon lens = 25.4mm
alpha = 5 * pi / 180; %axicon lens base angle 5 degrees
R_exp = 1.0 * 10 ^(-3); %radius of curvature of the hyperbole
e0 = 6.1 * 10 ^ (-3); %thickness of axicon at apex
n = 1.45; %refractive index of the axicon lens and converging lens
wz = w0 * sqrt( 1 + z^2 / zR^2); %beam radius
%Practical case Axicon doublet FFT
%[U15,U19] = propTF2dim_DoublePracAxicon(E_GB,X,Y,X_prime5,Y_prime5,X_prime6,Y_prime6,k,n,r_axi,alpha,R_exp,e0,z4,z5);
[U15,U19] = propTF2dim_DoubleIdealAxicon(E_GB,X,Y,X_prime5,Y_prime5,X_prime6,Y_prime6,k,n,alpha,z4,z5);
I15= abs(U15.^2); %Intensity profile at the entrance of the second axicon
I19= abs(U19.^2); %Intensity profile at the OBSERV plane
%Manual Integration(prac)
surf_element = xprime6_period * xprime6_period;
FFT_power_manual = sum(I19 .* surf_element,'all');
I19 = I19 .* power/FFT_power_manual; %Rescaled intensity profile at OBSERV plane so the total power = GB power
Power_check = sum(I19 .* surf_element,'all'); %should = GB power
%%%%%Comparing with the geometric approximation -- Edmund optics%%%%%
ring_thick_geo = wz * sqrt(1 - n^2 * (sin(alpha))^2)/((cos(alpha)) * (n*(sin(alpha))^2 + (cos(alpha)) * sqrt(1 - n^2 * (sin(alpha))^2)));
ring_outer_radius_geo = z4 * (sin(alpha) * (n* cos(alpha) - sqrt(1 - n^2*(sin(alpha))^2)))/(n*(sin(alpha))^2 + cos(alpha) * sqrt(1 - n^2 * (sin(alpha))^2));
ring_inner_radius_geo = ring_outer_radius_geo - ring_thick_geo;
%Fresnel Intensity Before Second PRAC Axicon
figure(5);
pcolor(X_prime5 * 1000,Y_prime5 * 1000, I15);shading interp;colorbar;axis equal;
title('Fresnel Intensity Before Second PRAC Axicon, zbef = 0,z4 = 120mm,z5 = 0.2m,Rexp = 0.6mm');
xlabel('xprime5(mm)')
ylabel('yprime5(mm)')
zlabel('Amplitude')
%Fresnel Intensity at OBSERV plane
figure(6);
pcolor(X_prime6 * 1000,Y_prime6 * 1000, I19);shading interp;colorbar;axis equal;
drawellipse('Center',[X_prime6(2001,2001) * 1000, Y_prime6(2001,2001) * 1000],'SemiAxes',[ring_outer_radius_geo * 1000, ring_outer_radius_geo * 1000],'StripeColor','w','HandleVisibility',"off","InteractionsAllowed","none");
drawellipse('Center',[X_prime6(2001,2001) * 1000, Y_prime6(2001,2001) * 1000],'SemiAxes',[ring_inner_radius_geo * 1000, ring_inner_radius_geo * 1000],'StripeColor','w','HandleVisibility',"off","InteractionsAllowed","none");
title('Fresnel Intensity at OBSERV plane, zbef = 0, z4 = 120mm,z5 = 0.2m,Rexp = 1.0mm');
xlabel('xprime6(mm)')
ylabel('yprime6(mm)')
zlabel('Amplitude')
caxis([0 0.25])
%1D Fresnel Intensity bef Second Axicon
figure(10);
plot(x_prime6 * 1000,I15(:,2001),'LineWidth',1);
xline(ring_outer_radius_geo * 1000, '-r','LineWidth',1)
xline(ring_inner_radius_geo * 1000, '-g','LineWidth',1)
title('1D Fresnel Intensity bef Second Axicon, IDEAL Axicon, zbef = 0, z4 = 120mm,z5 = 0.2m');
xlabel('xprime6(mm)')
ylabel('Amplitude')
%1D Fresnel Intensity at OBSERV plane
figure(7);
plot(x_prime6 * 1000,I19(:,2001),'LineWidth',1);
xline(ring_outer_radius_geo * 1000, '-r','LineWidth',1)
xline(ring_inner_radius_geo * 1000, '-g','LineWidth',1)
title('1D Fresnel Intensity at OBSERV plane,IDEAL Axicon, zbef = 0, z4 = 120mm,z5 = 0.2m');
xlabel('xprime6(mm)')
ylabel('Amplitude')
%%%%%%plot experimental data with simulation data%%%%%%%
load('Iexpdata','xaxis2','Icrop1vert');
load('Iexpdata2','Icrop950');
load('Iideal.mat','Iideal');
figure(8);
xaxis = linspace(-4.60 * 10 ^ (-3),4.50 * 10 ^ (-3),2010); %Reshape of exp data because the exp data ring size cannot be accurately measured
plot(xaxis*1000, 0.25/0.4 * Icrop1vert,'m','LineWidth',1) %Intensity of the exp data reduced to match the simulated data
hold on
plot(x_prime6 * 1000,I19(:,2001),'b','LineWidth',1);
hold on
plot(x_prime6 * 1000,0.25/0.12 * Iideal,'k--','LineWidth',1);
xline(ring_outer_radius_geo * 1000, '-r','LineWidth',1)
xline(ring_inner_radius_geo * 1000, '-g','LineWidth',1)
legend('Exp Seed Input Vertical','Simulation 2nd axicon decenter = 0.03axicon,CORRECTED axicon, zbef = 0, z4 = 120mm,z5 = 0.2m, Rexp = 1.0mm','Simulation no decenter,IDEAL axicon, zbef = 0, z4 = 120mm,z5 = 0.2m,')
title('Comparison between Exp and Sim with central maxima');
xlabel('xprime6(mm)')
ylabel('Amplitude')
xlim([-9.00,9.00])
ylim([0, 0.28])
%%%%%Ring power Content%%%%%
I19_crop = I19;
I19_crop(sqrt(X_prime6.^2 + Y_prime6.^2) < 0.3*ring_outer_radius_geo) = 0;
%Cropped Intensity at OBSERV plane
figure(9);
pcolor(X_prime6 * 1000,Y_prime6 * 1000, I19_crop);shading interp;colorbar;axis equal;
drawellipse('Center',[X_prime6(2001,2001) * 1000, Y_prime6(2001,2001) * 1000],'SemiAxes',[ring_outer_radius_geo * 1000, ring_outer_radius_geo * 1000],'StripeColor','w','HandleVisibility',"off","InteractionsAllowed","none");
drawellipse('Center',[X_prime6(2001,2001) * 1000, Y_prime6(2001,2001) * 1000],'SemiAxes',[0.3*ring_outer_radius_geo * 1000, 0.3*ring_outer_radius_geo * 1000],'StripeColor','w','HandleVisibility',"off","InteractionsAllowed","none");
title('Fresnel Intensity at OBSERV plane, ring power = 99.9%,zbef = 0, z4 = 120mm,z5 = 0.2m,Rexp = 0.1mm');
xlabel('xprime6(mm)')
ylabel('yprime6(mm)')
zlabel('Amplitude')
%Obtain ring power%
surf_element = xprime6_period * xprime6_period;
FFT_power_manual_crop = sum(I19_crop .* surf_element,'all');
ring_power = FFT_power_manual_crop/Power_check;
function[E_GB, I_GB, power] = getGB(A,z,lambda,w0,zR, t, rho, Rz)
%%%return GB field, intensity & total input power, power will be compared
%%%with the rescaled output FFT beam
%A: GB constant
%z: z_bef location of the beam at first optics relative to the beam
%waist
%w0: beam waist
%zR: Rayleigh range
%t: time
%rho: polor coordinate
%Rz: beam curvature
c = 3 * 10 ^8;
k = 2*pi/lambda; %wavevector
angularfreq = 2 * pi * c / lambda;
wz = w0 * sqrt( 1 + z^2 / zR^2); %beam radius
phi = atan(z/zR);
E_GB = A./sqrt(1 + z.^2/zR.^2) .* exp(1i .* (k.*z - angularfreq.*t)) .* exp( - rho.^2./wz.^2) .* exp(1i .* (k .* rho.^2./(2.*Rz))) .* exp(-1i .* phi);
I_GB = abs(E_GB.^2);
Integration_intensity = @(rho,theta) rho.*abs((A/sqrt(1 + z^2/zR^2) .* exp(1i * (k*z - angularfreq*t)) .* exp( - rho.^2./wz^2) .* exp(1i * (k * rho.^2./(2*Rz))) .* exp(-1i * phi)).^2);
power = integral2(Integration_intensity,0,Inf,0,2*pi);
end
function[U15,U19]=propTF2dim_DoubleIdealAxicon(u1,X_prime4,Y_prime4,X_prime5,Y_prime5,X_prime6,Y_prime6,k,n,alpha,z4,z5)
% propagation - transfer function approach
% Ideal Axicon Pair
% using fresnel approximation
% assumes equal side lengths for x and y
% uniform sampling
% u1 - source plane field
% X_prime4 - input beam coordinate
% y_prime4 - input beam coordinate
% X_prime5 - beam coordinate at second axicon
% y_prime5 - beam coordinate at second axicon
% X_prime6 - beam coordinate at OBSERV plane
% y_prime6 - beam coordinate at OBSERV plane
% k - wavevector (monochromatic approximation)
% n - refractive index of optics
% r_axi - radius of axicon
% alpha - conical lens base angle
% R_exp - curvature of the pratical axicon
% e0 - plane axicon thickness
% z4 - separation between axicons
% z5 - position of OBSERV plane
%%%Transmission_through_axicon_first
H2 = exp(-1i*k*(n-1)*(sqrt(X_prime4.^2 + Y_prime4.^2)).* tan(alpha)); %worktransfer function conical lens on paper
U12 = u1.*H2;
%%%Prop_to_second_axicon
factor4 = exp(0.5.*1i .* k ./z4.*(X_prime4.^2 + Y_prime4.^2));
U13 = fftshift(U12.*factor4); %field after first axicon
U14 = fftshift(fft2(U13)); %FFT of U2
U15 = -1i.*k./(2 .* pi).*exp(1i .* k .* z4)./z4 * exp((1i.*k./(2.*z4)).*(X_prime5.^2 + Y_prime5.^2)).*U14;
%%%Transmission_through_axicon_second
H4 = exp(-1i*k*(n-1)*(sqrt((X_prime5).^2 + Y_prime5.^2)).* tan(alpha)); %worktransfer function conical lens on paper
U16 = U15.*H4;
%%%Prop_to_observ
factor5 = exp(0.5.*1i .* k ./z5.*(X_prime5.^2 + Y_prime5.^2));
U17=fftshift(U16.*factor5); %multiply U2: field after optics H
U18=fftshift(fft2(U17)); %inverse fft and center observ field
U19 = -1i.*k./(2 .* pi).*exp(1i .* k .* z5)./z5 .* exp((1i.*k./(2.*z5)).*(X_prime6.^2 + Y_prime6.^2)).*U18;
end
function[U15,U19]=propTF2dim_DoublePracAxicon(u1,X_prime4,Y_prime4,X_prime5,Y_prime5,X_prime6,Y_prime6,k,n,r_axi,alpha,R_exp,e0,z4,z5)
% propagation - transfer function approach
% Hyperbolic corrected Axicon Pair
% using fresnel approximation
% assumes equal side lengths for x and y
% uniform sampling
% u1 - source plane field
% X_prime4 - input beam coordinate
% y_prime4 - input beam coordinate
% X_prime5 - beam coordinate at second axicon
% y_prime5 - beam coordinate at second axicon
% X_prime6 - beam coordinate at OBSERV plane
% y_prime6 - beam coordinate at OBSERV plane
% k - wavevector (monochromatic approximation)
% n - refractive index of optics
% r_axi - radius of axicon
% alpha - conical lens base angle
% R_exp - curvature of the pratical axicon
% e0 - plane axicon thickness
% z4 - separation between axicons
% z5 - position of OBSERV plane
%%%Transmission_through_axicon_first
%decenter = 0.03 * r_axi;
%thickness_function1 = e0 - R_exp .* (tan(alpha)).^2 .* sqrt(1 + ((X_prime4+decenter).^2 + Y_prime4.^2)./(R_exp .* tan(alpha)).^2);
thickness_function1 = e0 - R_exp .* (tan(alpha)).^2 .* sqrt(1 + ((X_prime4).^2 + Y_prime4.^2)./(R_exp .* tan(alpha)).^2);
H3 = exp(1i.*k.*(n-1).*thickness_function1); %transfer function of a single axicon
U12 = u1.*H3;
%%%Prop_to_second_axicon
factor4 = exp(0.5.*1i .* k ./z4.*(X_prime4.^2 + Y_prime4.^2));
U13 = fftshift(U12.*factor4); %field after first axicon
U14 = fftshift(fft2(U13)); %FFT of U2
U15 = -1i.*k./(2 .* pi).*exp(1i .* k .* z4)./z4 * exp((1i.*k./(2.*z4)).*(X_prime5.^2 + Y_prime5.^2)).*U14;
%%%Transmission_through_axicon_second
decenter = 0.03 * r_axi;
%thickness_function2 = e0 - R_exp .* (tan(alpha)).^2 .* sqrt(1 + ((X_prime5 + decenter).^2 + Y_prime5.^2)./(R_exp .* tan(alpha)).^2);
thickness_function2 = e0 - R_exp .* (tan(alpha)).^2 .* sqrt(1 + ((X_prime5).^2 + Y_prime5.^2)./(R_exp .* tan(alpha)).^2);
H5 = exp(1i.*k.*(n-1).*thickness_function2); %transfer function of a single axicon
U16 = U15.*H5;
%%%Prop_to_observ
factor5 = exp(0.5.*1i .* k ./z5.*(X_prime5.^2 + Y_prime5.^2));
U17=fftshift(U16.*factor5); %multiply U2: field after optics H
U18=fftshift(fft2(U17)); %inverse fft and center observ field
U19 = -1i.*k./(2 .* pi).*exp(1i .* k .* z5)./z5 .* exp((1i.*k./(2.*z5)).*(X_prime6.^2 + Y_prime6.^2)).*U18;
end