Split up files
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%all distances in m
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clear
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close all
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clc
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format compact
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% These calculations are based on Ashkin's article "Forces of a single-beam
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% gradient laser trap on a dielectric sphere in the ray optics regime".
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% There are axial forces only
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n1 = 1.33; % index of refraction of the immersion medium
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n2 = 1.6; % index of refraction of the fused silica at wavelength 523 nm
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n = n2/n1; % n2/n1
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c0 = 3e8; % speed of light
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NA = 1.25; % numerical aperture
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th_max = asin(NA/n1); % maximum angle of incidence
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f = 100.0e-3; % objective lens focus or WD
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r_max = f*tan(th_max); % radius of a Gaussian beam (1:1 with input aperture condition)
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Rsp = 1.0e-6; % sphere radius
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P = 20.0e-3; % power of the laser
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thr = @(th) asin(n1/n2*sin(th)); % refraction angle
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%reflectivity
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R = @(th,psi) (tan(th-thr(th)).^2./tan(th+thr(th)).^2).*cos(psi).^2+...
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(sin(th-thr(th)).^2./sin(th+thr(th)).^2).*sin(psi).^2;
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%transparency
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T = @(th,psi) 1-R(th,psi);
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% Factors
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Qs = @(th, psi) 1 + R(th, psi) .* cos(2*th) - T(th,psi).^2 .* (cos(2*th -...
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2*thr(th)) + R(th, psi) .* cos(2*th)) ./ (1 + R(th,psi).^2 +...
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2*R(th,psi) .* cos(2*thr(th)));
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Qg = @(th, psi) R(th, psi) .* sin(2*th) - T(th,psi).^2 .* (sin(2*th -...
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2*thr(th)) + R(th, psi) .* sin(2*th)) ./ (1 + R(th,psi).^2 +...
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2*R(th,psi) .* cos(2*thr(th)));
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Qmag = @(th, psi) sqrt(Qs(th, psi).^2 + Qg(th, psi).^2);
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% Average factors (circular polarization
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Qs_avg = @(th) 0.5*(Qs(th, 0) + Qs(th, pi/2));
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Qg_avg = @(th) 0.5*(Qg(th, 0) + Qg(th, pi/2));
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Qmag_avg = @(th) sqrt(Qs_avg(th).^2 + Qg_avg(th).^2);
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% Angles
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phi = @(r) atan(r/f);
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thi = @(r,z) asin(z/Rsp.*sin(phi(r)));
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Qgz = @(r,z) -Qg_avg(thi(r,z)).*sin(phi(r));
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Qsz = @(r,z) Qs_avg(thi(r,z)).*cos(phi(r));
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% Intensity profile
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a = 1.0;
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w0 = a*r_max;
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%I = @(r) P/(pi*r_max^2); % uniform distribution
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A = (1-exp(-2*r_max.^2/w0^2));
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I0 = P*2/(pi*w0^2)/A;
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I = @(r) I0*exp(-2*r.^2/w0^2); % Gaussian TEM00 beam
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%A = 2*pi*integral(@(r) r.*besselj(0,2.405/w0*r).^2,0,r_max)/P0;
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%w0_bb = 0.5*r_max;
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%I0 = P*2/(pi*w0^2);
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%I = @(r) I0*besselj(0,2.405/w0_bb*r).^2.*exp(-2*r.^2/w0^2); % Bessel beam
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% Intensity profile graphics
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rho = linspace(-r_max, r_max, 500);
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figure
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plot(rho, I(rho)/max(I(rho)),'k')
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grid
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xlabel('r, ì')
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ylabel('I(r)')
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sdf('my')
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% Integration
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Qres_g = @(z) 1/(pi*r_max^2)*2/(A*pi*w0^2)*integral2(@(beta,r) r.*I(r).*...
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iscomplex(Qgz(r,z)),0,2*pi,0,r_max,...
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'Method','iterated','AbsTol',1e-12,'RelTol',1e-6);
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Qres_s = @(z) 1/(pi*r_max^2)*2/(A*pi*w0^2)*integral2(@(beta,r) r.*I(r).*...
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iscomplex(Qsz(r,z)),0,2*pi,0,r_max,...
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'Method','iterated','AbsTol',1e-12,'RelTol',1e-6);
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% Calulation
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N = 200;
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z = linspace(-2*Rsp,2*Rsp,N);
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Axial_g = zeros(1,N);
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Axial_s = zeros(1,N);
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for ii = 1:N
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Axial_g(ii) = Qres_g(z(ii));
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Axial_s(ii) = Qres_s(z(ii));
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end
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F0 = n1*P/c0; % net force
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Axial_g = fliplr(Axial_g);
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Axial_s = fliplr(Axial_s);
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Axial = Axial_g + Axial_s;
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z = -fliplr(z);
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%Graphics
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figure
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plot(z,F0*Axial_g,'b-.',z,F0*Axial_s,'r--',z,F0*Axial,'k')
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legend('F_{g}','F_{s}','F_{t}')
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xlabel('r, ì')
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ylabel('F, Í')
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grid
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sdf('my')
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21
Matlab/angles.m
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21
Matlab/angles.m
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load_constants
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function angle = th_r(theta)
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angle = asin(n1 / n2 * sin(theta));
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end
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function angle = ph_i(r)
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angle = atan(r / f);
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end
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function angle = th_i(r, z)
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angle = asin(z / Rsp .* sin(ph_i(r)));
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end
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function angle = gamma(beta, r)
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angle = acos(cos(pi/2 - ph_i(r)) .* cos(beta));
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end
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function angle = theta(beta, r, y)
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angle = asin(y / Rsp .* sin(gamma(beta, r)));
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end
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65
Matlab/axial.m
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65
Matlab/axial.m
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clear
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close all
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clc
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format compact
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% These calculations are based on Ashkin's article "Forces of a single-beam
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% gradient laser trap on a dielectric sphere in the ray optics regime".
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% There are axial forces only
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load_constants
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import factors.*
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% Factors plots
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theta = linspace(0, pi/2, 500);
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figure
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plot(theta, Qs(theta, pi/4), theta, -Qg(theta, pi/4), theta, Qmag(theta, pi/4))
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grid
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xlabel('$\theta$, $^{\circ}$', 'Interpreter', 'latex')
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ylabel('Q')
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% Intensity profile plots
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rho = linspace(-r_max, r_max, 500);
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figure
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plot(rho, I(rho)/max(I(rho)),'k')
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grid
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xlabel('r, м')
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ylabel('I(r)')
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% Integration
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G0 = 2/(A*pi^2*r_max^2*w0^2);
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Qres_g = @(z) G0 * integral2(@(beta,r) r.*I(r).*...
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iscomplex(Qgz(r,z)),0,2*pi,0,r_max,...
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'Method','iterated','AbsTol',1e-6,'RelTol',1e-6);
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Qres_s = @(z) G0 * integral2(@(beta,r) r.*I(r).*...
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iscomplex(Qsz(r,z)),0,2*pi,0,r_max,...
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'Method','iterated','AbsTol',1e-6,'RelTol',1e-6);
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% Calulation
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N = 200;
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z = linspace(-2*Rsp,2*Rsp,N);
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Axial_g = zeros(1,N);
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Axial_s = zeros(1,N);
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wb = waitbar(0, 'Calculating...');
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for ii = 1:N
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Axial_g(ii) = Qres_g(z(ii));
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Axial_s(ii) = Qres_s(z(ii));
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waitbar(ii / N, wb, 'Calculating...');
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end
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close(wb);
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Axial_g = fliplr(Axial_g);
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Axial_s = fliplr(Axial_s);
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Axial = Axial_g + Axial_s;
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z = -fliplr(z);
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% Plots
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figure
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plot(z,F0*Axial_g,'b-.',z,F0*Axial_s,'r--',z,F0*Axial,'k')
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legend('F_{g}','F_{s}','F_{t}')
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xlabel('r, м')
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ylabel('F, Н')
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grid
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9
Matlab/coefficients.m
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9
Matlab/coefficients.m
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function reflectivity = R(th, psi)
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reflectivity = (tan(th - th_r(th)).^2 ./ tan(th + th_r(th)).^2) .* ...
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cos(psi).^2 + ...
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(sin(th - th_r(th)).^2 ./ sin(th + th_r(th)).^2) .* sin(psi).^2;
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end
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function transparency = T(th, psi)
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transparency = 1 - R(th, psi);
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end
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37
Matlab/factors.m
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37
Matlab/factors.m
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% Factors
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function factor = Qs(th, psi)
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factor = 1 + R(th, psi) .* cos(2*th) - T(th, psi).^2.*...
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(cos(2*th - 2*th_r(th)) + R(th, psi) .* cos(2*th)) ./ ...
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(1 + R(th, psi).^2 + 2*R(th, psi) .* cos(2*th_r(th)));
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end
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function factor = Qg(th, psi)
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factor = R(th, psi) .* sin(2*th) - T(th, psi).^2 .* ...
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(sin(2*th - 2*th_r(th)) + R(th, psi) .* sin(2*th)) ./ ...
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(1 + R(th, psi).^2 + 2*R(th, psi) .* cos(2*th_r(th)));
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end
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function factor = Qmag(th, psi)
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factor = sqrt(Qs(th, psi).^2 + Qg(th, psi).^2);
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end
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% Average factors
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function factor = Qs_avg(th)
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factor = 0.5*(Qs(th, 0) + Qs(th, pi/2));
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end
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function factor = Qg_avg(th)
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factor = 0.5*(Qg(th, 0) + Qg(th, pi/2));
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end
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function factor = Qmag_avg(th)
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factor = sqrt(Qs_avg(th).^2 + Qg_avg(th).^2);
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end
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function factor = Qgz(r, z)
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factor = -Qg_avg(th_i(r, z)) .* sin(ph_i(r));
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end
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function factor = Qsz(r, z)
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factor = Qs_avg(th_i(r, z)) .* cos(ph_i(r));
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end
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7
Matlab/intensity.m
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7
Matlab/intensity.m
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@@ -0,0 +1,7 @@
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load_constants
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function g = gauss(r)
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A = (1-exp(-2*r_max.^2 / w0^2));
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I0 = 2*P / (pi * w0^2 * A);
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g = I0 * exp(-2 * r.^2 / w0^2); % Gaussian TEM00 beam
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end
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@@ -1,11 +1,3 @@
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function S = iscomplex(A)
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[n,m] = size(A);
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S = zeros(n,m);
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for ii = 1:n
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for jj = 1:m
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S(ii,jj) = A(ii,jj)*double(isreal(A(ii,jj)));
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end
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end
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S = A .* double(imag(A) == 0);
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end
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13
Matlab/load_constants.m
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13
Matlab/load_constants.m
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Rsp = 1.0e-6; % sphere radius [m]
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n1 = 1.33; % index of refraction of the immersion medium
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n2 = 1.6; % index of refraction of the fused silica at wavelength 523 nm
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n = n2/n1; % relative refraction index
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c0 = 3e8; % speed of light [m/s]
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NA = 1.25; % numerical apertur e
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f = 100.0e-3; % objective lens focus or WD [m]
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P = 20.0e-3; % power of the laser [W]
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F0 = n1*P/c0; % resulting force [N]
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th_max = asin(NA/n1); % maximum incidence angle
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r_max = f * tan(th_max); % radius of a Gaussian beam (1:1 with input aperture condition)
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aperture = 1.0;
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w0 = aperture * r_max; % Gaussian beam waist radius [m]
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