Accordion Lattice simple fit
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GaussianBeams_simple_rayleigh.m
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243
GaussianBeams_simple_rayleigh.m
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%% Elementary constants
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clear;
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load qf
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epsilon=8.854187817e-12;
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c = 299792458; %Speed of light [m/s]
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h = 6.626070040e-34; %Planck constant [J/Hz]
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hbar = h/2/pi;
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u0 = 4*pi*1e-7; %Vacuum permeability [N/A^2]
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e0 = 1/u0/(c^2); %Vacuum permittivity [A^2*s^4/kg/m^3]
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qe = 1.6021766208e-19; %Elementary charge [C]
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G = 6.67408e-11; %Gravitational constant [m^3/kg/s^2]
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kB = 1.38064852e-23; %Boltzmann constant [J/K]
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me = 9.10938356e-31; %Electron rest mass [kg]
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mp = 1.672621898e-27; %Proton mass [kg]
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uB = qe*hbar /2/me; %Bohr magneton [J/T]
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uN = qe*hbar /2/mp; %Nuclear magnton [J/T]
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alpha = u0*qe^2*c/(2*h); %Fine structure constant
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u = 1.660539040e-27; %Atomic mass unit [kg]
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a0 = 4*pi*e0*hbar^2/me/qe^2; %Bohr radius [m]
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Eh = hbar^2/(me*a0^2); %Rydberg energy [eV]
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%% Laser Properties
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wavelength=532e-9; %Laser wavlength [m]
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k=2*pi/wavelength; %Wave vector [m^-1]
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P=10; %Power of each beam [W]
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wz0=60e-6; %Beam waist in z-theta-direction[m]
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wy0=250e-6;
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zRy=pi*wy0^2./wavelength; %Rayleigh wavelength
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zRz=pi*wz0^2./wavelength;
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theta=0.5/180*pi; %Half angle of interference
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x_ext =wz0./tan(theta);
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%Dysprosium Properties
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polar532=350*qe^2*a0^2/Eh; %Dynamical Polarizability of Dy at 532nm
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polar1064=193*qe^2*a0^2/Eh;
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mDy = 164*u; %Dysprosium mass [kg]
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%% Trapping frequency as Plane wave
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vz=sqrt(P.*polar532*k^2*sin(theta)^2/(pi^3*wz0*wy0*c*mDy*epsilon));
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d = wavelength/(2*tan(theta)); %Fringe distance
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focus = 100e-3;
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D = wavelength*focus/d;
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%% Gaussian beams
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%Coordinate System
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n=[2001,3,2001];
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xmax=(x_ext.*3/2);
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xmin=-(x_ext).*3/2;
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ymax=2.*wy0;
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ymin=-2.*wy0;
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zmax=2.*wz0;
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zmin=-2.*wz0;
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for i=1:3
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x=linspace(xmin,xmax,n(1));
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y=linspace(ymin,ymax,n(2));
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z=linspace(zmin,zmax,n(3));
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[X,Y,Z]=meshgrid(x,y,z);
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%Rotation of the coordinte system
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xTheta=X.*cos(theta)+Z.*sin(theta);
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xPhi=X.*cos(theta)-Z.*sin(theta);
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zTheta=-X.*sin(theta)+Z.*cos(theta);
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zPhi=X.*sin(theta)+Z.*cos(theta);
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%Electric field
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E0=sqrt(2*P/pi/(wz0*wy0)/c/epsilon); %Electric field at 0
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wz1=wz0*sqrt(1+(xTheta./zRz).^2); %Beam waist in z-direction for Beam 1
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wy1=wy0*sqrt(1+(xTheta./zRy).^2); %Beam waist in y-direction for Beam 1
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wz2=wz0*sqrt(1+(xPhi./zRz).^2); %Beam waist in z-direction for Beam 2
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wy2=wy0*sqrt(1+(xPhi./zRy).^2); %Beam waist in y-direction for Beam 1
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R1y=xTheta.*(1+(zRy./xTheta).^2); %Rayleigh length Beam 1
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R1z=xTheta.*(1+(zRz./xTheta).^2);
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R2y=xPhi.*(1+(zRy./xPhi).^2); %Rayleigh length Beam 2
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R2z=xPhi.*(1+(zRz./xPhi).^2);
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R1y(:,floor(n(1)/2)+1,floor(n(3)/2)+1)=Inf; %Rayleigh length ->0
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R1z(:,floor(n(1)/2)+1,floor(n(3)/2)+1)=Inf;
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R2y(:,floor(n(1)/2)+1,floor(n(3)/2)+1)=Inf;
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R2z(:,floor(n(1)/2)+1,floor(n(3)/2)+1)=Inf;
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gouyPhase1=atan(wavelength.*xTheta./(pi.*wz0.*wy0));
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gouyPhase2=atan(wavelength.*xPhi./(pi.*wz0.*wy0));
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E1=E0.*sqrt(wz0.*wy0)./sqrt(wz1.*wy1).*exp(-(zTheta./wz1).^2-(Y./wy1).^2).*exp(1i*(k.*xTheta-gouyPhase1)+1i.*k.*(zTheta.^2./2./R1z+Y.^2./2./R1y)); %Electric field Beam 1
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E2=E0.*sqrt(wz0.*wy0)./sqrt(wz2.*wy2).*exp(-(zPhi./wz2).^2-(Y./wy2).^2).*exp(1i*(k.*xPhi-gouyPhase2)+1i.*k.*(zPhi.^2./2./R2z+Y.^2./2./R2y)); %Electric field Beam 2
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%Intensity and trapping Potential
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Itot=abs(E1+E2).^2./2*c*epsilon;
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if i==1 %Set y=0
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Itot_y=squeeze(Itot(2,:,:));
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Imax=Itot_y(1001,1001);
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figure
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imagesc(x,z, Itot_y')
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title('x-z Interference Pattern for \theta=', theta*180/pi);
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colormap(QF);
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caxis([0 Imax]);
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%z-profile for x=0
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Udip=0.5.*Itot_y.*polar532./epsilon./c;
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Umid=squeeze(Udip(floor(n(1)/2)+1,:));
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zrange = find((-pi/(3*k*sin(theta))) < z & z < (pi./(3*k*sin(theta))));
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zfit = z(zrange(1):zrange(length(zrange)));
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Ufit = Umid(zrange(1):zrange(length(zrange)));
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zfit = zfit(:);
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Ufit = Ufit(:);
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[pks, locs]=findpeaks(Umid);
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p = polyfitn(zfit,Ufit,'constant x^2');
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% figure
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% plot(z,Umid, zfit, Ufit,'.', zfit, polyvaln(p,zfit))
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% title('Trapping potential in z-direction for x=y=0')
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% %Fitted Trapping frequency
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% nu_zfixed=sqrt(-p.Coefficients(2)*2/mDy)/2/pi;
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%FWHM of x-profile (actually 1/e^2)
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Uz0=squeeze(Udip(:,floor(n(1)/2)+1));
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Ufwhm= find(Uz0>Uz0(floor(n(1)/2)+1)/exp(2));
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x_ext_fwhm=abs(x(Ufwhm(1))-x(Ufwhm(length(Ufwhm))));
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x_range=x(Ufwhm);
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%x-profile for z=0
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xrange=find((-x_ext_fwhm/10)<x & x<(x_ext_fwhm/10));
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xfit=x(xrange(1):xrange(length(xrange)));
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Ufitx = Uz0(xrange(1):xrange(length(xrange)));
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px= polyfitn(xfit,Ufitx, 'constant x^2');
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% figure
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% plot(x,Uz0,xfit, Ufitx, '.',xfit, polyvaln(px,xfit) )
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% xline(x_ext_fwhm/2)
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% xline(-x_ext_fwhm/2)
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% title('x-Potential')
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% nu_xfixed=sqrt(-px.Coefficients(2)*2/mDy)/2/pi;
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%
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%z-profile x=x_ext
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Ux_ext=squeeze(Udip(Ufwhm(1),:));
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Ux_ext_fit=Ux_ext(zrange(1):zrange(length(zrange)));
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p_xext = polyfitn(zfit,Ux_ext_fit,'constant x^2');
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% figure
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% plot(z,Ux_ext, zfit, Ux_ext_fit,'.', zfit, polyvaln(p_xext,zfit))
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% %Fitted Trapping Frequency
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% nu_zxext=sqrt(-p_xext.Coefficients(2)*2/mDy)/2/pi;
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else
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if i==2
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Itot_z=squeeze(Itot(:,:,2));
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%
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% figure
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% imagesc(x,y,Itot_z)
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% title('x-y Interference Pattern for \theta=', theta*180/pi);
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% colormap(QF);
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% %caxis([min(min(Itot_z)) max(max(Itot_z))])
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%Gaussian profile y
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Udip_y=0.5.*Itot_z.*polar532./epsilon./c;
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Umid_y=squeeze(Udip_y(:,floor(n(1)/2)+1));
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Ufwhm_y= find(Umid_y>Umid_y(floor(n(1)/2)+1)/exp(2));
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y_ext_fwhm=abs(y(Ufwhm_y(1))-y(Ufwhm_y(length(Ufwhm_y))));
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y_range=y(Ufwhm_y);
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yrange = find( -y_ext_fwhm/20 < y & y < y_ext_fwhm/20);
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yfit = y(yrange(1):yrange(length(yrange)));
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Ufit_y = Umid_y(yrange(1):yrange(length(yrange)));
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yfit = yfit(:);
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Ufit_y = Ufit_y(:);
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p_ext_y = polyfitn(yfit,Ufit_y, 'constant x^2');
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% figure
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% plot(y,Umid_y, yfit, Ufit_y,'.', yfit, polyvaln(p_ext_y,yfit))
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% title('y profile for \theta=', theta*180/pi);
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nu_y=sqrt(-p_ext_y.Coefficients(2)*2/mDy)/2/pi;
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%Gaussian profile x
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Udip_x=0.5.*Itot_z.*polar532./epsilon./c;
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Umid_x=squeeze(Udip_x(floor(n(1)/2)+1,:));
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Ufwhm_x= find(Umid_x>Umid_x(floor(n(1)/2)+1)/exp(2));
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x_ext_fwhm=abs(x(Ufwhm_x(1))-x(Ufwhm_x(length(Ufwhm_x))));
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x_range=x(Ufwhm_x);
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xrange = find( -x_ext_fwhm/20 < x & x < x_ext_fwhm/20);
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xfit = x(xrange(1):xrange(length(xrange)));
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Ufit_x = Umid_x(xrange(1):xrange(length(xrange)));
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xfit = xfit(:);
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Ufit_x = Ufit_x(:);
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p_ext_x = polyfitn(xfit,Ufit_x,'constant x^2');
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% figure
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% plot(x,Umid_x, xfit, Ufit_x,'.', xfit, polyvaln(p_ext_x,xfit))
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% title('x profile for \theta=', theta*180/pi);
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nu_x=sqrt(-p_ext_x.Coefficients(2)*2/mDy)/2/pi;
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else
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Itot_x=squeeze(Itot(:,2,:));
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% figure
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% imagesc(y,z,Itot_x')
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% title('y-z Interference Pattern for \theta=', theta*180/pi);
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% colormap(QF);
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% %caxis([min(min(Itot_x)) max(max(Itot_x))])
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end
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end
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tempn=n(3);
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n(3)=n(2);
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n(2)=n(1);
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n(1)=tempn;
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end
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Trap_f= table(theta*180/pi,P, wavelength, vz, nu_zfixed,nu_zxext, nu_xfixed,nu_x, nu_y,'VariableNames',{'Theta', 'Laser Power','Laser Wavelength', 'vz Plane wave','vz Gaussian Beam', 'vz Gaussian Beam at x_ext','vx in x-z-Plane','vx in x-y-Plane', 'vy in x-y-Plane'});
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%% Peaks for shallow angle
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npeaks=length(pks);
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smax=(npeaks-1)/2;
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fringeratios=zeros(smax,3);
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for s=1:1:smax+1
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ratio=pks((npeaks+1)/2+s-1)/pks((npeaks+1)/2);
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fringeratios(s,1)=s-1;
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fringeratios(s,2)=pks((npeaks+1)/2+s-1);
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fringeratios(s,3)=ratio;
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end
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