set timestamp offset graph 1.11,0.7 rotate font 'Verdana,6' set label 'fixed_pitch/pitch.gnu' at graph 1.02,0.02 rotate left font 'Verdana,6' noenhance amu=0.00116592 R0=7.112 c_light = 2.99792458e2 #meters/microsecond gamma = 29.3 betaxy0 = (gamma**2-1)**.5/gamma omega_c = c_light*betaxy0/R0 betaz0 = 0.01 index = 0.108 omega_v = index**.5 * omega_c omega_a = omega_c * gamma * amu bomega0 = -betaz0*omega_a*gamma/(gamma+1) *betaxy0 bpi0 = -betaz0*omega_a*(1+1/amu/(gamma+1))*omega_v/omega_c *betaxy0 bpi0f(x) = -betaz0*omega_a*(1+1/amu/(gamma+1))*x/omega_c *betaxy0 omega0p = omega_a*(1+1/amu/gamma-gamma/(gamma+1)*betaz0**2/2) pi0p = omega_a*gamma/(gamma+1)*betaz0**2/2 omega_p = omega0p - omega_c X0 = -(bomega0 +bpi0)/2 Y0 = -(bomega0-bpi0)/2 X0f(x) = -(bomega0 + bpi0f(x))/2 Y0f(x) = -(bomega0 - bpi0f(x))/2 X0mm(x) = -0.5*betaz0*omega_a*(gamma/(gamma+1)+gamma*x/omega_c)*betaxy0 Y0mm(x) = -0.5*betaz0*omega_a*(gamma/(gamma+1)-gamma*x/omega_c)*betaxy0 X0mm2_m_Y0mm2(x) = betaz0**2*omega_a**2*gamma**2/(gamma+1)*x/omega_c*betaxy0**2 X0mm2_p_Y0mm2(x) = 0.5*betaz0**2*omega_a**2*(gamma**2/(gamma+1)**2+gamma**2*x**2/omega_c**2)*betaxy0**2 X02 = X0**2 Y02 = Y0**2 A1(x)=pi0p/2 * sin(2*omega_v *x)/2/omega_v B1_I(x) = 0.5*X0* 2*(sin((omega_v+omega_p)*x/2))**2/(omega_v+omega_p) - Y0*2*(sin((omega_v-omega_p)*x/2))**2/(omega_v-omega_p) B1_R(x) = 0.5*(X0* sin((omega_v+omega_p)*x)/(omega_v+omega_p)) +0.5* Y0*sin((omega_v-omega_p)*x)/(omega_v-omega_p) B1_R1(x) = 0.5*(X0* sin((omega_v+omega_p)*x)/(omega_v+omega_p)) B1_R2(x)= 0.5* Y0*sin((omega_v-omega_p)*x)/(omega_v-omega_p) B1_star_2_R(x)= B1_R(x)*B1_R(x) - B1_I(x)*B1_I(x) B1_star_2_I(x) = -2* B1_R(x)* B1_I(x) F2_R(x) = 0.25*(2*X0*Y0*( (cos(omega_p*x)*cos(omega_v*x) -1)/(omega_v**2-omega_p**2) + sin(omega_v *x)/omega_v *(omega_v*sin(omega_v*x))/(omega_v**2-omega_p**2) ) - X02*(cos((omega_v+omega_p)*x) -1)/(omega_v+omega_p)**2 -Y02*(cos((omega_v-omega_p)*x) -1)/(omega_v-omega_p)**2) F2_R1(x) = 0.25*(2*X0*Y0*((cos(omega_p*x))**2 -1)/(omega_v**2-omega_p**2)) F2_R2(x) = 0.25*(2*X0*Y0*(sin(omega_v *x)/omega_v *(omega_v*sin(omega_v*x))/(omega_v**2-omega_p**2))) F2_R3(x) = 0.25*(-X02*(cos((omega_v+omega_p)*x) -1)/(omega_v+omega_p)**2 -Y02*(cos((omega_v-omega_p)*x) -1)/(omega_v-omega_p)**2) F2_I1(x) = 0.25*x*(omega_v*(X02-Y02)-omega_p*(X02+Y02))/(omega_v**2-omega_p**2) F2_I2(x) = 0.25*(2*X0*Y0*(sin(omega_p*x)*cos(omega_v*x) /(omega_v**2-omega_p**2) - sin(omega_v *x)/omega_v *(omega_p*cos(omega_v*x))/(omega_v**2-omega_p**2))-X02*(sin((omega_v+omega_p)*x))/(omega_v+omega_p)**2 + Y02*(sin((omega_v-omega_p)*x))/(omega_v-omega_p)**2) F2_I(x) = F2_I1(x)+F2_I2(x) F2_I_star(x) = -F2_I(x) a2_b2(x) = -2*2*B1_I(x) asb_iot_R(x) = 0.5*(cos(omega_p*x) + (-2*A1(x) - B1_star_2_I(x)+ 2*F2_I_star(x))*sin(omega_p*x) ) + (B1_star_2_R(x) -2*F2_R(x))*cos(omega_p*x) asb_iot_R1(x) = 0.5*(cos(omega_p*x)) asb_iot_R2(x) = 0.5*(-2*A1(x))*sin(omega_p*x) asb_iot_R3(x) = 0.5*(cos(omega_p*x) + ( - B1_star_2_I(x) )*sin(omega_p*x) ) + (B1_star_2_R(x) -2*F2_R(x))*cos(omega_p*x) asb_iot_I(x) = 0.5*(sin(omega_p*x) + (B1_star_2_R(x) - 2*F2_R(x))*sin(omega_p*x) ) + (2*A1(x)+ B1_star_2_I(x) - 2*F2_I_star(x))*cos(omega_p*x) lambda(x)= 0.5*(x*(X0f(x)**2-Y0f(x)**2)-omega_p*(X0f(x)**2+Y0f(x)**2))/(x**2-omega_p**2) lambdamm(x) = 0.5*(x*(X0mm(x)**2-Y0mm(x)**2)-omega_p*(X0mm(x)**2+Y0mm(x)**2))/(x**2-omega_p**2) lambdamm2(x) = 0.5*(x*X0mm2_m_Y0mm2(x)-omega_p*X0mm2_p_Y0mm2(x))/(x**2-omega_p**2) lambdaomegavmax=0.5*(betaz0*omega_a*betaxy0)**2*(2*gamma**2/(gamma+1)/omega_c - omega_a*gamma**2/omega_c**2) sx_0(x) = cos((omega_p+omega_c+lambda)*x) sx_1(x) = pi0p * sin(omega_v*x)/omega_v * sin((omega_p+omega_c +lambda)*x) sx_2(x) = X02*(sin((omega_v +omega_p)*x/2)/(omega_v+omega_p))**2 * cos((-omega_v+omega_c+lambda)*x) sx_3(x) = Y02*(sin((omega_v -omega_p)*x/2)/(omega_v-omega_p))**2 * cos((omega_v+omega_c+lambda)*x) sx_4(x) = -4*X0*Y0*(cos(omega_p*x)-cos(omega_v*x))/(omega_v**2-omega_p**2)*cos((omega_c+lambda)*x) Sx(x) = sx_0(x)+sx_1(x) + sx_2(x) + sx_3(x) + sx_4(x) sy_0(x) = sin((omega_p+omega_c+2*lambda)*x) sy_1(x) = -2*pi0p * sin(omega_v*x)/omega_v * cos((omega_p+omega_c +2*lambda)*x) sy_2(x) = 4*ABS_X02*(sin((omega_v +omega_p)*x/2)/(omega_v+omega_p))**2 * sin((-omega_v+omega_c+2*lambda)*x) sy_3(x) = 4*ABS_Y02*(sin((omega_v -omega_p)*x/2)/(omega_v-omega_p))**2 * sin((omega_v+omega_c+2*lambda)*x) sy_4(x) = -4*X0Y0*(cos(omega_p*x)-cos(omega_v*x))/(omega_v**2-omega_p**2)*sin((omega_c+2*lambda)*x) Sy(x) = sy_0(x) + sy_1(x) + sy_2(x) + sy_3(x) + sy_4(x) Sz(x) = -(2*IMX0*sin((omega_v+omega_p)*x)/(omega_v+omega_p) + IMY0*sin((omega_v-omega_p)*x)/(omega_v-omega_p)) betaz(x) = -betaz0*sin(omega_v*x) betax(x) = -betaxy0*cos(betaz0*sin(omega_v*x))*cos(omega_c*x) betay(x) = -betaxy0*cos(betaz0*sin(omega_v*x))*sin(omega_c*x) betadots(x) = betax(x)*Sx(x) + betay(x)*Sy(x) + betaz(x)* Sz(x) bdsz(x)=-2*(IMX0*sin((omega_v+omega_p)*x)/(omega_v+omega_p)**2 +IMY0*sin((omega_v-omega_p)*x)/(omega_v-omega_p)**2)*betaz0*sin(omega_v*x) bdsxy1(x) = cos((omega_p+2*lambda)*x) bdsxy2(x) = 2*pi0p*sin(omega_v*x)/omega_v * sin((omega_p+2*lambda)*x) bdsxy3(x) = 4*ABS_X02*cos((omega_v-2*lambda)*x) * (sin((omega_v+omega_p)/2*x)/(omega_v+omega_p))**2 bdsxy4(x) = 4*ABS_Y02*cos((omega_v+2*lambda)*x) * (sin((omega_v-omega_p)/2*x)/(omega_v-omega_p))**2 bdsxy5(x) = 4*IMX0*IMY0*cos(2*lambda*x)*(cos(omega_p *x)-cos(omega_v*x))/(omega_v**2-omega_p**2) bds(x) = bdsz(x)-(bdsxy1(x) +bdsxy2(x)+bdsxy3(x)+bdsxy4(x)+bdsxy5(x))*betaxy0*cos(betaz0*sin(omega_v*x)) lambdaov(x) = -(betaz0*omega_a*betaxy0*(1/(gamma+1)))**2*((gamma*(gamma+1)+gamma/amu)*(x*omega_c)**2/omega_c-0.5*(gamma**2+(x*omega_c)**2/omega_c**2*((gamma+1)+1/amu)**2)*omega_p)/((x*omega_c)**2-omega_p**2) bds_true(x) = a2_b2(x)*betaz0*sin(omega_v*x) + 2*asb_iot_R(x)*betaxy0*cos(betaz0*sin(omega_v *x)) #bds_true(x) = 2*asb_iot_R(x)*betaxy0*cos(betaz0*sin(omega_v *x)) set samples 10000 set xrange [0:2.2] set xtics 0,0.2 set yrange [-25.4:-24.1] set ytics -25.4,.1 set xlabel 'f_v [MHz]' set ylabel 'C_p [ppm]' plot (lambda(2*pi*x)-pi0p)/omega_a*1.e6 set term x11 1 set yrange [-26:-24] set ytics -26,0.2 op = 1.43934193108689 plot 'omega_vs_volts.dat' u ($5/2/pi):(($2-op)/op)*1.e6 w p pt 5 t 'numerical', (lambda(2*pi*x)-pi0p)/omega_a*1.e6 t 'analytic' set term pdf fontscale 0.5 size 6in,4in set output 'pitch_numerical_analytic.pdf' plot 'omega_vs_volts.dat' u ($5/2/pi):(($2-op)/op)*1.e6 w p pt 7 ps 0.5 t 'numerical', (lambda(2*pi*x)-pi0p)/omega_a*1.e6 t 'analytic' set term x11