def gaussian_beam(fwhm=10):
"""Display a Gaussian beam at the given fwhm.
Parameters
----------
fwhm : float
Full width at half maximum (for intensity not amplitude) (µm).
"""
w = fwhm / np.sqrt(2 * np.log(2))
bpm = Bpm(1, 1, 1, 1, 1, 1, 1)
bpm.x = np.linspace(-1.5*fwhm, 1.5*fwhm, 500)
x = bpm.x
plt.figure("Beam")
plt.title("Width definition of a gaussian beam with FWHM=%s µm" % fwhm)
plt.xlabel("x (µm)")
plt.plot(x, bpm.gauss_light(fwhm, 0), label='field')
plt.plot(x, (bpm.gauss_light(fwhm, 0))**2, label='intensity')
plt.plot(x, [1/2]*x.size, '-.', label='1/2')
plt.plot([fwhm/2]*x.size, np.linspace(0, 1, x.size),
'--', label='fwhm/2')
plt.plot(x, [np.exp(-1)]*x.size, '-.', label='1/e')
plt.plot(x, [np.exp(-2)]*x.size, '-.', label='1/$e^2$')
plt.plot([w]*x.size, np.linspace(0, 1, x.size),
'--', label='$w_0$=%.2f' % w)
plt.legend()
plt.show()
def test_kerr():
"""More test than example. Show different approximations for the BPM
implementation of the Kerr effect."""
no = 1
lo = 1.5
length_z = 50
dist_z = 0.1
nbr_z_disp = 1
dist_x = 0.01
length_x = 400
bpm = Bpm(no, lo,
length_z, dist_z, nbr_z_disp,
length_x, dist_x)
[length_z, nbr_z, nbr_z_disp, length_x, nbr_x, x] = bpm.create_x_z()
dn = np.zeros((nbr_z, nbr_x))
fwhm = 6
field = bpm.gauss_light(fwhm)
irrad = 260e13 # if too high, see big difference between method
theta_ext = 0
[progress_pow] = bpm.init_field(field, theta_ext, irrad)
# Need to overwrite those variables due to changes
theta = asin(sin(radians(theta_ext)) / no) # angle in the guide
nu_max = 1 / (2 * dist_x) # max frequency due to sampling
# Spacial frequencies over x (1/µm)
nu = np.linspace(-nu_max,
nu_max*(1 - 2/nbr_x),
nbr_x)
intermed = no * cos(theta) / lo
fr = -2 * pi * nu**2 / (intermed + np.sqrt(
intermed**2
- nu**2
+ 0j))
bpm.phase_mat = fftshift(np.exp(1j * dist_z * fr))
bpm.phase_mat_demi = fftshift(np.exp(1j * dist_z / 2 * fr))
bpm.nl_mat = bpm.ko * bpm.dist_z * dn
# End overwrite
kerr_loop = 10
chi3 = 10 * 1E-20
nbr_step = nbr_z-1
print("\n dz/2+lens+dz/2")
field = bpm.field
for i in range(nbr_step):
# Linear propagation over dz/2
field = ifft(bpm.phase_mat_demi * fft(field))
# Influence of the index modulation on the field
field = field * np.exp(1j * bpm.nl_mat[nbr_step, :]) # No changes if
# no initial guide (exp=1)
# Linear propagation over dz/2
field = ifft(bpm.phase_mat_demi * fft(field))
cur_pow = bpm.epnc * abs2(field)
field_ref = field
cur_ref = cur_pow
print("\n dz+kerr")
field = bpm.field
for i in range(nbr_step):
# Linear propagation over dz
field = ifft(bpm.phase_mat * fft(field))
# Influence of the index modulation on the field
field_x = field * np.exp(1j * bpm.nl_mat[nbr_step, :])
cur_pow = bpm.epnc * abs2(field_x)
for k in range(1): # insensitive to correction loop
prev_pow = cur_pow
# influence of the beam intensity on the index modulation
dn_temp = dn[nbr_step, :] + (3*chi3)/(8*no)*(prev_pow/bpm.epnc)
nl_mat = bpm.ko * bpm.dist_z * dn_temp
# influence of the index modulation on the field
field_x = field * np.exp(1j*nl_mat) # No changes for the pow
cur_pow = bpm.epnc * abs2(field_x)
try:
bpm.variance(prev_pow, cur_pow)
except ValueError as error:
print(error)
field = field_x
field_1 = field
cur_1 = cur_pow
dn_1 = dn_temp
print("\n dz/2+kerr+dz/2")
field = bpm.field
for i in range(nbr_step):
# Linear propagation over dz/2
field = ifft(bpm.phase_mat_demi * fft(field))
# Influence of the index modulation on the field
field_x = field * np.exp(1j * bpm.nl_mat[nbr_step, :])
# Linear propagation over dz/2
field_x = ifft(bpm.phase_mat_demi * fft(field_x))
cur_pow = bpm.epnc * abs2(field_x)
for k in range(kerr_loop):
prev_pow = cur_pow
# influence of the beam intensity on the index modulation
dn_temp = dn[nbr_step, :] + (3*chi3)/(8*no)*(prev_pow/bpm.epnc)
nl_mat = bpm.ko * bpm.dist_z * dn_temp
# influence of the index modulation on the field
field_x = field * np.exp(1j * nl_mat)
# Linear propagation over dz/2
field_x = ifft(bpm.phase_mat_demi * fft(field_x))
cur_pow = bpm.epnc * abs2(field_x)
try:
bpm.variance(prev_pow, cur_pow)
except ValueError as error:
print(error)
field = field_x
field_2 = field
cur_2 = cur_pow
dn_2 = dn_temp
print("\n kerr+dz")
field = bpm.field
for i in range(nbr_step):
# Influence of the index modulation on the field
field_x = field * np.exp(1j * bpm.nl_mat[nbr_step, :])
# Linear propagation over dz
field_x = ifft(bpm.phase_mat * fft(field_x))
cur_pow = bpm.epnc * abs2(field_x)
for k in range(kerr_loop):
prev_pow = cur_pow
# influence of the beam intensity on the index modulation
dn_temp = dn[nbr_step, :] + (3*chi3)/(8*no)*(prev_pow/bpm.epnc)
nl_mat = bpm.ko * bpm.dist_z * dn_temp
# influence of the index modulation on the field
field_x = field * np.exp(1j * nl_mat)
# Linear propagation over dz
field_x = ifft(bpm.phase_mat * fft(field_x))
cur_pow = bpm.epnc * abs2(field_x)
try:
bpm.variance(prev_pow, cur_pow)
except ValueError as error:
print(error)
field = field_x
field_3 = field
cur_3 = cur_pow
dn_3 = dn_temp
plt.figure(num="Impact order kerr")
ax1 = plt.subplot(211)
ax2 = plt.subplot(212)
ax1.set_title("phase: comparison")
ax2.set_title("power: comparison")
ax1.set_xlim(-15, 15)
ax2.set_xlim(-15, 15)
ax1.plot(x, np.angle(field_ref), label="no kerr")
ax1.plot(x, np.angle(field_1), label="dz+kerr")
ax1.plot(x, np.angle(field_2), label="dz/2+kerr+dz/2")
ax1.plot(x, np.angle(field_3), label="kerr+dz")
ax2.plot(x, cur_ref, label="no kerr")
ax2.plot(x, cur_1, label="dz+kerr")
ax2.plot(x, cur_2, label="dz/2+kerr+dz/2")
ax2.plot(x, cur_3, label="kerr+dz")
ax1.legend(loc="upper right")
ax2.legend(loc="upper right")
plt.show()
dn_ref = dn[nbr_step, :]
plt.figure(num="Impact on dn order kerr")
ax1 = plt.subplot(111)
ax1.set_title("dn: comparison")
ax1.set_xlim(-15, 15)
ax1.plot(x, dn_ref, label="no kerr")
ax1.plot(x, dn_1, label="dz+kerr")
ax1.plot(x, dn_2, label="dz/2+kerr+dz/2")
ax1.plot(x, dn_3, label="kerr+dz")
ax1.legend(loc="upper right")
plt.show()
def stability():
"""Show the possible BPM approximations for implementing a refractive
index variation"""
width = 6
no = 2.14
wavelength = 1.55
delta_no = 0.0014
length_z = 200
dist_z = 1
nbr_z_disp = 1
dist_x = 0.2
length_x = 1000
bpm = Bpm(no, wavelength,
length_z, dist_z, nbr_z_disp,
length_x, dist_x)
[length_z, nbr_z, nbr_z_disp, length_x, nbr_x, x] = bpm.create_x_z()
shape = bpm.squared_guide(width)
nbr_p = 1
p = 0
[peaks, dn] = bpm.create_guides(shape, delta_no, nbr_p, p)
fwhm = 6
lo = 1.5
field = bpm.gauss_light(fwhm)
irrad = 1
theta_ext = 0
[progress_pow] = bpm.init_field(field, theta_ext, irrad)
nbr_step = nbr_z-1
# Need to overwrite those variables due to changes
theta = asin(sin(radians(theta_ext)) / no) # angle in the guide
nu_max = 1 / (2 * dist_x) # max frequency due to sampling
# Spacial frequencies over x (1/µm)
nu = np.linspace(-nu_max,
nu_max * (1 - 2/nbr_x),
nbr_x)
intermed = no * cos(theta) / lo
fr = -2 * pi * nu**2 / (intermed + np.sqrt(
intermed**2
- nu**2
+ 0j))
bpm.phase_mat = fftshift(np.exp(1j * dist_z * fr))
bpm.phase_mat_demi = fftshift(np.exp(1j * dist_z / 2 * fr))
bpm.nl_mat = bpm.ko * bpm.dist_z * dn
# End overwrite
field = bpm.field
for i in range(nbr_step):
field = ifft(bpm.phase_mat * fft(field))
field *= np.exp(1j * bpm.nl_mat[nbr_step, :])
test1 = field
field = bpm.field
for i in range(nbr_step):
field = ifft(bpm.phase_mat_demi * fft(field))
field *= np.exp(1j * bpm.nl_mat[nbr_step, :])
field = ifft(bpm.phase_mat_demi * fft(field))
test2 = field
field = bpm.field
for i in range(nbr_step):
field *= np.exp(1j * bpm.nl_mat[nbr_step, :])
field = ifft(bpm.phase_mat * fft(field))
test3 = field
plt.figure("Power")
plt.title("Power: Impact of the chosen algorithm on the free propagation")
plt.xlim(-20, 20)
# plt.ylim(-1, 350)
plt.plot(x, abs2(test1), label='dz+lens')
plt.plot(x, abs2(test2), label='dz/2+lens+dz/2')
plt.plot(x, abs2(test3), label='lens+dz')
plt.legend()
plt.show()
plt.figure("Phase")
plt.title("Phase: Impact of the chosen algorithm on the free propagation")
plt.xlim(-30, 30)
plt.plot(x, np.angle(test1), label='dz+lens')
plt.plot(x, np.angle(test2), label='dz/2+lens+dz/2')
plt.plot(x, np.angle(test3), label='lens+dz')
plt.legend()
plt.show()
def benchmark_kerr():
"""Kerr benchmark by looking at the critical power at which a beam become
a soliton. Several test were done with this function and for now, it seems
that the critical power find by the simulations is about 85% of the
theorical value. Meaning a 15% error. Further tests are needed to
understand the observed differences."""
width = 0
no = 1
wavelength = 4
delta_no = 0.000
length_z = 1000
dist_z = 0.5
nbr_z_disp = 200
dist_x = 0.05
length_x = 500
nbr_p = 1
p = 20
fwhm = 12
theta_ext = 0
n2 = 2.44e-20
alpha = 1.8962 # gaussian beam
# See Self-focusing on wiki
P_c = alpha*(wavelength*1e-6)**2/(4*pi*no*n2)
print("critical power = %.2e W" % P_c)
# irrad = 5e3*1e13
# power = irrad * (fwhm*1e-6)**2 * pi / (4*np.log(2))
# print("power", power)
# See Gaussian_beam on wiki on beam power
w = fwhm*1e-6 / sqrt(2*log(2))
irrad_cri = 2*P_c/(pi*w**2)
print("Critical irradiance %.2e W/m^2" % irrad_cri)
bpm = Bpm(no, wavelength,
length_z, dist_z, nbr_z_disp,
length_x, dist_x)
[length_z, nbr_z, nbr_z_disp, length_x, nbr_x, x] = bpm.create_x_z()
shape = bpm.gauss_guide(width, 4)
sweep = (1,
0.2*irrad_cri, 0.5*irrad_cri, 0.7*irrad_cri, 0.75*irrad_cri,
0.8*irrad_cri, 0.85*irrad_cri, # 0.85 seem correct
0.9*irrad_cri, irrad_cri
)
for i, irrad in enumerate(sweep):
power = irrad/2 * (pi*w**2)
print(i+1, "/", len(sweep), "\tpower= %.2e W" % power, sep="")
[peaks, dn] = bpm.create_guides(shape, delta_no, nbr_p, p)
field = bpm.gauss_light(fwhm)
[progress_pow] = bpm.init_field(field, theta_ext, irrad)
[progress_pow] = bpm.main_compute(dn, n2=n2, kerr_loop=2,
disp_progress=False)
x_pos = int(length_x/2/dist_x)
z_disp = np.linspace(0, length_z/1000, nbr_z_disp+1)
plt.figure("Benchmark kerr")
plt.title(r"Central irradition for a theorical $P_c$ = %.2e W" % P_c)
plt.xlabel("z (µm)")
plt.ylabel(r"I (a.u)")
# plt.ylim(-0.05, 2)
plt.plot(z_disp*1e3,
progress_pow[:, x_pos]/progress_pow[0, x_pos],
label="Power=%.2e" % power)
plt.legend()
plt.show()
# plt.savefig("kerr_irrad.png", bbox="tight", dpi=250)
# x_beg = np.array([None]*nbr_z)
# x_end = np.array([None]*nbr_z)
# P = np.zeros(nbr_z_disp+1)
# x_beg, x_end = bpm.guide_position(peaks, 0, p)
# P = bpm.power_guide(x_beg, x_end)
# plt.figure("Benchmark kerr2")
# plt.title("Power for a theorical critical power = %.2e" % P_c)
# plt.xlabel("z (µm)")
# plt.ylabel(r"Power (a.u)")
# plt.ylim(-0.05, 2)
# plt.plot(z_disp, P, label="Power=%.2e" % power)
# plt.legend()
# plt.show()
# xv, zv = np.meshgrid(x, z_disp)
# plt.figure("check borders")
# plt.title("check if beam don't reach the borders")
# plt.pcolormesh(xv, zv, progress_pow, cmap='gray')
# plt.show()
plt.figure("Benchmark beam profil")
plt.title("Beam profil at z=%.0f µm" % length_z)
plt.xlabel("x (µm)")
plt.ylabel(r"I/$I_0$")
plt.xlim(-30, 30)
plt.plot(x, progress_pow[-1]/np.max(progress_pow[0]),
label="Power=%.2e" % power)
plt.legend()
plt.show()
def free_propag(dist_x=0.1, length_x=1000, length_z=10000, no=1, lo=1):
"""Show the free propagation of a beam and compare Beampy results with
the theorical values.
Parameters
----------
dist_x : float
Step over x (µm).
length_x : float
Size of the compute window over x (µm).
length_z : float
Size of the compute window over z (µm).
no : float
Refractive index of the cladding
lo : float
Wavelength of the beam in vaccum (µm).
"""
dist_z = length_z
nbr_z_disp = 1
bpm = Bpm(no, lo,
length_z, dist_z, nbr_z_disp,
length_x, dist_x)
[length_z, nbr_z, nbr_z_disp, length_x, nbr_x, x] = bpm.create_x_z()
dn = np.zeros((nbr_z, nbr_x))
fwhm = 20
field = bpm.gauss_light(fwhm)
irrad = 1E13
theta_ext = 0
[progress_pow] = bpm.init_field(field, theta_ext, irrad)
[progress_pow] = bpm.main_compute(dn)
intensity = progress_pow[-1]
index = np.where(intensity >= (np.max(intensity)/2))[0][0]
fwhm_final = abs(2 * x[index])
w0_final = fwhm_final / np.sqrt(2 * np.log(2))
print("Beampy: width w = %f µm" % w0_final)
w0 = fwhm / np.sqrt(2 * np.log(2))
z0 = np.pi * no * w0**2 / lo
w = w0 * np.sqrt(1 + (length_z / z0)**2)
print("Theory: width w = %f µm" % w)
diff = abs(w - w0_final)/w*100
print("Relative difference: %f" % diff, "%")
if diff > 1:
print("Check the dist_x or length_x parameters")
irrad_theo = irrad*w0/w # in 2D: irrad*(w0/w)**2
print("Beampy: irradiance I =", np.max(intensity))
print("Theory: irradiance I =", irrad_theo)
diff = abs(np.max(intensity) - irrad_theo)/irrad_theo*100
print("Relative difference: %f" % diff, "%")
if diff > 1:
print("Check the dist_x or length_x parameters")
fwhm_theo = w * np.sqrt(2 * np.log(2))
profil_theo = bpm.gauss_light(fwhm_theo)
intensity_theo = irrad_theo * abs2(profil_theo)
plt.figure()
plt.title("Beam profil after %.0f mm of free propagation" % (length_z/1e3))
plt.xlabel("x (µm)")
plt.ylabel(r"irradiance (W.m$^{-2}$)")
plt.plot(x, intensity_theo, "-", label="Theory")
plt.plot(x, intensity, "--", label="Beampy")
plt.legend()
plt.show()