11. Comparison between Rayleigh-Sommerfeld (RS) and Chirped Z-transform (CZT)

In this example, we will analyze how CZT algorithm produces better results than in RS when the region of interest in the focus of a lens.

11.1. X Scheme

[1]:

from diffractio import degrees, mm, plt, sp, um, np
from diffractio.scalar_sources_X import Scalar_source_X
from diffractio.scalar_masks_X import Scalar_mask_X

[2]:

focal = 25 * mm
diameter = 4 * mm

# Initial parameters
x0 = np.linspace(-diameter / 2 - 50 * um, diameter / 2 + 50 * um, 4096)
wavelength = 0.6238 * um

# Definition of source
u0 = Scalar_source_X(x=x0, wavelength=wavelength)
u0.plane_wave(A=1)

t0 = Scalar_mask_X(x=x0, wavelength=wavelength)
t0.lens(x0=0.0, focal=focal, radius=diameter / 2)

u1 = t0 * u0

RS

[3]:

%%time
u2_rs = u1.RS(z=focal, verbose=False)
u2_rs.draw()
plt.xlim(-25, 25)

CPU times: user 248 ms, sys: 3.67 ms, total: 251 ms
Wall time: 277 ms

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_6_1.png

CZT

[4]:

%%time
xout=np.linspace(-25,25,512)
u2_czt = u1.CZT(z=focal, xout=xout)
u2_czt.draw()

CPU times: user 59.5 ms, sys: 0 ns, total: 59.5 ms
Wall time: 71.4 ms

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_8_1.png

11.2. XZ scheme

[5]:

from diffractio import degrees, mm, plt, sp, um, np
from diffractio.scalar_sources_X import Scalar_source_X
from diffractio.scalar_masks_X import Scalar_mask_X
from diffractio.scalar_masks_XZ import Scalar_mask_XZ

[6]:

focal = 15 * mm
diameter = 4 * mm

x0 = np.linspace(-diameter / 2, diameter / 2, 1024)
wavelength = 0.6238 * um

u0 = Scalar_source_X(x=x0, wavelength=wavelength)
u0.plane_wave(A=1, theta=0 * degrees)

t0 = Scalar_mask_X(x=x0, wavelength=wavelength)
t0.lens(x0=0.0, radius=diameter / 2, focal=focal)

u1 = u0 * t0

RS

[7]:

%%time
z0 = np.linspace(14.8 * mm, 15.2 * mm, 512)
u2_rs = Scalar_mask_XZ(x=x0, z=z0, wavelength=wavelength)
u2_rs.incident_field(u1)
u2_rs.RS(num_processors=4);
u2_rs.draw(logarithm=1e-1, z_scale='mm')
plt.ylim(-30,30)

CPU times: user 805 ms, sys: 305 ms, total: 1.11 s
Wall time: 2.44 s

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_13_1.png

CZT

[8]:

%%time
xout=np.linspace(-30,30,256)
u2_czt = u1.CZT(z=z0, xout=xout)
u2_czt.draw(logarithm=1e-1, z_scale='mm')

CPU times: user 2.43 s, sys: 7.56 ms, total: 2.44 s
Wall time: 3.9 s

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_15_1.png

11.2.1. XY scheme

[9]:

from diffractio import degrees, mm, plt, sp, um, np
from diffractio.scalar_fields_XY import Scalar_field_XY
from diffractio.scalar_masks_XY import Scalar_mask_XY
from diffractio.scalar_sources_XY import Scalar_source_XY

[10]:

diameter = 2 * mm
focal = 25 * mm

x0 = np.linspace(-diameter / 2, diameter / 2, 1024)
y0 = np.linspace(-diameter / 2, diameter / 2, 1024)
wavelength = 0.6238 * um

[11]:

u0 = Scalar_source_XY(x=x0, y=y0, wavelength=wavelength)
u0.plane_wave()

t0 = Scalar_mask_XY(x=x0, y=y0, wavelength=wavelength)
t0.lens(r0=(0 * um, 0 * um),
        radius=(diameter / 2, diameter / 2),
        focal=(focal, focal))

t0.draw('phase')

u1 = u0 * t0

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_19_0.png

RS

[12]:

%%time
u2_rs = u1.RS(z=focal)

u2_rs.cut_resample(x_limits=(-40, 40),
                   y_limits=(-40, 40),
                   num_points=(128, 128),
                   new_field=False,
                   interp_kind=(3, 1))

u2_rs.draw(logarithm=1e-2)

CPU times: user 3.84 s, sys: 346 ms, total: 4.19 s
Wall time: 4.03 s

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_21_1.png

CZT

[13]:

%%time
xout=np.linspace(-40,40,128)
yout=np.linspace(-40,40,128)
u2_czt = u1.CZT(z=focal, xout=xout, yout=yout)

u2_czt.draw(logarithm=1e-2)

CPU times: user 774 ms, sys: 185 ms, total: 959 ms
Wall time: 709 ms

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_23_1.png

11.3. Vector scheme

[14]:

from diffractio import degrees, mm, plt, sp, um, np
from diffractio.scalar_fields_XY import Scalar_field_XY
from diffractio.scalar_masks_XY import Scalar_mask_XY
from diffractio.scalar_sources_XY import Scalar_source_XY

from diffractio.vector_sources_XY import Vector_source_XY
from diffractio.vector_masks_XY import Vector_mask_XY
from diffractio.vector_fields_XY import Vector_field_XY

[15]:

diameter = 2 * mm
x0 = np.linspace(-diameter / 2, diameter / 2, 512)
y0 = np.linspace(-diameter / 2, diameter / 2, 512)
wavelength = 0.6238 * um

[16]:

focal = 25 * mm
limit = 80 * um

[17]:

u0 = Scalar_source_XY(x0, y0, wavelength)
u0.plane_wave(A=1)

t0 = Scalar_mask_XY(x=x0, y=y0, wavelength=wavelength)
t0.lens(r0=(0 * um, 0 * um),
        radius=(diameter / 2, diameter / 2),
        focal=(focal, focal))

u1 = u0 * t0

[18]:

EM0 = Vector_source_XY(x0, y0, wavelength)
EM0.constant_polarization(u1, v=[(1 - 1j) / 2, (1 + 1j) / 2])

RS

[19]:

%%time
EM1_rs = EM0.VRS(z=focal)
EM1_rs.cut_resample(x_limits=(-limit / 2, limit / 2),
                          y_limits=(-limit / 2, limit / 2),
                          num_points=(128, 128),
                          new_field=False,
                          interp_kind=(3, 1))

EM1_rs.draw('intensities', logarithm=1e-1)

CPU times: user 2.61 s, sys: 388 ms, total: 3 s
Wall time: 2.37 s

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_31_1.png

CZT

[20]:

%%time
xout = np.linspace(-limit / 2, limit / 2, 128)
yout = np.linspace(-limit / 2, limit / 2, 128)
EM1_czt = EM0.CZT(z=focal, xout=xout, yout=yout)
EM1_czt.draw('intensities', logarithm=1e-1)

CPU times: user 1.55 s, sys: 372 ms, total: 1.92 s
Wall time: 1.39 s

../../../../_images/source_tutorial_algorithms_comparison_CZT_vs_RS_33_1.png