def test07_rotate_stokes_basis(variant_scalar_rgb):
from mitsuba.core import Transform4f
from mitsuba.render.mueller import stokes_basis, rotate_stokes_basis
# Each Stokes vector describes light polarization only w.r.t. a certain reference
# frame determined by unit vector perpendicular to the direction of travel.
#
# Here we test a duality between rotating the polarization direction (e.g. from
# linear polarization at 0˚ to 90˚) and rotating the Stokes reference frame
# in the opposite direction.
# We start out with horizontally polarized light (Stokes vector [1, 1, 0, 0])
# and perform several transformations.
w = [0, 0, 1] # Optical axis / forward direction
s_00 = [1, 1, 0, 0] # Horizontally polarized light
b_00 = stokes_basis(w) # Corresponding Stokes basis
def rotate_vector(v, axis, angle):
return Transform4f.rotate(axis, angle).transform_vector(v)
# Switch to basis rotated by 90˚.
b_90 = rotate_vector(b_00, w, 90.0)
assert ek.allclose(b_90, [0, 1, 0], atol=1e-3)
# Now polarization should be at 90˚ / vertical (with Stokes vector [1, -1, 0, 0])
s_90 = rotate_stokes_basis(w, b_00, b_90) @ s_00
assert ek.allclose(s_90, [1.0, -1.0, 0.0, 0.0], atol=1e-3)
# Switch to basis rotated by +45˚.
b_45 = rotate_vector(b_00, w, +45.0)
assert ek.allclose(b_45, [0.70712, 0.70712, 0.0], atol=1e-3)
# Now polarization should be at -45˚ (with Stokes vector [1, 0, -1, 0])
s_45 = rotate_stokes_basis(w, b_00, b_45) @ s_00
assert ek.allclose(s_45, [1.0, 0.0, -1.0, 0.0], atol=1e-3)
# Switch to basis rotated by -45˚.
b_45_neg = rotate_vector(b_00, w, -45.0)
assert ek.allclose(b_45_neg, [0.70712, -0.70712, 0.0], atol=1e-3)
# Now polarization should be at +45˚ (with Stokes vector [1, 0, +1, 0])
s_45_neg = rotate_stokes_basis(w, b_00, b_45_neg) @ s_00
assert ek.allclose(s_45_neg, [1.0, 0.0, +1.0, 0.0], atol=1e-3)
def test08_rotate_mueller_basis(variant_scalar_rgb):
from mitsuba.core import Transform4f
from mitsuba.render.mueller import stokes_basis, rotated_element, rotate_mueller_basis, \
rotate_mueller_basis_collinear, linear_polarizer
# If we have an optical element such as a linear polarizer, its rotation around
# the optical axis can also be interpreted as a change of both incident and
# outgoing Stokes reference frames:
w = [0, 0, 1] # Optical axis / forward direction
s_00 = [1, 1, 0, 0] # Horizontally polarized light
b_00 = stokes_basis(w) # Corresponding Stokes basis
M = linear_polarizer()
def rotate_vector(v, axis, angle):
return Transform4f.rotate(axis, angle).transform_vector(v)
# As reference, rotate the element directly by -45˚
M_rotated_element = rotated_element(-45 * ek.pi/180, M)
# Now rotate the two reference bases by 45˚ instead and make sure results are
# equivalent.
# More thorough explanation of what's going on here:
# Mueller matrix 'M' works in b_00 = [1, 0, 0], and we want to construct
# another matrix that works in b_45 =[0.707, 0.707, 0] instead.
# 'rotate_mueller_basis' does exactly that:
# R(b_00 -> b_45) * M * R(b_45 -> b_00) = R(+45˚) * M * R(-45˚) which is
# equivalent to rotating the elemnt by -45˚.
b_45 = rotate_vector(b_00, w, +45.0)
M_rotated_bases = rotate_mueller_basis(M,
w, b_00, b_45,
w, b_00, b_45)
assert ek.allclose(M_rotated_element, M_rotated_bases, atol=1e-5)
# Also test alternative rotation method that assumes collinear incident and
# outgoing directions.
M_rotated_bases_aligned = rotate_mueller_basis_collinear(M, w, b_00, b_45)
assert ek.allclose(M_rotated_element, M_rotated_bases_aligned, atol=1e-5)
def test07_path_tracer_half_wave(variant_scalar_mono_polarized):
from mitsuba.core import Spectrum
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode
from mitsuba.render.mueller import stokes_basis, rotate_stokes_basis_m
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i, 0] = v[i]
return res
# Another test involving a half-wave plate, this time inside an actual
# polarized path tracer.
# (Serves also as test case for the polarized path tracer itself.)
#
# Following "Polarized Light - Fundamentals and Applications" by Edward Collett
# Chapter 5.3:
#
# Case 1 & 2) Switch between diagonal linear polarization states (-45˚ & + 45˚)
#
# In this test, a polarizer and half-wave plate are placed between light
# source and camera.
# The polarizer is used to convert emitted light into a +-45˚ linearly
# polarized state. The subsequent retarder will flip the polarization
# axis to the opposite one.
linear_pos = spectrum_from_stokes([1, 0, +1, 0])
linear_neg = spectrum_from_stokes([1, 0, -1, 0])
angles = [+45.0, -45.0]
expected = [linear_neg, linear_pos]
observed = []
for angle in angles:
scene_str = """<scene version='2.0.0'>
<integrator type="path"/>
<sensor type="perspective">
<float name="fov" value="0.00001"/>
<transform name="to_world">
<lookat origin="0, 10, 0"
target="0, 0, 0"
up ="0, 0, 1"/>
</transform>
<film type="hdrfilm">
<integer name="width" value="1"/>
<integer name="height" value="1"/>
<rfilter type="gaussian"/>
<string name="pixel_format" value="rgb"/>
</film>
</sensor>
<!-- Light source -->
<shape type="rectangle">
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/>
<translate y="-10"/>
</transform>
<emitter type="area">
<spectrum name="radiance" value="1"/>
</emitter>
</shape>
<!-- Polarizer at given angle -->
<shape type="rectangle">
<bsdf type="polarizer">
</bsdf>
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/>
<rotate y="1" angle="{}"/> <!-- Rotate around optical axis -->
<translate y="-5"/>
</transform>
</shape>
<!-- Half-wave plate. -->
<shape type="rectangle">
<bsdf type="retarder">
<spectrum name="delta" value="180"/>
</bsdf>
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/>
<translate y="0"/>
</transform>
</shape>
</scene>""".format(angle)
scene = load_string(scene_str)
integrator = scene.integrator()
sensor = scene.sensors()[0]
sampler = sensor.sampler()
# Sample ray from sensor
ray, _ = sensor.sample_ray_differential(0.0, 0.5, [0.5, 0.5],
[0.5, 0.5])
# Call integrator
value, _, _ = integrator.sample(scene, sampler, ray)
# Normalize Stokes vector
value /= value[0, 0]
# Align output stokes vector (based on ray.d) with optical table. (In this configuration, this is a no-op.)
forward = -ray.d
basis_cur = stokes_basis(forward)
basis_tar = [-1, 0, 0]
R = rotate_stokes_basis_m(forward, basis_cur, basis_tar)
value = R @ value
observed.append(value)
# Make sure observations match expectations
for k in range(len(expected)):
assert ek.allclose(observed[k], expected[k], atol=1e-3)
def test05_sample_half_wave_world(variant_scalar_mono_polarized):
from mitsuba.core import Ray3f, Spectrum
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode
from mitsuba.render.mueller import stokes_basis, rotate_mueller_basis_collinear
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i, 0] = v[i]
return res
# Test polarized implementation. Special case of delta = 180˚, also known
# as a half-wave plate. (In world coordinates.)
#
# Following "Polarized Light - Fundamentals and Applications" by Edward Collett
# Chapter 5.3:
#
# Case 1 & 2) Switch between diagonal linear polarization states (-45˚ & + 45˚)
# Case 3 & 4) Switch circular polarization direction
linear_pos = spectrum_from_stokes([1, 0, +1, 0])
linear_neg = spectrum_from_stokes([1, 0, -1, 0])
circular_right = spectrum_from_stokes([1, 0, 0, +1])
circular_left = spectrum_from_stokes([1, 0, 0, -1])
# Incident direction
forward = [0, -1, 0]
ctx = BSDFContext()
ctx.mode = TransportMode.Importance
ray = Ray3f([0, 100, 0], forward, 0.0, 0.0)
# Build scene with given polarizer rotation angle
scene_str = """<scene version='2.0.0'>
<shape type="rectangle">
<bsdf type="retarder">
<spectrum name="delta" value="180"/>
</bsdf>
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/> <!-- Rotate s.t. it is no longer aligned with local coordinates -->
</transform>
</shape>
</scene>"""
scene = load_string(scene_str)
# Intersect ray against geometry
si = scene.ray_intersect(ray)
bs, M_local = si.bsdf().sample(ctx, si, 0.0, [0.0, 0.0])
M_world = si.to_world_mueller(M_local, -si.wi, bs.wo)
# Make sure we are measuring w.r.t. to the optical table
M = rotate_mueller_basis_collinear(M_world, forward, stokes_basis(forward),
[-1, 0, 0])
# Case 1)
assert ek.allclose(M @ linear_pos, linear_neg, atol=1e-3)
# Case 2)
assert ek.allclose(M @ linear_neg, linear_pos, atol=1e-3)
# Case 3)
assert ek.allclose(M @ circular_right, circular_left, atol=1e-3)
# Case 4)
assert ek.allclose(M @ circular_left, circular_right, atol=1e-3)
def test03_sample_world(variant_scalar_mono_polarized):
from mitsuba.core import Ray3f, Spectrum
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode
from mitsuba.render.mueller import stokes_basis, rotate_mueller_basis_collinear
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i,0] = v[i]
return res
# Test polarized implementation, version in world coordinates. This involves
# additional coordinate changes to the local reference frame and back.
#
# The polarizer is rotated to different angles and hit with fully
# unpolarized light (Stokes vector [1, 0, 0, 0]).
# We then test if the outgoing Stokes vector corresponds to the expected
# rotation of linearly polarized light.
# Incident direction
forward = [0, 1, 0]
stokes_in = spectrum_from_stokes([1, 0, 0, 0])
ctx = BSDFContext()
ctx.mode = TransportMode.Importance
ray = Ray3f([0, -100, 0], forward, 0.0, 0.0)
# Polarizer rotation angles
angles = [0, 90, +45, -45]
# Expected outgoing Stokes vector
expected_states = [spectrum_from_stokes([0.5, 0.5, 0, 0]),
spectrum_from_stokes([0.5, -0.5, 0, 0]),
spectrum_from_stokes([0.5, 0, +0.5, 0]),
spectrum_from_stokes([0.5, 0, -0.5, 0])]
for k in range(len(angles)):
angle = angles[k]
expected = expected_states[k]
# Build scene with given polarizer rotation angle
scene_str = """<scene version='2.0.0'>
<shape type="rectangle">
<bsdf type="polarizer"/>
<transform name="to_world">
<rotate x="0" y="0" z="1" angle="{}"/> <!-- Rotate around optical axis -->
<rotate x="1" y="0" z="0" angle="-90"/> <!-- Rotate s.t. it is no longer aligned with local coordinates -->
</transform>
</shape>
</scene>""".format(angle)
scene = load_string(scene_str)
# Intersect ray against geometry
si = scene.ray_intersect(ray)
bs, M_local = si.bsdf().sample(ctx, si, 0.0, [0.0, 0.0])
M_world = si.to_world_mueller(M_local, -si.wi, bs.wo)
# Make sure we are measuring w.r.t. to the optical table
M = rotate_mueller_basis_collinear(M_world,
forward,
stokes_basis(forward), [-1, 0, 0])
stokes_out = M @ stokes_in
assert ek.allclose(stokes_out, expected, atol=1e-3)
def test04_path_tracer_polarizer(variant_scalar_mono_polarized):
from mitsuba.core import Spectrum
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode
from mitsuba.render.mueller import stokes_basis, rotate_stokes_basis_m
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i,0] = v[i]
return res
# Test polarizer implementation inside of an actual polarized path tracer.
# (Serves also as test case for the polarized path tracer itself.)
#
# The polarizer is place in front of a light source that emits unpolarized
# light (Stokes vector [1, 0, 0, 0]). The filter is rotated to different
# angles and the transmitted polarization state is direclty observed with
# a sensor.
# We then test if the outgoing Stokes vector corresponds to the expected
# rotation of linearly polarized light.
horizontal_pol = spectrum_from_stokes([1, +1, 0, 0])
vertical_pol = spectrum_from_stokes([1, -1, 0, 0])
pos_diagonal_pol = spectrum_from_stokes([1, 0, +1, 0])
neg_diagonal_pol = spectrum_from_stokes([1, 0, -1, 0])
angles = [0, 90, +45, -45]
expected = [horizontal_pol,
vertical_pol,
pos_diagonal_pol,
neg_diagonal_pol]
for k, angle in enumerate(angles):
scene_str = """<scene version='2.0.0'>
<integrator type="path"/>
<sensor type="perspective">
<float name="fov" value="0.00001"/>
<transform name="to_world">
<lookat origin="0, 10, 0"
target="0, 0, 0"
up ="0, 0, 1"/>
</transform>
<film type="hdrfilm">
<integer name="width" value="1"/>
<integer name="height" value="1"/>
<rfilter type="gaussian"/>
<string name="pixel_format" value="rgb"/>
</film>
</sensor>
<!-- Light source -->
<shape type="rectangle">
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/>
<translate y="-10"/>
</transform>
<emitter type="area">
<spectrum name="radiance" value="1"/>
</emitter>
</shape>
<!-- Polarizer -->
<shape type="rectangle">
<bsdf type="polarizer">
</bsdf>
<transform name="to_world">
<rotate x="1" y="0" z="0" angle="-90"/>
<rotate y="1" angle="{}"/>
</transform>
</shape>
</scene>""".format(angle)
scene = load_string(scene_str)
integrator = scene.integrator()
sensor = scene.sensors()[0]
sampler = sensor.sampler()
# Sample ray from sensor
ray, _ = sensor.sample_ray_differential(0.0, 0.5, [0.5, 0.5], [0.5, 0.5])
# Call integrator
value, _, _ = integrator.sample(scene, sampler, ray)
# Normalize Stokes vector
value /= value[0, 0]
# Align output stokes vector (based on ray.d) with optical table. (In this configuration, this is a no-op.)
forward = -ray.d
basis_cur = stokes_basis(forward)
basis_tar = [-1, 0, 0]
R = rotate_stokes_basis_m(forward, basis_cur, basis_tar)
value = R @ value
assert ek.allclose(value, expected[k], atol=1e-3)
def test02_sample_pol_local(variant_scalar_mono_polarized):
from mitsuba.core import Frame3f, Transform4f, Spectrum, UnpolarizedSpectrum, Vector3f
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode, SurfaceInteraction3f, fresnel_conductor
from mitsuba.render.mueller import stokes_basis, rotate_mueller_basis
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i, 0] = v[i]
return res
# Create a Silver conductor BSDF and reflect different polarization states
# at a 45˚ angle.
# All tests take place directly in local BSDF coordinates. Here we only
# want to make sure that the output of this looks like what you would
# expect from a Mueller matrix describing specular reflection on a mirror.
eta = 0.136125 + 4.010625j # IoR values of Ag at 635.816284nm
bsdf = load_string("""<bsdf version='2.0.0' type='conductor'>
<spectrum name="eta" value="{}"/>
<spectrum name="k" value="{}"/>
</bsdf>""".format(eta.real, eta.imag))
theta_i = 45 * ek.pi / 180
wi = Vector3f([-ek.sin(theta_i), 0, ek.cos(theta_i)])
ctx = BSDFContext()
ctx.mode = TransportMode.Importance
si = SurfaceInteraction3f()
si.p = [0, 0, 0]
si.wi = wi
n = [0, 0, 1]
si.sh_frame = Frame3f(n)
bs, M_local = bsdf.sample(ctx, si, 0.0, [0.0, 0.0])
wo = bs.wo
# Rotate into standard coordinates for specular reflection
bi_s = ek.normalize(ek.cross(n, -wi))
bi_p = ek.normalize(ek.cross(-wi, bi_s))
bo_s = ek.normalize(ek.cross(n, wo))
bo_p = ek.normalize(ek.cross(wo, bi_s))
M_local = rotate_mueller_basis(M_local, -wi, stokes_basis(-wi), bi_p, wo,
stokes_basis(wo), bo_p)
# Apply to unpolarized light and verify that it is equivalent to normal Fresnel
a0 = M_local @ spectrum_from_stokes([1, 0, 0, 0])
F = fresnel_conductor(ek.cos(theta_i),
ek.scalar.Complex2f(eta.real, eta.imag))
a0 = a0[0, 0]
assert ek.allclose(a0[0], F, atol=1e-3)
# Apply to horizontally polarized light (linear at 0˚)
# Test that it is..
# 1) still fully polarized (though with reduced intensity)
# 2) still horizontally polarized
a1 = M_local @ spectrum_from_stokes([1, +1, 0, 0])
assert ek.allclose(a1[0, 0], ek.abs(a1[1, 0]), atol=1e-3) # 1)
a1 /= a1[0, 0]
assert ek.allclose(a1, spectrum_from_stokes([1, 1, 0, 0]), atol=1e-3) # 2)
# Apply to vertically polarized light (linear at 90˚)
# Test that it is..
# 1) still fully polarized (though with reduced intensity)
# 2) still vertically polarized
a2 = M_local @ spectrum_from_stokes([1, -1, 0, 0])
assert ek.allclose(a2[0, 0], ek.abs(a2[1, 0]), atol=1e-3) # 1)
a2 /= a2[0, 0]
assert ek.allclose(a2, spectrum_from_stokes([1, -1, 0, 0]),
atol=1e-3) # 2)
# Apply to (positive) diagonally polarized light (linear at +45˚)
# Test that..
# 1) The polarization is flipped to -45˚
# 2) There is now also some (left) circular polarization
a3 = M_local @ spectrum_from_stokes([1, 0, +1, 0])
assert ek.all(a3[2, 0] < UnpolarizedSpectrum(0.0))
assert ek.all(a3[3, 0] < UnpolarizedSpectrum(0.0))
# Apply to (negative) diagonally polarized light (linear at -45˚)
# Test that..
# 1) The polarization is flipped to +45˚
# 2) There is now also some (right) circular polarization
a4 = M_local @ spectrum_from_stokes([1, 0, -1, 0])
assert ek.all(a4[2, 0] > UnpolarizedSpectrum(0.0))
assert ek.all(a4[3, 0] > UnpolarizedSpectrum(0.0))
# Apply to right circularly polarized light
# Test that the polarization is flipped to left circular
a5 = M_local @ spectrum_from_stokes([1, 0, 0, +1])
assert ek.all(a5[3, 0] < UnpolarizedSpectrum(0.0))
# Apply to left circularly polarized light
# Test that the polarization is flipped to right circular
a6 = M_local @ spectrum_from_stokes([1, 0, 0, -1])
assert ek.all(a6[3, 0] > UnpolarizedSpectrum(0.0))
def test02_sample_pol_world(variant_scalar_mono_polarized):
from mitsuba.core import Frame3f, Spectrum, UnpolarizedSpectrum
from mitsuba.core.xml import load_string
from mitsuba.render import BSDFContext, TransportMode, SurfaceInteraction3f, fresnel_conductor
from mitsuba.render.mueller import stokes_basis, rotate_mueller_basis
def spectrum_from_stokes(v):
res = Spectrum(0.0)
for i in range(4):
res[i, 0] = v[i]
return res
# Create a Silver conductor BSDF and reflect different polarization states
# at a 45˚ angle.
# This test takes place in world coordinates and thus involves additional
# coordinate system rotations.
#
# The setup is as follows:
# - The mirror is positioned at [0, 0, 0], angled s.t. the surface normal
# is along [1, 1, 0].
# - Incident ray w1 = [-1, 0, 0] strikes the mirror at a 45˚ angle and
# reflects into direction w2 = [0, 1, 0]
# - The corresponding Stokes bases are b1 = [0, 1, 0] and b2 = [1, 0, 0].
# Setup
eta = 0.136125 + 4.010625j # IoR values of Ag at 635.816284nm
bsdf = load_string("""<bsdf version='2.0.0' type='conductor'>
<spectrum name="eta" value="{}"/>
<spectrum name="k" value="{}"/>
</bsdf>""".format(eta.real, eta.imag))
ctx = BSDFContext()
ctx.mode = TransportMode.Importance
si = SurfaceInteraction3f()
si.p = [0, 0, 0]
si.n = ek.normalize([1.0, 1.0, 0.0])
si.sh_frame = Frame3f(si.n)
# Incident / outgoing directions and their Stokes bases
w1 = ek.scalar.Vector3f([-1, 0, 0])
b1 = [0, 1, 0]
w2 = ek.scalar.Vector3f([0, 1, 0])
b2 = [1, 0, 0]
# Perform actual reflection
si.wi = si.to_local(-w1)
bs, M_tmp = bsdf.sample(ctx, si, 0.0, [0.0, 0.0])
M_world = si.to_world_mueller(M_tmp, -si.wi, bs.wo)
# Test that outgoing direction is as predicted
assert ek.allclose(si.to_world(bs.wo), w2, atol=1e-5)
# Align reference frames s.t. polarization is expressed w.r.t. b1 & b2
M_world = rotate_mueller_basis(M_world, w1, stokes_basis(w1), b1, w2,
stokes_basis(w2), b2)
# Apply to unpolarized light and verify that it is equivalent to normal Fresnel
a0 = M_world @ spectrum_from_stokes([1, 0, 0, 0])
F = fresnel_conductor(si.wi[2], ek.scalar.Complex2f(eta.real, eta.imag))
a0 = a0[0, 0]
assert ek.allclose(a0[0], F, atol=1e-3)
# Apply to horizontally polarized light (linear at 0˚)
# Test that it is..
# 1) still fully polarized (though with reduced intensity)
# 2) still horizontally polarized
a1 = M_world @ spectrum_from_stokes([1, +1, 0, 0])
assert ek.allclose(a1[0, 0], ek.abs(a1[1, 0]), atol=1e-3) # 1)
a1 /= a1[0, 0]
assert ek.allclose(a1, spectrum_from_stokes([1, 1, 0, 0]), atol=1e-3) # 2)
# Apply to vertically polarized light (linear at 90˚)
# Test that it is..
# 1) still fully polarized (though with reduced intensity)
# 2) still vertically polarized
a2 = M_world @ spectrum_from_stokes([1, -1, 0, 0])
assert ek.allclose(a2[0, 0], ek.abs(a2[1, 0]), atol=1e-3) # 1)
a2 /= a2[0, 0]
assert ek.allclose(a2, spectrum_from_stokes([1, -1, 0, 0]),
atol=1e-3) # 2)
# Apply to (positive) diagonally polarized light (linear at +45˚)
# Test that..
# 1) The polarization is flipped to -45˚
# 2) There is now also some (left) circular polarization
a3 = M_world @ spectrum_from_stokes([1, 0, +1, 0])
assert ek.all(a3[2, 0] < UnpolarizedSpectrum(0.0))
assert ek.all(a3[3, 0] < UnpolarizedSpectrum(0.0))
# Apply to (negative) diagonally polarized light (linear at -45˚)
# Test that..
# 1) The polarization is flipped to +45˚
# 2) There is now also some (right) circular polarization
a4 = M_world @ spectrum_from_stokes([1, 0, -1, 0])
assert ek.all(a4[2, 0] > UnpolarizedSpectrum(0.0))
assert ek.all(a4[3, 0] > UnpolarizedSpectrum(0.0))
# Apply to right circularly polarized light
# Test that the polarization is flipped to left circular
a5 = M_world @ spectrum_from_stokes([1, 0, 0, +1])
assert ek.all(a5[3, 0] < UnpolarizedSpectrum(0.0))
# Apply to left circularly polarized light
# Test that the polarization is flipped to right circular
a6 = M_world @ spectrum_from_stokes([1, 0, 0, -1])
assert ek.all(a6[3, 0] > UnpolarizedSpectrum(0.0))