def f_qbfs(n):
"""f(m) from oe-18-19-19700 eq. (A.16)."""
if n == 0:
return 2
elif n == 1:
return np.sqrt(19) / 2
else:
term1 = n * (n + 1) + 3
term2 = g_qbfs(n - 1)**2
term3 = h_qbfs(n - 2)**2
return np.sqrt(term1 - term2 - term3)
def sphere_sag(c, rhosq, phi=None):
"""Sag of a spherical surface.
Parameters
----------
c : float
surface curvature
rhosq : numpy.ndarray
radial coordinate squared
e.g. for a 15 mm half-diameter optic,
rho = 0 .. 15
rhosq = 0 .. 225
there is no requirement on rectilinear sampling or array
dimensionality
phi : numpy.ndarray, optional
(1 - c^2 r^2)^.5
computed if not provided
many surface types utilize phi; its computation can be
de-duplicated by passing the optional argument
Returns
-------
numpy.ndarray
surface sag
"""
if phi is None:
csq = c * c
phi = np.sqrt(1 - csq * rhosq)
return (c * rhosq) / (1 + phi)
def refract(n, nprime, S, r):
"""Use Newton-Raphson iteration to solve Snell's law for the exitant direction cosines.
Parameters
----------
n : float
preceeding index of refraction
nprime : float
following index of refraction
S : numpy.ndarray
shape (3,) or (N,3), any float dtype
(k,l,m) incident direction cosines
r : numpy.ndarray
shape (3,) or (N,3), any float dtype
surface normals (Fx, Fy, 1)
Returns
-------
numpy.ndarray
Sprime, a length 3 vector containing the exitant direction cosines
"""
mu = n/nprime
musq = mu * mu
cosI = _multi_dot(r, S)
cosIsq = cosI * cosI
# the inline newaxis-es are terrible for readability, but serve a performance purpose
# broadcast the square root to 2D, so that fewer very expensive sqrt ops are done
# then, in the second term, broadcast cosI for compatability with S and r
# since it is needed there
first_term = np.sqrt(1 - musq * (1 - cosIsq))[:, np.newaxis] * r
second_term = mu * (S - cosI[:, np.newaxis] * r)
return first_term + second_term
def noll_to_nm(idx):
"""Convert Noll Z to (n, m) two-term index."""
# I don't really understand this code, the math is inspired by POPPY
# azimuthal order
n = int(np.ceil((-1 + np.sqrt(1 + 8 * idx)) / 2) - 1)
if n == 0:
m = 0
else:
# this is sort of a rising factorial to use that term incorrectly
nseries = int((n + 1) * (n + 2) / 2)
res = idx - nseries - 1
if is_odd(idx):
sign = -1
else:
sign = 1
if is_odd(n):
ms = [1, 1]
else:
ms = [0]
for i in range(n // 2):
ms.append(ms[-1] + 2)
ms.append(ms[-1])
m = ms[res] * sign
return n, m
def conic_sag_der(c, kappa, rho, phi=None):
"""Sag of a spherical surface.
Parameters
----------
c : float
surface curvature
kappa : float
conic constant, 0=sphere, 1=parabola, etc
rho : numpy.ndarray
radial coordinate
e.g. for a 15 mm half-diameter optic,
rho = 0 .. 15
there is no requirement on rectilinear sampling or array
dimensionality
phi : numpy.ndarray, optional
(1 - (1+kappa) c^2 r^2)^.5
computed if not provided
many surface types utilize phi; its computation can be
de-duplicated by passing the optional argument
Returns
-------
numpy.ndarray
surface sag
"""
if phi is None:
csq = c**2
rhosq = rho * rho
phi = np.sqrt(1 - (1 + kappa) * csq * rhosq)
return (c * rho) / phi
def fringe_to_nm(idx):
"""Convert Fringe Z to (n, m) two-term index."""
m_n = 2 * (np.ceil(np.sqrt(idx)) - 1) # sum of n+m
g_s = (m_n /
2)**2 + 1 # start of each group of equal n+m given as idx index
n = m_n / 2 + np.floor((idx - g_s) / 2)
m = (m_n - n) * (1 - np.mod(idx - g_s, 2) * 2)
return int(n), int(m)
def off_axis_conic_sigma(c, kappa, r, t, dx, dy=0):
"""Lowercase sigma (direction cosine projection term) for an off-axis conic.
See Eq. (5.2) of oe-20-3-2483.
Parameters
----------
c : float
axial curvature of the conic
kappa : float
conic constant
r : numpy.ndarray
radial coordinate, where r=0 is centered on the off-axis section
t : numpy.ndarray
azimuthal coordinate
dx : float
shift of the surface in x with respect to the base conic vertex,
mutually exclusive to dy (only one may be nonzero)
use dx=0 when dy != 0
dy : float
shift of the surface in y with respect to the base conic vertex
Returns
-------
sigma(r,t)
"""
if dy != 0 and dx != 0:
raise ValueError('only one of dx/dy may be nonzero')
if dx != 0:
s = dx
oblique_term = 2 * s * r * np.cos(t)
else:
s = dy
oblique_term = 2 * s * r * np.sin(t)
aggregate_term = r * r + oblique_term + s * s
csq = c * c
num = np.sqrt(1 - (1 + kappa) * csq * aggregate_term)
den = np.sqrt(1 - kappa * csq * aggregate_term) # flipped sign, 1-kappa
return num / den
def f_q2d(n, m):
"""Lowercase f term for 2D-Q polynomials. oe-20-3-2483 Eq. (A.18b).
Parameters
----------
n : int
radial order
m : int
azimuthal order
Returns
-------
float
f
"""
if n == 0:
return np.sqrt(F_q2d(n=0, m=m))
else:
return np.sqrt(F_q2d(n, m) - g_q2d(n - 1, m)**2)
def off_axis_conic_sag(c, kappa, r, t, dx, dy=0):
"""Sag of an off-axis conicoid.
Parameters
----------
c : float
axial curvature of the conic
kappa : float
conic constant
r : numpy.ndarray
radial coordinate, where r=0 is centered on the off-axis section
t : numpy.ndarray
azimuthal coordinate
dx : float
shift of the surface in x with respect to the base conic vertex,
mutually exclusive to dy (only one may be nonzero)
use dx=0 when dy != 0
dy : float
shift of the surface in y with respect to the base conic vertex
Returns
-------
numpy.ndarray
surface sag, z(x,y)
"""
if dy != 0 and dx != 0:
raise ValueError('only one of dx/dy may be nonzero')
if dx != 0:
s = dx
oblique_term = 2 * s * r * np.cos(t)
else:
s = dy
oblique_term = 2 * s * r * np.sin(t)
aggregate_term = r * r + oblique_term + s * s
num = c * aggregate_term
csq = c * c
den = 1 + np.sqrt(1 - (1 + kappa) * csq * aggregate_term)
return num / den
def phi_spheroid(c, k, rhosq):
"""'phi' for a spheroid.
phi = sqrt(1 - c^2 rho^2)
Parameters
----------
c : float
curvature, reciprocal radius of curvature
k : float
kappa, conic constant
rhosq : numpy.ndarray
squared radial coordinate (non-normalized)
Returns
-------
numpy.ndarray
phi term
"""
csq = c * c
return np.sqrt(1 - (1 + k) * csq * rhosq)
def _establish_axis(P1, P2):
"""Given two points, establish an axis between them.
Parameters
----------
P1 : numpy.ndarray
shape (3,), any float dtype
first point
P2 : numpy.ndarray
shape (3,), any float dtype
second point
Returns
-------
numpy.ndarray, numpy.ndarray
P1 (same exact PyObject) and direction cosine from P1 -> P2
"""
diff = P2 - P1
euclidean_distance = np.sqrt(diff**2).sum()
num = diff
den = euclidean_distance
return num / den
def zernikes_to_magnitude_angle_nmkey(coefs):
"""Convert Zernike polynomial set to a magnitude and phase representation.
Parameters
----------
coefs : list of tuples
a list looking like[(1,2,3),] where (1,2) are the n, m indices and 3 the coefficient
Returns
-------
dict
dict keyed by tuples of (n, |m|) with values of (rho, phi) where rho is the magnitudes, and phi the phase
"""
def mkary(): # default for defaultdict
return list()
combinations = defaultdict(mkary)
# for each name and coefficient, make a len 2 array. Put the Y or 0 degree values in the first slot
for n, m, coef in coefs:
m2 = abs(m)
key = (n, m2)
combinations[key].append(coef)
for key, value in combinations.items():
if len(value) == 1:
magnitude = value[0]
angle = 0
else:
magnitude = np.sqrt(sum([v**2 for v in value]))
angle = np.degrees(np.arctan2(*value))
combinations[key] = (magnitude, angle)
return dict(combinations)
def generate_finite_ray_fan(nrays,
na,
P=0,
min_na=None,
azimuth=90,
yangle=0,
xangle=0,
n=1,
distribution='uniform'):
"""Generate a 1D fan of rays.
Parameters
----------
nrays : int
the number of rays in the fan
na : float
object-space numerical aperture
P : numpy.ndarray
length 3 vector containing the position from which the rays emanate
min_na : float, optional
minimum NA for the beam, -na if None
azimuth: float
angle in the XY plane, degrees. 0=X ray fan, 90=Y ray fan
yangle : float
propagation angle of the chief/gut ray with respect to the Y axis, clockwise
xangle : float
propagation angle of the gut ray with respect to the X axis, clockwise
n : float
refractive index at P (1=vacuum)
distribution : str, {'uniform', 'random', 'cheby'}
The distribution to use when placing the rays
a uniform distribution has rays which are equally spaced from minr to maxr,
random has rays randomly distributed in minr and maxr, while cheby has the
Cheby-Gauss-Lobatto roots as its locations from minr to maxr
Returns
-------
numpy.ndarray, numpy.ndarray
"P" and "S" variables, positions and direction cosines of the rays
"""
# TODO: revisit this; tracing a parabola from the focus, the output
# ray spacing is not uniform as it should be. Or is this some manifestation
# of the sine condition?
# more likely it's the square root since it hides unless the na is big
P = _ensure_P_vec(P)
distribution = distribution.lower()
if min_na is None:
min_na = -na
max_t = np.arcsin(na / n)
min_t = np.arcsin(min_na / n)
if distribution == 'uniform':
t = np.linspace(min_t, max_t, nrays, dtype=config.precision)
elif distribution == 'random':
t = np.random.uniform(low=min_t, high=max_t,
size=nrays).astype(config.precision)
# use the even function for the y direction cosine,
# use trig identity to compute the z direction cosine
l = np.sin(t) # NOQA
m = np.sqrt(1 - l * l) # NOQA
k = np.array([0.], dtype=t.dtype)
k = np.broadcast_to(k, (nrays, ))
if azimuth == 0:
k, l = l, k # NOQA swap Y and X axes
S = np.stack([k, l, m], axis=1)
if yangle != 0 and xangle != 0:
R = make_rotation_matrix((0, yangle, -xangle))
# newaxis for batch matmul, squeeze needed for size 1 dim after
S = np.matmul(R, S[..., np.newaxis]).squeeze()
# need to see a copy of P for each ray, -> add empty dim and broadcast
P = P[np.newaxis, :]
P = np.broadcast_to(P, (nrays, 3))
return P, S
def Qbfs(n, x):
"""Qbfs polynomial of order n at point(s) x.
Parameters
----------
n : int
polynomial order
x : numpy.array
point(s) at which to evaluate
Returns
-------
numpy.ndarray
Qbfs_n(x)
"""
# to compute the Qbfs polynomials, compute the auxiliary polynomial P_n
# recursively. Simultaneously use the recurrence relation for Q_n
# to compute the intermediary Q polynomials.
# for input x, transform r = x ^ 2
# then compute P(r) and consequently Q(r)
# and scale outputs by Qbfs = r*(1-r) * Q
# the auxiliary polynomials are the jacobi polynomials with
# alpha,beta = (-1/2,+1/2),
# also known as the chebyshev polynomials of the third kind, V(x)
# the first two Qbfs polynomials are
# Q_bfs0 = x^2 - x^4
# Q_bfs1 = 1/19^.5 * (13 - 16 * x^2) * (x^2 - x^4)
rho = x**2
# c_Q is the leading term used to convert Qm to Qbfs
c_Q = rho * (1 - rho)
if n == 0:
return c_Q # == x^2 - x^4
if n == 1:
return 1 / np.sqrt(19) * (13 - 16 * rho) * c_Q
# c is the leading term of the recurrence relation for P
c = 2 - 4 * rho
# P0, P1 are the first two terms of the recurrence relation for auxiliary
# polynomial P_n
P0 = np.ones_like(x) * 2
P1 = 6 - 8 * rho
Pnm2 = P0
Pnm1 = P1
# Q0, Q1 are the first two terms of the recurrence relation for Qm
Q0 = np.ones_like(x)
Q1 = 1 / np.sqrt(19) * (13 - 16 * rho)
Qnm2 = Q0
Qnm1 = Q1
for nn in range(2, n + 1):
Pn = c * Pnm1 - Pnm2
Pnm2 = Pnm1
Pnm1 = Pn
g = g_qbfs(nn - 1)
h = h_qbfs(nn - 2)
f = f_qbfs(nn)
Qn = (Pn - g * Qnm1 - h * Qnm2) * (
1 / f) # small optimization; mul by 1/f instead of div by f
Qnm2 = Qnm1
Qnm1 = Qn
# Qn is certainly defined (flake8 can't tell the previous ifs bound the loop
# to always happen once)
return Qn * c_Q # NOQA
def Qbfs_sequence(ns, x):
"""Qbfs polynomials of orders ns at point(s) x.
Parameters
----------
ns : Iterable of int
polynomial orders
x : numpy.array
point(s) at which to evaluate
Returns
-------
generator of numpy.ndarray
yielding one order of ns at a time
"""
# see the leading comment of Qbfs for some explanation of this code
# and prysm:jacobi.py#jacobi_sequence the "_sequence" portion
ns = list(ns)
min_i = 0
rho = x**2
# c_Q is the leading term used to convert Qm to Qbfs
c_Q = rho * (1 - rho)
if ns[min_i] == 0:
yield np.ones_like(x) * c_Q
min_i += 1
if min_i == len(ns):
return
if ns[min_i] == 1:
yield 1 / np.sqrt(19) * (13 - 16 * rho) * c_Q
min_i += 1
if min_i == len(ns):
return
# c is the leading term of the recurrence relation for P
c = 2 - 4 * rho
# P0, P1 are the first two terms of the recurrence relation for auxiliary
# polynomial P_n
P0 = np.ones_like(x) * 2
P1 = 6 - 8 * rho
Pnm2 = P0
Pnm1 = P1
# Q0, Q1 are the first two terms of the recurrence relation for Qbfs_n
Q0 = np.ones_like(x)
Q1 = 1 / np.sqrt(19) * (13 - 16 * rho)
Qnm2 = Q0
Qnm1 = Q1
for nn in range(2, ns[-1] + 1):
Pn = c * Pnm1 - Pnm2
Pnm2 = Pnm1
Pnm1 = Pn
g = g_qbfs(nn - 1)
h = h_qbfs(nn - 2)
f = f_qbfs(nn)
Qn = (Pn - g * Qnm1 - h * Qnm2) * (
1 / f) # small optimization; mul by 1/f instead of div by f
Qnm2 = Qnm1
Qnm1 = Qn
if ns[min_i] == nn:
yield Qn * c_Q
min_i += 1
if min_i == len(ns):
return
def ansi_j_to_nm(idx):
"""Convert ANSI single term to (n,m) two-term index."""
n = int(np.ceil((-3 + np.sqrt(9 + 8 * idx)) / 2))
m = 2 * idx - n * (n + 2)
return n, m
def off_axis_conic_sigma_der(c, kappa, r, t, dx, dy=0):
"""Derivatives of 1/off_axis_conic_sigma.
See Eq. (5.2) of oe-20-3-2483.
Parameters
----------
c : float
axial curvature of the conic
kappa : float
conic constant
r : numpy.ndarray
radial coordinate, where r=0 is centered on the off-axis section
t : numpy.ndarray
azimuthal coordinate
dx : float
shift of the surface in x with respect to the base conic vertex,
mutually exclusive to dy (only one may be nonzero)
use dx=0 when dy != 0
dy : float
shift of the surface in y with respect to the base conic vertex
Returns
-------
numpy.ndarray, numpy.ndarray
d/dr(z), d/dt(z)
"""
if dy != 0 and dx != 0:
raise ValueError('only one of dx/dy may be nonzero')
cost = np.cos(t)
sint = np.sin(t)
if dx != 0:
s = dx
oblique_term = 2 * s * r * cost
ddr_oblique = 2 * r + 2 * s * cost
# I accept the evil in writing this the way I have
# to deduplicate the computation
ddt_oblique_ = r * (-s) * sint
else:
s = dy
oblique_term = 2 * s * r * sint
ddr_oblique = 2 * r + 2 * s * sint
ddt_oblique_ = r * s * cost
aggregate_term = r * r + oblique_term + s * s
csq = c * c
# d/dr first
phi_kernel = (1 + kappa) * csq * aggregate_term
phi = np.sqrt(1 - phi_kernel)
notquitephi_kernel = kappa * csq * aggregate_term
notquitephi = np.sqrt(1 + notquitephi_kernel)
num = csq * (1 + kappa) * ddr_oblique * notquitephi
den = 2 * (1 - phi_kernel)**(3 / 2)
term1 = num / den
num = csq * kappa * ddr_oblique
den = 2 * phi * notquitephi
term2 = num / den
dr = term1 + term2
# d/dt
num = csq * (1 + kappa) * ddt_oblique_ * notquitephi
den = (1 - phi_kernel)**(3 / 2) # phi^3?
term1 = num / den
num = csq * kappa * ddt_oblique_
den = phi * notquitephi
term2 = num / den
dt = term1 + term2 # minus in writing, but sine/cosine
return dr, dt
def off_axis_conic_der(c, kappa, r, t, dx, dy=0):
"""Radial and azimuthal derivatives of an off-axis conic.
Parameters
----------
c : float
axial curvature of the conic
kappa : float
conic constant
r : numpy.ndarray
radial coordinate, where r=0 is centered on the off-axis section
t : numpy.ndarray
azimuthal coordinate
dx : float
shift of the surface in x with respect to the base conic vertex,
mutually exclusive to dy (only one may be nonzero)
use dx=0 when dy != 0
dy : float
shift of the surface in y with respect to the base conic vertex
Returns
-------
numpy.ndarray, numpy.ndarray
d/dr(z), d/dt(z)
"""
if dy != 0 and dx != 0:
raise ValueError('only one of dx/dy may be nonzero')
cost = np.cos(t)
sint = np.sin(t)
if dx != 0:
s = dx
oblique_term = 2 * s * r * cost
ddr_oblique = 2 * r + 2 * s * cost
# I accept the evil in writing this the way I have
# to deduplicate the computation
ddt_oblique_ = r * (-s) * sint
ddt_oblique = 2 * ddt_oblique_
else:
s = dy
oblique_term = 2 * s * r * sint
ddr_oblique = 2 * r + 2 * s * sint
ddt_oblique_ = r * s * cost
ddt_oblique = 2 * ddt_oblique_
aggregate_term = r * r + oblique_term + s * s
csq = c * c
c3 = csq * c
# d/dr first
num = c * ddr_oblique
phi_kernel = (1 + kappa) * csq * aggregate_term
phi = np.sqrt(1 - phi_kernel)
phip1 = 1 + phi
phip1sq = phip1 * phip1
den = phip1
term1 = num / den
num = c3 * (1 + kappa) * ddr_oblique * aggregate_term
den = (2 * phi) * phip1sq
term2 = num / den
dr = term1 + term2
# d/dt
num = c * ddt_oblique
den = phip1
term1 = num / den
num = c3 * (1 + kappa) * ddt_oblique_ * aggregate_term
den = phi * phip1sq
term2 = num / den
dt = term1 + term2
return dr, dt