CoCalc -- polarization tensors in detector's frame.ipynb

Polarization Tensors in Detector's Frame

Project: sogro

Path: polarization tensors in detector's frame.ipynb

Views: ²³⁷

Kernel: SageMath 8.2

In [1]:

%display latex

In [2]:

theta, phi, psi = var('theta, phi, psi')

In [3]:

R_z = matrix([[cos(x),-sin(x),0],[sin(x),cos(x),0],[0,0,1]])

In [4]:

R_y = matrix([[cos(x),0,sin(x)],[0,1,0],[-sin(x),0,cos(x)]])

In [5]:

R1 = R_z(x=phi); R1

In [6]:

R2 = R_y(x=theta); R2

In [7]:

R3 = R_z(x=psi); R3

In [8]:

Lambda = R1*R2*R3; Lambda

In [9]:

Lambda_inverse = R3(psi=-psi)*R2(theta=-theta)*R1(phi=-phi); Lambda_inverse.simplify_trig()

In [10]:

(Lambda*Lambda_inverse).simplify_trig()

In [11]:

e = [
    matrix([[1,0,0],[0,-1,0],[0,0,0]]),
    matrix([[0,1,0],[1,0,0],[0,0,0]]),
    matrix([[1,0,0],[0,1,0],[0,0,0]]),
    matrix([[0,0,0],[0,0,0],[0,0,sqrt(2)]]),
    matrix([[0,0,1],[0,0,0],[1,0,0]]),
    matrix([[0,0,0],[0,0,1],[0,1,0]])
]; e

In [12]:

e_bar = []
for i in range(6):
    e_bar.append(Lambda*e[i]*Lambda_inverse)
    show(e_bar[i].simplify_trig())

In [13]:

a = matrix(SR, 6,6)
for i in range(6):
    for j in range(6):
        a[i,j] = sum(sum(e[i].elementwise_product(Lambda*e[j]*Lambda.transpose()/2))).simplify_trig()
a

In [14]:

T = Spherical('radius', ['inclination', 'azimuth'])

In [15]:

r = var('r')

In [16]:

T.transform(radius=r, azimuth=phi, inclination=theta)

In [17]:

# cm = colormaps.hsv
# def c(x,y):
#     return float((x+y+x*y)/15) % 1
# pol = ['+', 'cross', 'b', 'l', 'x', 'y']
# for j,k in [(0,0),(1,1),(2,2),(0,1),(1,2),(2,0)]:
#     for i in range(2):
#         print('polarization: '+pol[i]+', ('+str(j+1)+','+str(k+1)+') component')
#         show(plot3d(e_bar[i][j][k], (theta, 0, pi), (phi, 0, 2*pi), transformation=T, color=(c,cm)))

In [0]: