Instrument Science Report NICMOS 98 - 017
Methodologies to Calibrating NICMOS
Polarimetry Characteristics
Lisa Mazzuca, Bill Sparks, Dave Axon
November 24, 1998
ABSTRACT
Astronomical imaging polarimetry commonly utilizes a sequence of images taken with a
polarizing filter oriented at different angles. For example, each of the Faint Object Camera (FOC), Near Infra-Red Camera and Multi Object Spectrometer (NICMOS) and the
Advanced Camera for Surveys (ACS) instruments on the Hubble Space Telescope use three
separate polarizing spectral elements oriented at approximately 60 degrees to one another
to obtain a set of images that may subsequently be processed to derive polarization information on the astronomical object.
Here we present the general linear theory of analysis for the case of three non-ideal polarizers. There are several ways to express the calibration of such polarizing elements and a
variety of commonly used conventions. We present a summary of three of the more commonly encountered conventions, along with transformations between the different
calibration systems. Specific examples are also cited for reference based on the short
wavelength NICMOS polarizer characteristics. We present the transformations between
three images taken with independent polarizers and the corresponding set of Stokes
parameter images.
1. Introduction
In order to calibrate data obtained with a set of polarizers, there are three quantities
needed. They are the throughput to unpolarized light, the efficiency of the element as a
polarizer, and the orientation of the polarizer. These elements can be expressed in a polarization reduction algorithm to form a solution containing the polarization characteristics
of the incoming beam (i.e., the Stokes parameters I, Q, and U).
1
The general form of the equation for polarimetry data reduction is expressed as1
′
I k = A k I + ε k ( B k Q cos 2φ k + C k U sin 2φ k )
(1)
where I’k is the emerging light intensity from the kth polarizer, Ak, Bk, and Ck are the transmission coefficients, εk is the polarizing efficiency, and φk is the position angle of the kth
polarizer. This linear equation captures the observed signal from a polarized source of
intensity I and linear Stokes parameters Q and U, which describe the state of polarization
for each polarizer. The transmittance expressions for Ak, Bk, and Ck will differ depending
on the characteristics of the polarizing elements.
For a set of three polarizers, as used with the FOC, NICMOS, and ACS instruments, equation (1) becomes a straightforward linear system of three equations with three unknowns.
After solving the system of equations to derive the Stokes parameters, the degree of polarization P and the position angle θ can be calculated as
2
2
Q +U
P = ------------------------,
I
1
U
θ = --- atan ---2
Q
(2)
There are three conventions presented below, each with differing transmittance expressions, that can be used to characterize the polarizing elements. The differences in the
expressions for the transmission coefficients Ak, Bk, and Ck are dependent on the method
chosen to characterize the polarizer. Although each convention involves different expressions, the solution is the same for a given polarizing set.
2. Theory and Convention
Any one polarizer is characterized by two calibration parameters which are the parallel
light transmission and the perpendicular light transmission, and a third parameter which is
its position angle. Both the transmission expressions and the polarizer efficiency expression are composed of these two quantities. The material presented below shows the
transformations between three commonly used conventions.
In the natural theoretical approach, the polarizer splits the unpolarized light into two
orthogonally polarized beams, one of which passes straight through onto the detector. The
representation for the transmission coefficients in this case is presented in Hines, et al
2
paper entitled “The Polarimetric Capabilities of NICMOS2”. The transmission coefficients, including the polarizing efficiency, are:
(tk )
(tk )
A k = -------- ( 1 + l k ) , B k = ε k -------- ( 1 + l k ) cos 2φ,
2
2
(tk )
C k = ε k --------( 1 + l k )sin 2 φ ( 3 )
2
where tk is the fraction of light transmitted for a 100% polarized input aligned with the
polarizer’s axis, and lk is the fraction transmitted when the incoming light is perpendicular
to the axis of the polarizer.
The polarizer efficiency ε is defined as the following:
ε = ( S par – S perp ) ⁄ ( S par + S perp )
(4)
where Spar = tk and Sperp = tk lk. Therefore, the equation for εk becomes
1 – lk
t k – ( t k ⋅ lk )
ε k = -------------------------- = ------------1 + lk
t k – ( t k ⋅ lk )
(5)
and the reduction equation (1) becomes:
I
′
k
tk
1 – lk t k
tk
= ---- ( 1 + l k )I + ------------- ⋅  ---- ( 1 + l k ) ⋅ cos 2φ k Q + ---- ( 1 + l k ) sin 2φ k U ( 6 )

2
1 + lk  2
2
which reduces to
tk
1 – lk
′
I k = ---- ( 1 + l k )  I + ------------- ( cos 2φ k Q + sin 2φ k U )


2
1 + lk
(7)
The full expression for the transmission coefficients from equation (3) is then
(tk )
1 – lk
A k = -------- ( 1 + l k ) , B k = ------------- cos 2φ,
2
1 + lk
1 – lk
C k = ------------- sin 2 φ
1 + lk
(8)
The second convention contains transmission coefficient expressions that apply to a pair of
identical polarizers in series. This convention can be used as a practical way of characterizing a polarizer set in the laboratory (e.g., Serkowski, 1962).
3
In this case the values for Ak, Bk, and Ck, including the polarizing efficiency, are the
following3:
Ak =
T k + Lk
T k + Lk
T k + Lk
-----------------, B k = ε k ----------------- cos 2φ, C k = ε k ----------------- sin 2 φ
2
2
2
(9)
where Tk is the transmittance of unpolarized light by the two identical analyzers oriented
parallel, and Lk is the transmittance by the two identical perpendicularly oriented analyzers. The polarizer efficiency ε k is defined as
εk =
T k – Lk
----------------T k + Lk
( 10 )
The reduction equation (1) now becomes
T k + Lk
T k + Lk T k – Lk
T k + Lk T k – Lk
′
- I + ----------------- ------------------ ⋅ cos 2φ k Q + ----------------- ------------------ sin 2φ k U
I k = ----------------2
2
T k + Lk
2
T k + Lk
( 11 )
and reduces to
′
Ik =
T k – Lk
T k + Lk
( Q cos 2φ k + U sin 2φ k )
----------------- ⋅ I + ----------------2
2
( 12 )
Therefore, the transmission expressions in equation (9) become
T k – Lk
T k + Lk
cos 2φ, C k =
A k = -----------------, B k = ----------------2
2
T k – Lk
----------------- sin 2 φ
2
( 13 )
Another generic approach is to simply let the transmittances be defined as zk with a polarizing efficiency of εk. The expressions for Ak, Bk, and Ck, including the polarizing
efficiency, then become
1
1
1
A k = --- z k , B k = --- ε k z k cos 2φ, C k = --- ε k z k sin 2 φ
2
2
2
4
( 14 )
with Equation (1) transforming to
I
′
k
1
= --- z k ( I + ε k Q cos 2φ k + ε k U sin 2φ k )
2
( 15 )
3. Conversions
Equating the transmission coefficients from equations (8), (13), and (14) will yield general
transformations between the polarization conventions stated. That is, by setting any two
expressions for Ak, Bk, and Ck (accounting for the polarizing efficiency) equal to each
other and then solving the system of equations for the various transmission variables, a
straightforward relation can be made. The relations are as follows:
tk in terms of Tk and Lk:
tk =
T k + Lk
T k – Lk
------------------ + ----------------2
2
( 16 )
lk in terms of Tk and Lk:
T k + Lk – T k – Lk
l k = ------------------------------------------------------------T k + Lk + T k – Lk
( 17 )
Tk in terms of tk and lk:
2
2
T k = 0.5t k ( 1 + l k )
( 18 )
Lk in terms of tk and lk:
2
( 19 )
Lk = t k lk
zk in terms of Tk or Lk:
Tk
z k = 2 ------------------- 1 + ε 2
k

or
Lk
z k = 2 ------------------- 1 – ε 2
k

( 20 )
zk in terms of tk and/or lk:
t
k
z k = 2 ------------------( 1 + εk )
or
t k lk
z k = 2 ------------------( 1 – εk )
( 21 )
5
tk in terms of zk:
1
t k = --- z k ( 1 + ε k )
2
( 22 )
lk in terms of zk:
1
l k = ------- z k ( 1 – ε k )
2t k
( 23 )
Tk in terms of zk:
1 2
2
T k = --- z k ( 1 + ε k )
4
( 24 )
Lk in terms of zk:
1 2
2
L k = --- z k ( 1 – ε k )
4
( 25 )
4. Solutions for 3-polarizer case
All three conventions used to express the polarization reduction algorithm reduce to a set
of three equations with three unknowns. The solution results in the Stokes parameters I, Q,
and U for the incoming light. Mathematically, the solution is derived from a matrix product of the inverse of the transmission coefficients matrix with the vector of intensities
observed through the three polarizers.
Let x be the incoming Stokes vector (I, Q, U), and let the vector of observed intensities be
b = (I1, I2, I3). Then analytically, the linear equation [A]x = b can be solved for the incoming Stokes vector x, which yields the equation x=[A-1]b. The matrix [A] contains the
transmission coefficients for each Stokes parameter I, Q, and U. In general the matrix [A]
corresponding to equation (1) is
A1
ε1 B1
ε1 C 1
A = A2
ε2 B2
ε2 C 2
A3
( 26 )
ε3 B3 ε3 C 3
6
More explicitly, for the single polarizer convention, the matrix [A] is
t1
---- ( 1 + l 1 )
2
1–l t
------------1⋅- ---1-( 1 + l 1 ) cos 2φ 1
1 + l1 2
1–l t
------------1⋅- ---1- ( 1 + l 1 ) sin ( 2φ 1 )
1 + l1 2
t
A = ---2- ( 1 + l )
2
2
1–l t
------------2⋅- ---2-( 1 + l 2 )cos 2φ 2
1 + l2 2
1–l t
------------2⋅- ---2- ( 1 + l 2 ) sin ( 2φ 2 )
1 + l2 2
t3
---- ( 1 + l 3 )
2
1–l t
------------⋅3- ---3- ( 1 + l 3 )cos 2φ 3
1 + l3 2
1–l t
------------3⋅- ---3- ( 1 + l 3 ) sin ( 2φ 3 )
1 + l3 2
( 27 )
For the second convention of dual polarizers [A] is
A =
T 1 + L1
-----------------2
T 1 – L1
----------------⋅- cos 2φ 1
2
T 1 – L1
----------------⋅- sin 2 φ 1
2
T 2 + L2
-----------------2
T 2 – L2
----------------⋅- cos 2φ 2
2
T 2 – L2
----------------⋅- sin 2 φ 2
2
T 3 + L3
-----------------2
T 3 – L3
----------------⋅- cos 2φ 3
2
T 3 – L3
-----------------⋅- sin 2 φ 3
2
( 28 )
and for the third convention [A] is
z
----1
2
z2
A = ---2
z
----3
2
z 1 ε 1cos 2φ 1
--------------------------2
z 2 ε 2cos 2φ 2
--------------------------2
z 3 ε 3cos 2φ 3
--------------------------2
z 1 ε 1 sin 2φ 1
-------------------------2
z 2 ε 2 sin 2φ 2
-------------------------2
z 3 ε 3 sin 2φ 3
-------------------------2
( 29 )
All three expressions for the matrix [A]can then be inverted to find the same solution to
the original reduction equation. For simplicity, we present the analytical solution to the
third convention using the matrix from equation (29).
Using the reduction algorithm according to the convention in equation (15), let the following conditions exist:
2 ′
- I be the set of three observed intensities, and
let I k∗ = ---z k
k
define an incoming Stokes vector to be
{ S1 } { S2 } { S3 } ≡ I Q U
7
.
The matrix [A] from equation (29) becomes
1 ε 1cos 2φ 1 ε 1sin 2 φ 1
( 30 )
A = 1 ε 2cos 2φ 2 ε 2sin 2 φ 2
1 ε 3cos 2φ 3 ε 3sin 2 φ 3
Solving for the incoming Stokes vector yields
3
Sk =
∑ akj I j∗
( 31 )
j=1
where akj represent the elements of the inverted matrix
a kj
ε 2 ε 3sin( – 2φ 2 + 2φ 3 ) ε 3ε 1sin( – 2φ 3 + 2φ 1 )
1
= ---- ε 2 sin 2φ 2 – ε 3 sin 2φ 3 ε 3 sin 2φ 3 – ε 1 sin 2φ 1
Ω
ε 3 cos 2φ 3 – ε 2 cos 2φ 2 ε 1 cos 2φ 1– ε 3 cos 2φ 3
ε 1ε 2sin ( – 2φ 1 + 2φ 2 )
ε 1 sin 2φ 1 – ε 2 sin 2φ 2 ( 32 )
ε 2 cos 2φ 2– ε 1 cos 2φ 1
where
Ω = ε 1 ε 2 sin ( – 2φ 1 + 2φ 2 ) + ε 2 ε 3 sin ( – 2φ 2 + 2φ 3 ) + ε 3 ε 1 sin ( – 2φ 3 + 2φ 1 ) ( 33 )
The solution is a vector consisting of the Stokes parameters for the incoming beam.
Note that for the ideal configuration when ε 1 = ε 2 = ε 3 = 1 and
°
°
°
( φ 1 = 180 ) ,( φ 2 = 60 ) ,( φ 3 = 120 ) this reduces to the equation given in the NICMOS
handbook4 with the exception of a sign convention difference
I ∗
1
I
1 1
1
1
Q = --3- 2 – 1 – 1 • I 2∗
0 3 – 3
U
I ∗
( 34 )
3
8
5. Numerical Example: NICMOS Polarizers
In the case of NICMOS, the three polarizers have been characterized by Hines (Hines, et
al,1997). Referring to the Hines’ paper previously cited, let the characteristics of the polarizers for POL0S, POL120S, and POL240S be the following:
FILTER
ϕk
εk
tk
lk
POL0S
1.42
0.9717
0.7760
0.0144
POL120S
116.30
0.4771
0.5946
0.3540
POL240S
258.72
0.7682
0.7169
0.1311
Table 1. Filter Characteristics for Convention One
The resulting coefficient matrix (as computed from equation 27) is
0.3936 0.3819 0.0189
A = 0.4025 – 0.1166 – 0.1526
0.4054 – 0.2876 0.1195
( 35 )
which can be used to compute the expected (I1, I2, I3) for a given set (I, Q, U). The
inverted matrix is numerically calculated to be
0.8754 0.7735 0.8488
–1
A = 1.6647 – 0.5957 – 1.0246
1.0368 – 4.0582 3.0227
( 36 )
which can be used to compute the Stokes parameters (I, Q, U) from a set of observations
(I1, I2, I3).
To complete this example, let the incoming Stokes vector be a beam fully polarized at a
position angle of 45 degrees
x =
1
1
=
cos ( 2 ⋅ 45° )
0
sin ( 2 ⋅ 45° )
1
( 37 )
Then the observed intensity vector, b, is
0.3936 0.3819 0.0189
1
0.4125
b = A x = 0.4025 – 0.1166 – 0.1526 ⋅ 0 = 0.2500
0.4054 – 0.2876 0.1195
1
0.5249
9
( 38 )
Conversely, let the observed intensity vector be
0.4125
b = 0.2500
0.5249
( 39 )
Then
x = A
–1
0.8867 0.7735 0.8487
0.4125
1
b = 1.6864 – 0.5957 – 1.0246 ⋅ 0.2500 = 0
1.0504 – 4.0582 3.0227
0.5249
1
( 40 )
thus recovering the incoming Stokes vector (I, Q, U).
Of course with the mathematics now presented for two other conventions, the same matrix
can be found using the same initial coefficients, but with the expressions from the respective conventions. From equations (18) through (21), Tk,Lk, and zk are computed from the
original values of tk and lk to be:
FILTER
Tk
Lk
zk
POL0S
0.3012
0.0087
0.7871
POL120S
0.1989
0.1252
0.8051
POL240S
0.2614
0.0674
0.8109
Table 2. Filter Characteristics for Conventions Two and Three
The matrix with respect to the second convention stated uses the coefficient matrix from
equation (28), and is calculated to be
0.3936 0.3819 0.0189
A = 0.4025 – 0.1166 – 0.1526
0.4054 – 0.2876 0.1195
( 41 )
which is the same as with the first convention, as shown in equation (35).
10
Using the third convention stated, the coefficient matrix from equation (29) is used, and is
also calculated to be
0.3936 0.3819 0.0189
=
A
0.4025 – 0.1166 – 0.1526
0.4054 – 0.2876 0.1195
( 42 )
which again corresponds to the inverted matrices from the first two conventions from
equations (35) and (41).
6. Summary
Several conventions exist that can equally be used to solve the reduction equation for polarimetry. The solution produces the Stokes parameters (I,Q,U) for the incoming beam,
which once known, can be used to calculate the degree and percentage of polarization and
the position angle.
7. References
1N.Carleton,ed.
Methods of Experimental Physics, Polarization Techniques. Academic Press, New
York, 1974, p. 364.
2D.Hines., et al. “The Polarimetric Capabilities of NICMOS”. (Steward Observatory, University of Arizona), 1997.
3Z.Kopal, ed. Advances in Astronomy and Astrophysics, Polarization of Starlight. Academic Press,
New York, 1962, p295.
4D. Axon, et al. NICMOS Instrument Handbook. Version 2.0 (Baltimore: STScI), 1997.
11