---
title: "Why transmitted and thermal polarization turn opposite ways"
description: "Build Stokes I, Q and U from two electric-field modes, then derive the perpendicular axes of transmission and thermal emission."
canonical_url: "https://sandboxphysics.com/stories/interstellar-dust/aligned-grains-and-two-polarizations/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/interstellar-dust/aligned-grains-and-two-polarizations/"
last_modified: "2026-11-04"
---

[Interstellar dust, infrared light and polarization](https://sandboxphysics.com/stories/interstellar-dust/)

Interstellar dust · L2

# Why transmitted and thermal polarization turn opposite ways

Sandbox Physics · 2026-11-04

An aligned material can absorb one electric-field direction more strongly than another. The surviving beam favors the weakly absorbed direction; thermal emission along that same sightline favors the strongly emitting direction. To turn this picture into a polarization angle, we first need a ray and two declared sky axes.

## Before you begin

Read the grain-temperature chapter. You need exponential attenuation and two-dimensional vectors. The polarization matrix and Stokes parameters are defined here.

## Declare the sky before declaring an angle

Choose a propagation direction $\boldsymbol n$. The electric field of a transverse wave lies in the plane perpendicular to it. Choose orthonormal axes $\boldsymbol e_1,\boldsymbol e_2$ in that plane. In the Lab’s infrared observation, light travels along $+z$, so the sky axes are $x,y$. The main scene instead shows an $x,z$ side projection; moving the display viewpoint does not redefine the observer.

For many incoherent waves, average the products of their complex electric-field amplitudes. This gives a two-by-two coherency matrix $J$. We normalize its trace to the relevant intensity or ray power. Its diagonal elements describe the two linear modes; its off-diagonal terms retain the information needed when the preferred direction lies between our axes.

Stokes $I$ is the total. Positive $Q$ means more power along the first sky axis; negative $Q$ means more along the second. $U$ distinguishes the two diagonal directions. The present real, absorbing tensor does not generate circular polarization, so we do not infer a missing circular signal from it.

$$
I=J_{11}+J_{22},\qquad Q=J_{11}-J_{22},\qquad U=2\operatorname{Re}J_{12}
$$

The sign convention belongs to the declared sky basis. The degree of linear polarization is the length of the Q/U vector divided by I.

Reading: Hensley & Draine, arXiv:2208.12365v1 (2022) · Dust absorption, heat and polarization: standalone science

## Represent the material’s orientation dependence

Let $\boldsymbol b$ be a unit vector specifying an unoriented alignment axis. The teaching material has scalar opacity $\kappa_0(\nu)$ multiplied by a positive tensor $K$. The contrast $\eta$ lies between zero and one half. Along $\boldsymbol b$ the eigenvalue is $1-\eta$; in the two perpendicular directions it is $1+\eta/2$.

Only electric fields transverse to the ray matter, so project $K$ into its sky plane. For a uniform column $\Sigma$ the amplitude propagation matrix is $A=\exp[-\kappa_0\Sigma K_\perp/2]$. Then $J_{\mathrm{out}}=AJ_{\mathrm{in}}A^\dagger$. The half belongs in the amplitude exponent; intensity attenuation already follows after multiplying by its conjugate. Squaring an intensity attenuation again would remove too much power.

The tensor prescribes an absorption difference. It does not calculate how torques align grains, how alignment varies with size, or how a magnetic field evolves. The field-direction handle changes this input axis. Its length is a direction guide, not a measure of magnetic strength.

$$
K=\left(1+\frac\eta2\right)\mathbf1-\frac{3\eta}{2}\boldsymbol b\boldsymbol b^{\mathsf T},\qquad\operatorname{tr}K=3
$$

The fixed trace preserves the scalar angular-mean cooling opacity, while individual viewing directions remain anisotropic.

Reading: Dust absorption, heat and polarization: standalone science · Hensley & Draine, arXiv:2208.12365v1 (2022)

## Worked slab: the two modes exchange their advantage

Put $\boldsymbol b$ along the first sky axis and take $\eta=0.35$. The two mode opacities are $0.65\kappa_0$ and $1.175\kappa_0$. Illuminate a uniform slab with unpolarized unit intensity, half in each mode. At reference optical depth $\kappa_0\Sigma=1$, the outgoing mode intensities are $e^{-0.65}/2$ and $e^{-1.175}/2$.

Their sum is $I=0.415432$ and their fractional difference is $Q/I=0.256632$. Transmission favors the first direction because that direction loses less light. This is a homogeneous-slab analytic check, not a fit to a particular star.

Now consider optically thin thermal emission from the same material along the same direction. Its mode powers are proportional to their absorptivities, not their surviving transmissions. Thus $Q/I=(0.65-1.175)/(0.65+1.175)=-0.287671$. The negative sign places the thermal electric-field axis perpendicular to the transmitted one. It is not a negative total intensity.

A whole cloud adds cells with different temperatures and weights. It also sees source rays that need not point toward our observer. The slab explains the local sign and provides an independent limit; it does not permit us to compare angles measured in two different ray planes without projecting them first.

$$
\left(\frac QI\right)_{\mathrm{trans}}=\tanh\!\left[\frac{\tau_2-\tau_1}{2}\right]
$$

Unpolarized incident light, a homogeneous diagonal absorbing slab, and the first mode less opaque than the second.

![Transmission favors the less absorbed mode; thin thermal emission favors the perpendicular mode.](https://sandboxphysics.com/stories/interstellar-dust/materials/two-mode-slab.svg)

Independent two-mode analytic slab. The horizontal thermal reference assumes a separate optically thin emission wavelength; it is not a thick-slab thermal solution at the transmission optical depth.

Reading: Dust absorption, heat and polarization: standalone science

## A direction has two Stokes angles

The observable linear polarization fraction is $p=\sqrt{Q^2+U^2}/I$. When it is nonzero, the electric-field position angle is $\chi=\tfrac12\operatorname{atan2}(U,Q)$. The half-angle reflects a physical symmetry: a line and the same line rotated by $180^\circ$ describe the same electric-field axis.

As an analytic check, actively rotate an entire physical configuration by $\alpha$ around the observer direction while keeping the sky axes fixed. The Stokes vector rotates by $2\alpha$. A $90^\circ$ rotation reverses both $Q$ and $U$; the image rotates and $I$ is unchanged for this globally rotated source–cloud problem. This is a whole-configuration test in the scientific replay, not a claim that rotating only the field in a fixed illuminated cloud leaves heating unchanged.

Reversing $\boldsymbol b$ leaves $\boldsymbol b\boldsymbol b^{\mathsf T}$ unchanged. Looking along it makes the two transverse tensor eigenvalues equal, so polarization vanishes. At zero polarization the position angle is undefined, so the direction bar disappears.

$$
\begin{pmatrix}Q'\\U'\end{pmatrix}=\begin{pmatrix}\cos2\alpha&-\sin2\alpha\\\sin2\alpha&\cos2\alpha\end{pmatrix}\begin{pmatrix}Q\\U\end{pmatrix}
$$

Active rotation of the physical configuration in fixed sky axes. Rotating the coordinate basis instead gives the opposite rotation.

Reading: Dust absorption, heat and polarization: standalone science

## Why an angle is not a three-dimensional field measurement

A small polarization fraction has several possible causes: a direction close to the line of sight, weak alignment, different orientations canceling within a beam, or extra unpolarized emission. The toy cloud uses a single prescribed axis, but real clouds need not. One observed fraction cannot distinguish all these explanations.

The native Astrodust model also has wavelength-dependent material properties, a size distribution and a fitted alignment function. Its signed polarization cross sections can behave differently from the simple positive-contrast tensor. Keep those material-model predictions separate from this derivation.

The Lab lets us isolate projection: choose the line-of-sight preset and then the diagonal-direction preset. The handle sets the input axis; the Stokes output shows its projected radiative effect. In an observation, the inverse problem needs additional geometry and alignment assumptions, with their uncertainties stated.

Reading: Hensley & Draine, arXiv:2208.12365v1 (2022) · Dust absorption, heat and polarization: standalone science

## Exercises and explanations

1. The slab has $\tau_1=0.65$ and $\tau_2=1.175$. Which transmitted electric-field axis wins, and why does the thermal sign differ?

   Read the explanation

   Transmission favors axis one, giving $Q/I=+0.256632$, because less of that mode is removed. Optically thin thermal power favors the more absorptive mode by Kirchhoff’s law, giving $Q/I=-0.287671$ in the same sky basis. The axes are perpendicular; neither result requires negative total power.

2. The infrared polarization disappears. Does the magnetic field have to be zero?

   Read the explanation

   No. Even this prescribed model gives zero polarization when the alignment axis is along the observer direction. In real data, weak alignment, unresolved cancellation and unpolarized components add other possibilities. The model does not infer field strength at all.

## Try the prediction

### [Follow the same dust from absorbed light to thermal emission](https://sandboxphysics.com/labs/interstellar-dust-polarization/)

1. Load the diagonal-direction preset. Find a nonzero U contribution in the saved thermal Stokes data and relate its sign to the declared sky axes.
2. Move the inclination to the line of sight. Confirm that polarization falls to zero and that the image does not display a meaningful electric-field direction there.
3. Drag the direction handle with the cloud fixed. Compare the thermal map and intercepted-power readout; distinguish this input change from rotating the entire source–cloud configuration.

[Previous: From absorbed starlight to grain temperature](https://sandboxphysics.com/stories/interstellar-dust/grain-absorption-and-equilibrium-temperature/) [Next: How many temperatures do three infrared colours reveal?](https://sandboxphysics.com/stories/interstellar-dust/three-bands-and-many-temperatures/)

## Sources and calculation scope

1. [Hensley & Draine, arXiv:2208.12365v1 (2022)](https://arxiv.org/abs/2208.12365v1)

   Fixed first manuscript: sections 2.1–2.3, 4–6; equations 1–11/27 and Figures 6/11/12/18. Material cross sections, temperature distributions and prescribed size-dependent alignment are distinct from the teaching tensor. Its 2022 forecasts are not current-status claims.

2. [Dust absorption, heat and polarization: standalone science](https://sandboxphysics.com/stories/interstellar-dust/materials/dust-transfer-science-v1.zip)

   MIT scientific code plus separately licensed CC0 native inputs. Ten genuinely recomputed offline steps, retained coarse-grid failures, independent volume/Planck references and optional first-network real exercise. The site supplement records a later successful real first acquisition; the frozen archive remains unchanged.