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The temperature and polarization distribution of the radiation
is in general a function of both spatial position and angle
defining the propagation direction.
In flat space, we know that
plane waves form a complete basis for the spatial dependence.
Thus a spin- field like the temperature may be expanded in

| |
(8) |

where the normalization is chosen to agree with the standard
Legendre polynomial conventions for *m*=0.
Likewise a spin-2 field like the polarization may be expanded in
| |
(9) |

The plane wave itself also carries an angular dependence of course,

| |
(10) |

where and
(see Fig. 2).
The sign convention for the direction is opposite
to direction on the sky to be in accord with the direction of
propagation of the radiation to the observer. Thus the extra factor
of comes from the parity relation Eqn. (6).

**Figure 2:**
Projection effects. A plane wave can be decomposed into and
hence carries an ``orbital'' angular dependence. A
plane wave source at distance *r* thus contributes
angular power to at but also
to larger angles at which is
encapsulated into the structure of (see Fig. 3). If the source has an intrinsic
angular dependence, the distribution of power is altered. For
an aligned dipole
(`figure 8's) power at or
is suppressed. These arguments are generalized
for other intrinsic angular dependences in the text.

The separation of the mode functions into an intrinsic angular
dependence and plane-wave spatial dependence is essentially
a division into spin () and orbital
() angular momentum. Since only
the total angular dependence is observable, it is instructive
to employ the Clebsch-Gordan relation of
Eqn. (8) to add the angular momenta.
In general this couples the states between
and . Correspondingly
a state of definite total will correspond to a weighted
sum of to in its radial
dependence. This can be reexpressed in terms of
the using the recursion
relations of spherical Bessel functions,

We can then rewrite
| |
(11) |

where the lowest radial functions are
| |
(12) |

with primes representing derivatives with respect to the argument
of the radial function *x*=*kr*.
These modes are shown in Fig. 3.

Similarly for the spin functions with *m*>0 (see
Fig. 4),

| |
(13) |

where
which corresponds to the coupling and
which corresponds to the coupling.
The corresponding relation for negative *m* involves a reversal
in sign of the -functions
These functions are plotted in Fig. 4.
Note that is displayed
in Fig. 3.

**Figure 4:**
Radial spin-2 (polarization) modes. Displayed is
the angular power in a plane-wave spin-2 source. The top panel
shows that vector (*m*=1, upper panel)
sources are dominated by *B*-parity
contributions, whereas tensor (*m*=2, lower panel) sources have
comparable but less power in the *B*-parity. Note that the power is strongly
peaked at for the *B*-parity vectors and *E*-parity
tensors.
The argument of the radial functions *kr*=100 here.

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*Wayne Hu*

*9/9/1997*