How do you make a spherical radio wave?

A result known as Birkhoff's theorem forbids spherical electromagnetic radiation. The statement of the theorem is that any spherically symmetric vacuum solution to Maxwell's equations must be static. It is rather simple to prove. In a spherically symmetric solution $\mathbf E$ and $\mathbf B$ must be radial. Make an Ansatz, $$\mathbf E = E_0 \exp(i(\mathbf k\cdot\mathbf r-\omega t)) \hat r \quad \mathbf B = B_0 \exp(i(\mathbf k\cdot\mathbf r-\omega t)) \hat r $$ The wavevector $\mathbf k$ must be $\mathbf k = k\hat r$ for spherical symmetry. Now Ampere's law is $$\nabla\times \mathbf B = i\mathbf k \times \mathbf B = 0 = \partial_t \mathbf E = -i\omega \mathbf E$$ which implies $\omega = 0$, so that the field is static, or $E_0 = 0$. From Faraday's law $\nabla\times\mathbf E =- \partial_t \mathbf B$ you can see that if $E_0 = 0$ but $\omega \neq 0$, then also $B_0 = 0$.

The most general result for electromagnetic radiation is that in Coulomb gauge, in the radiation zone, the vector potential is $$\mathbf A(\mathbf x, t) = \frac{\mu_0}{4\pi }\frac{e^{i(kr-\omega t)}}{r} \int \mathbf J(\mathbf x') e^{-ik\hat{x} \cdot \mathbf x'} \, dx'$$ where $\mathbf J(\mathbf x')$ is the current in the source region, e.g., your antenna, and the current is assumed to have sinusoidal (harmonic) time dependence. [This is not a restriction because Maxwell's equations are linear and Fourier transform exists.]

The angular dependence is entirely in the integral over the source current. Thus to achieve some desired angular profile of the radiation, one needs to design $\mathbf J$ appropriately.

Your particular case of an oscillating sphere of charge actually does not radiate because it has only a monopole moment and there is no monopole radiation. A spheroidal charge distribution is treated by Jackson Classical Electrodynamics, Sec. 9.3. There Jackson shows that this arrangement leads to quadrupole radiation with a four-lobed distribution of radiated power. For a more in-depth discussion, read Ch. 9 in Jackson, which treats radiation in detail, including the angular distribution of radiated power from various sources.

"If that is possible, how do you produce a spherical EM radiation?"

A spherically symmetric transverse field is topologically impossible - if it is required to be coherent and linearly polarized everywhere. This is the case for usual dipole or higher multipole radiation, as has already been pointed out in another answer.

On the other hand, an incoherent transverse field, otherwise known as an incoherent superposition of coherent fields, may well be spherically symmetric, because it no longer has a well-defined polarization. This is what we casually refer to as a "spherically symmetric e.m. radiation field". But this is boring.

The far more interesting stuff is that if we still require a coherent field, but

  1. relax the requirement for uniformity in polarization,


  1. make good use of already available metamaterials technology,

it is possible to put together something arbitrarily close to "spherically symmetric intensity". The corresponding devices are known as isotropic radiators (not to be confused with old omnidirectional antennas, which generally produce a doughnut-shaped field).

For alternative (1), example antenna arrangements that may generate virtually isotropic far-field intensity are given in arXiv:physics/0312023. They are variants of $\lambda / 4$ narrow U-shaped antennas or linear arrays of turnstile-antennas operated in phase. According to the paper, even for one pair of turnstile antennas the maximum intensity is only 1.08 times larger than minimum intensity. It is also pointed out that a spherical shell radiator may produce isotropic output from certain patterns of finite oscillating currents (see refs. therein).

Alternative (2) has been described in Phys. Rev. Lett. 111, 133901 (2013) (also available here). The idea is to reshape a dipole radiation pattern by means of suitably arranged metamaterial structures such that even the near-field becomes isotropic. Similar reshaping techniques are applicable in principle to any antenna output.

Bottom line is, looks like there are always possibilities... ;)