Microwave Diagnostics of Ultracold Neutral Plasma

The diagnostics methods of UNP are mainly developed from some well defined technique of optic probes, such as laser induced fluorescence imaging [4], optical absorption imaging [5] and even recombination fluorescence [6]. The optic probes can excite the fluorescence or be absorbed in the case of ions and Rydberg atoms. However, free electrons in UNP can not respond to the laser beam except the Thomason scattering which is too weak to be measured in the current techniques. Furthermore, most traditional plasma diagnostics are not accessible due the size-limited UNP. In this paper, we suggest a new method of using microwave radiation for the study of UNP. Using this method, we can measure the amount of electrons N e and the recombination rate of plasma, which is extremely important in the research of UNP.

Normally, when the microwave wavelength λ is much smaller than the size plasma size L, microwave-based diagnostics such as interferometry and reflectometry have found very broad application in classical plasma diagnostics.

However, the UNP size limited by the beamwidth of the cooling laser is usually very small around the range of mm . In the small plasma objects situation λ/L ≫ 1, those microwave diagnostics based on propagation, absorption or reflection fail to work. Shneider and Miles develop the theory in the case of small plasma objects by measuring the radiation scattered by the effective oscillating plasma dipole [7] [8]. In their work [7], the plasma is static uniform plasma column, and the microwave frequency used is smaller than plasma frequency ω < ω pe . However, UNP is an expanding spherical plasma cloud with radial inhomogeneity, the density profile of UNP decay radically quickly in the few of mm. There is still no any discussion about microwave diagnostics on this special plasma so far. In this paper we investigate the dipole radiation of UNP in an incident microwave as the physical scheme shown in Fig. 1.

From linearized electron motion equation, we can easily derive the high frequency conductivity σ and dielectric constant ǫ of plasma [9]:

where ω pe = n e e 2 /m e ǫ 0 is the plasma frequency,m e is the electron mass and n e is eletron density .

We calculate the dipole of an uniform dielectric ball with the radius r and dielectric constant epsilon ǫ in the first step. The dielectric ball responds to the incident electric field E and the electric dipole is induced . we can get that outside the sphere the potential is equivalent to that of the applied field plus the field of a point electric dipole p unif orm , so the equivalent dipole of the dielectric ball is

Next, we consider a thin spherical shell with spherical shell of radius r, thickness dr and dielectric constant ǫ(r), From Eq.3, the corresponding dipole of the shell is

So the integral on the Eq.4 along the radius yields the equivalent dipole of the UNP ball

Because the size of UNP is much less than the wavelength of microwave, we can neglect the field variance across the UNP ball. We set E = E 0 e r exp iωt , the equivalent dipole of the UNP ball will vibrate in the same frequency ω of the incident electric field. The oscillation mainly comes from the oscillation of E, so we get total dipole radiation power in space

and the effective scattering cross section

ω 2 pe (r) can be written as ω 2 pe0 f e (r), where ω pe0 is plasma frequency at the center of UNP and f e (r) is the radial density profile.

In Eq.6 and Eq.7, the dipole radiation power and the effective scattering cross section tend to a constant when the microwave frequency is much greater than the plasma frequency. But it is worthy to note the corresponding constant reflects the crucial properties of plasma. The UNP has a typical guassianlike radial density profile f e (r) = exp(-r 2 /2r 2 0 ). In the UNP situation, Fig. 2 and Fig. 3 illuminate the constant tendency when the ratio of microwave and plasma frequency is greater than 5 in three different density cases. In the two figures, the dipole radiation power P (ω) and the effective scattering cross section σ(ω) are scaled vertically by ω 4 pe0 E 2 0 and ω 4 pe0 respectively. The overlapping horizontal lines at large frequency ratio indicate the radiation power and cross section never change while ω increases at large ω. The classical ω 4

dependence of dipole radiation is not satisfied here. Specially, when the microwave frequency is large enough it shows the frequency independence as we explained above. The underlying reason is the frequency dependence of the equivalent dipole instead of the assumption of constant dipole in the classical description.

For large ω/ω pe0 , we can ignore the term of ω 2 pe (r)/ω

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