# Detecting Axial Ratio of Microwave Field with High Resolution Using NV Centers in Diamond
## 2 Principle  
The axial ratio is an important parameter to describe a polarized MW field. Broadly, an arbitrarily-polarized MW field can be treated as an elliptically-polarized wave. The arbitrarily- polarized MW field spread along the z direction can be described as a combination of two circularly-polarized microwave fields, \(\textbf{B}_{+}\) and \(\textbf{B}_{-}\), spread along the z direction with the same frequency as the source MW field that rotate in different directions in the xoy plane (Figure 1). \[\textbf{B}=\textbf{B}_{+}+\textbf{B}_{-}, \tag{1}\]  
where \(B_{+}\) (\(B_{-}\)) indicates the magnetic field strength of the \(\sigma^{+}\) (\(\sigma^{-}\)) circularly-polarized field. The axial ratio can be described as the ratio of the magnitudes of the major and minor axes defined by the magnetic field vector. It can be figured out that the length of the major (minor) axis of the ellipse equals \(2|B_{+}+B_{-}|\) (\(2|B_{+}-B_{-}|\)). \[AR=|\frac{B_{+}+B_{-}}{B_{+}-B_{-}}|. \tag{2}\]  
Obviously, \(AR=1\) for a circularly-polarized MW field, and \(AR=\infty\) for a linearly-polarized MW field. Axial ratio detection of MW fields with NV centers in diamond is possible as the strength of two counter-rotating MW magnetic fields can be selectively acquired by a spin-1 system. NV electron spin in its ground state can be optically pumped to the excited state and emit red fluorescence while decaying  
Figure 1: At a fixed point in space (or for fixed z), the magnetic vector **B** of a polarized microwave (MW) field traces out an ellipse in the xoy plane. to the ground spin state.The spin state selective intersystem crossing (ISC) process [22] in the NV center enables the state read out and state initialization of NV electron spin. The NV electron in the \(m_{s}=0\) state emits more photons then the NV center in the \(m_{s}=\pm 1\) state. Successive illumination initializes the NV center to the \(m_{s}=0\) state. As shown in Figure 2a, the electron spin ground triplet state of NV centers has an energy splitting of \(D=2.87\) GHz between the \(m_{s}=\pm 1\) state and \(m_{s}=0\) state under zero magnetic field. The spin transition \(m_{s}=0\leftrightarrow m_{s}=-1(+1)\), which can be simply treated as a two-level system (TLS) and can be selectively driven by \(\sigma^{-}(\sigma^{+})\) circularly-polarized microwave field perpendicular to the NV axis [20]. According to the semi-classical theory of light-matter interaction, the transition frequency, i.e., Rabi frequency \(\Omega_{R}\), and the optically-detected signal contrast \(C_{R}\) that corresponds to the electron spin state population of the TLS under a driving MW magnetic field can be written as:  
\[\Omega_{R}=\sqrt{\bigtriangleup^{2}+\Omega_{0}^{2}},\ \ C_{R}=C_{R0}\frac{ \Omega_{0}^{2}}{\Omega_{0}^{2}+\bigtriangleup^{2}}, \tag{3}\]  
where \(\bigtriangleup=\omega_{0}-\omega_{m}\) is the frequency difference between the frequency corresponding to the energy separation \(\hbar\omega_{0}\) of the TLS and the frequency of the driving microwave field \(\omega_{m}\). \(\Omega_{0}=\gamma_{e}B_{mw}\) is the Rabi frequency at resonance, i.e., \(\omega_{0}=\omega_{m}\); \(\gamma_{e}=2.8\) MHz/G is the gyromagnetic ratio of NV electron spin. \(B_{mw}\) is the amplitude of the resonant component of the driving microwave magnetic field. \(C_{R0}\) is the maximum signal contrast between the \(m_{s}=0\) state and \(m_{s}=\pm 1\) state. Thus, the MW magnetic field strength of particular polarization can be deduced from Rabi frequency. Furthermore, due to the Zeeman effect, the resonance frequency of transition \(m_{s}=0\leftrightarrow m_{s}=\pm 1\) moves to \(\omega_{\pm}=D_{gs}\pm\gamma_{e}B_{z}\) under a bias magnetic field \(B_{z}\) along the NV axis (Figure 2a). This enables wide-band detecting of the microwave magnetic field.