## 1 Introduction  
Precise measurements of the strength, phase, polarity, etc. of electromagnetic (EM) fields are of great importance for science, communications, and biomedicine, having prominent roles in everyday life. Microwaves--electromagnetic waves with very high frequency (300 MHz to 3000 GHz)--play a very important role. In this study, microwaves are emitted from a horn. The radiation field of such a horn can be divided into three parts: the induction near field, the radiation far field, and an intermediate zone. Measurement of the induction near field is advantageous, as these measurements are convenient, have high fidelity, and can provide a large amount of characteristic information. The electric field distribution of a microwave horn can be used to understand the propagation of the microwaves as well as the electric field distribution in the far field, which can be difficult to measure. Traditionally, measurements of the microwave electric field are made using a dipole antenna as a probe. Probes are calibrated prior to measurement [1; 2; 3]. The accuracy is limited by the probe geometry, material, sensitivity, etc. With the development of atomic physics and detection technology, one can improve field measurement accuracy and sensitivity with an atom-based method [4; 5; 6; 7]. Our atom-based field measurement has clear advantages over such traditional methods because it is based on invariant atomic properties and is therefore calibration free. Rydberg atoms, with principal quantum number \(n\gtrsim 10\), are an ideal candidate for measurement of the microwave field due to their huge dc polarisability (\(\sim\)\(n^{7}\)) and microwave-transition dipole moments (\(\sim\)\(n^{2}\)). This provides Rydberg atoms with extreme sensitivities to dc and ac electric fields [8]. At a high principle quantum number, the energy difference of Rydberg levels (\(\sim\)\(n^{-3}\)) approaches the microwave-frequency range, leading to even stronger interaction between Rydberg atoms and microwave fields. When the microwave field couples two close lying Rydberg states, the coupling strength is expressed with Rabi frequency \(\Omega_{MW}=\mu_{MW}E/\hbar\), with microwave coupled dipole transition element \(\mu_{MW}\), electric field \(E\), and reduced Planck constant \(\hbar\). The \(\Omega_{MW}\) can be large when the electric field is weak because the \(\mu_{MW}\) between neighboring Rydberg states scales as \(n^{2}\). For the caesium transition \(56\mathrm{D}_{5/2}\to 57\mathrm{P}_{3/2}\) used in this work, calculated \(\mu_{MW}\) is \(\sim\)4258 \(ea_{0}\). Coupling between these two states due to the microwave electric field, \(\Omega_{MW}\), causes the electromagnetically induced transparency (EIT) peak to split into two, known as Autler-Townes (AT) splitting [9]. The magnitude of the AT splitting, \(\gamma_{AT}\), is proportional to \(\Omega_{MW}\), and so the magnitude of the splitting can be used to make an atom-based microwave field measurement. Rydberg EIT [10; 11; 12], a quantum coherence effect of the interaction between Rydberg atoms and lasers, will be employed to optically detect the microwave dressed AT splitting \(\gamma_{AT}\) induced by the microwave field [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. Additionally, we can use this kind of resonant Rydberg transition and fluorescence to detect THz radiation [30; 31]. In this work, we first investigate the microwave near field distribution produced by a standard microwave horn employing microwave dressed Rydberg EIT-AT spectroscopy in a room temperature vapor cell. A 3.99-GHz microwave field emitted by the horn couples the Rydberg transition, \(56D_{5/2}\to 57P_{3/2}\), leading to AT splitting of the Rydberg EIT peak. The microwave EM near field distribution is extracted using measurements of the EIT-AT splitting as a function of the position of the atomic sensor in the near field of the horn. This measurement is compared to measurements taken with a traditional metal probe using a vector network analyzer (VNA).