Far-field superresolution of thermal sources by double homodyne or double array homodyne detection

We propose two schemes for estimating the separation of two thermal sources via double homodyne and double array homodyne detection considering the joint measurement of conjugate quadratures of the image plane field.By using the Cramér-Rao bound, we demonstrate that the two schemes can estimate the separation well below the Rayleigh limit and have an advantage over direct imaging when the average photon number per source exceeds five.For arbitrary source strengths, double homodyne detection is superior to homodyne detection when the separation is above 2^5/4 σ/N , σ is the beam width, N_s is the average photon number per source.A larger separation can be estimated better via double array homodyne detection with the superiority of flexible operation compared with other schemes. High-speed and high-efficiency detection enables the measurement schemes with potential practical applications in fluorescence microscopy, astronomy and quantum imaging.