## 2 Proposed Fast Method for Recovering Reference-Wave Intensity  
In two-step quadrature phase-shifting holography, two phase-shifting on-axis holograms with a zero phase shift and a \(\pi/2\) phase shift are achieved, as follows:  
\[I_{I}=\ \left|R+O\right|^{2}=I_{0}+2R\cdot\mathrm{Re}(O), \tag{1}\]  
\[I_{Q}=\ \left|R\cdot\mathrm{e}^{\mathrm{i}\cdot\pi/2}+O\right|^{2}=I_{0}+2R \cdot\mathrm{Im}(O), \tag{2}\]  
where \(O\) is the complex amplitude of the object wave on the recording plane, \(R\) is the reference wave on the recording plane, which is a plane wave and regarded as real amplitude for simplicity. \(I_{I}\) is called the in-phase hologram and \(I_{Q}\) is the quadrature hologram. \(I_{0}\) is the bias or the background term in the hologram that is given by:  
\[I_{0}=R^{2}+\left|O\right|^{2}. \tag{3}\]  
Upon optical illumination of hologram \(I_{I}\) or \(I_{Q}\), \(I_{0}\) gives rise to a zeroth-order beam that overlaps the reconstructed object beam on the reconstruction plane. Now, based on Equations (1) and (2), the complex amplitude of the object wave is found to be:  
\[O=\frac{(I_{I}-I_{0})+\mathrm{i}\cdot(I_{Q}-I_{0})}{2R}. \tag{4}\]  
After obtaining the complex amplitude of the object wave, we can perform backward propagation from the recording plane to the initial plane, and obtain the three-dimensional (3D) object distribution. In order to extract the object wave, \(O\), on the recording plane, the intensity of \(R\) and \(O\) should be recorded before holographic recordings as in traditional QPSH, as [PERSON] and [PERSON] have done experimentally [10]. [PERSON] and [PERSON] have proposed a method called two-step-only quadrature phase-shifting holography (TSO-QPSH) to extract the object wave by estimating the reference-wave intensity using iterative computations instead of an actual recording of the intensity map. The iterative computation equation for the bias term, \(I_{0}\), is given by [13]:  
\[I_{0}=\frac{2R^{2}+I_{I}+I_{Q}}{2}-\frac{\sqrt{\left(2R^{2}+I_{I}+I_{Q} \right)^{2}-2\left({I_{I}}^{2}+{I_{Q}}^{2}+4R^{4}\right)}}{2}. \tag{5}\]  
In the above equation, \(I_{I}\) and \(I_{Q}\) are obtained experimentally and \(R^{2}\) is obtained through iteration. Hence, Equation (4) is completely determined. In our previous published paper [17], we have recorded two phase-shifted holograms and constructed a complex hologram given by:  
\[I_{+\mathrm{i}}=I_{I}+\mathrm{i}\cdot I_{Q}=(1+\mathrm{i})\cdot R^{2}(1+\beta)+2 RO, \tag{6}\]  
where \(\beta=1\,O\,|\,^{2}\!/\!R^{2}\) is the intensity ratio of the object light and the reference light. Note that performing complex addition, as shown above, allows us to eliminate the twin image, \(O*\). However, the zeroth-order term of the complex hologram still exists and is complex, being different from the two on-axis holograms in which the zeroth-order term is real and positive. So, if we want to obtain the exact phase information of the object, the complex zeroth-order term in the complex hologram needs to be removed. Equation (6) can be rewritten as:  
\[I_{+\mathrm{i}}=I_{I}+\mathrm{i}\cdot I_{Q}=A\cdot(1+\mathrm{i})+2 RO, \tag{7}\]  
where \(A=R^{2}(1+\beta)\). Note that in this case, we have a complex bias term, \(A\cdot(1+\mathrm{i})\), which gives rise to a zeroth-order beam upon reconstruction. To obtain also zeroth-order-beam-free reconstruction, we perform the following calculation:  
\[I=I_{+\mathrm{i}}-A\cdot(1+\mathrm{i})=2 RO\propto O. \tag{8}\]  
Since \(I_{+\mathrm{i}}\) is known, the key factor is to find \(A\). We propose a simple and yet effective method to estimate \(A\). We select a particular area in the complex hologram where there is negligible diffraction on the CCD from the test sample to estimate \(A\). We then adopt two criteria. First, we assume that the geometric projection of the sample onto the recording plane falls well within the boundaries of the CCD array.