In conventional holography, an optical lens is used to focus on an object of interest; however, our techniques are based on lensless digital holography in which reconstructions and focusing of the holograms are numerically obtained by Fresnel–Kirchhoff integrals.^{13} Using temporal phase-stepping algorithms,^{14}^{,}^{15} the complex amplitude of the hologram, $h(k,l)$, is obtained with Display Formula
$h(k,l)=[I3(k,l)\u2212I1(k,l)]+i[I4(k,l)\u2212I2(k,l)],$(1)
where $I1$ to $I4$ are intensity patterns of four consecutive phase-stepped frames of the camera with an induced phase step of $\pi /2$ between them, and $k$ and $l$ are the coordinates of the pixel in the CCD (hologram plane). As shown in Fig. 1, numerical reconstruction algorithms are based on two-dimensional (2-D) Fast Fourier Transform (FFT) of the product of a reconstruction reference wave, $R(k,l)$, complex amplitude of the hologram, $h(k,l)$, and a chirp function, $\psi (k,l)$, that can be obtained with Display Formula$\Gamma (m,n)=Q(m,n)\xd7FFT2[R(k,l)h(k,l)\psi (k,l)],$(2)
where $\Gamma (m,n)$ is the complex reconstructed hologram at coordinates $m$ and $n$ in the reconstruction plane, $R(k,l)$ is the complex amplitude of the reference wave, $Q(m,n)$ and $\psi (k,l)$ are the quadratic phase factor and 2-D chirp function, respectively, and are defined by Display Formula$Q(m,n)=exp[\u2212i\pi \lambda d(m2N2\Delta x2+n2N2\Delta y2)],$(3)
and Display Formula$\psi (k,l)=exp[\u2212i\pi \lambda d(k2\Delta x2+l2\Delta y2)],$(4)
where $\Delta x$ and $\Delta y$ are the pixel size of the CCD sensor, $N2$ is the number of pixels, $\lambda $ is the laser wavelength, and $d$ is the reconstruction distance. As shown in Fig. 1, the chirp function is a complex 2-D oscillatory signal, where the frequency of oscillation linearly varies with the spatial coordinate and is used for numerical reconstruction of the hologram at different distances of $d$. The reconstructed hologram, $\Gamma (m,n)$, is a complex function that contains both the amplitude and optical phase, $\phi (m,n)$, that is defined by Display Formula$\phi (m,n)=arg[\Gamma (m,n)].$(5)