Scanning holography is a two-pupil interaction method [8] by which incoherent imaging with complex point-spread-functions (PSF) is possible. The method has recently been applied to the recording of high resolution holographic images of incoherent objects and fluorescent biological specimens [6, 7]. A single-sideband in-line Fresnel hologram is obtained by a 2D raster scan of the object with the superposed 3D diffraction distributions of two pupils, as sketched in fig. 1. The two pupil distributions {\tilde{P}}_{1}(\overrightarrow{\rho})
and {\tilde{P}}_{2}(\overrightarrow{\rho})
from the same source (for example, but not necessarily, a laser) are combined by a beam splitter in the pupil plane of the objective where they interfere, forming a Fresnel pattern with a depth-dependent Fresnel number. To obtain the conventional point-spread function of wide field imaging, *P*
_{1} is chosen as a point source, and *P*
_{2} as a spherical wave with appropriate curvature. The 3D specimen is placed in the focal region of the objective, and scattered lights (transmitted, reflected, and fluorescent) are collected by non-imaging detectors. The hologram data can be obtained by heterodyne detection with one of the pupils shifted in frequency (as done in this work), or by a homodyne method requiring the capture of at least three frames with different relative phases between the two pupils [9]. The amplitude distribution of the illuminating beam, in a transverse plane at an axial distance *z* from the focal plane of the objective, is the Fourier transform of the combined pupil distributions [10]. Namely:

S(\overrightarrow{r},z)={F}^{-1}\left\{\left[{\tilde{P}}_{1}(\overrightarrow{\rho};z)+{\tilde{P}}_{2}(\overrightarrow{\rho};z)\mathrm{exp}(-i\Omega t\right]\right\},\left(1\text{a}\right)

where

{\tilde{P}}_{1,2}(\overrightarrow{\rho};z)={\tilde{P}}_{1,2}(\overrightarrow{\rho})\mathrm{exp}(i\pi \lambda z{\rho}^{2})\left(1\text{b}\right)

are the generalized defocused pupils [11]. *F*^{-1} stands for inverse Fourier transform, and Ω is the frequency shift of one of the pupils. The transverse spatial frequency vector \overrightarrow{\rho}
= \overrightarrow{r}
_{
P
}/*λf*
_{0} is proportional to the real space coordinate \overrightarrow{r}
_{
P
}in the pupil plane [10]. *λ* is the wavelength of the illumination, and *f*
_{0} is the focal length of the objective. The two pupils used to obtain an in-line Fresnel hologram are, respectively, a spherical wave filling the pupil of the objective, and a point at the center of that pupil:

{\tilde{P}}_{1}(\overrightarrow{\rho})

(1)

= exp(*iπλz*
_{0}
*ρ*^{2})*circ*(*ρ*/*ρ*
_{
MAX
})

{\tilde{P}}_{2}(\overrightarrow{\rho})=\delta (\overrightarrow{\rho}).\left(2\right)

*circ*(*x*) is a disc function of unit radius. *ρ*
_{
MAX
}= sin*α*/*λ* is the cutoff frequency of the objective, where sin*α* = *NA* is its numerical aperture. The Fresnel pattern projected on the object is the interference of the Fourier transforms of the two pupils, namely a spherical wave and a plane wave in the paraxial approximation. The Fresnel number of this pattern is determined by the free parameter *z*
_{0}, which is the distance from the objective's focal plane to the point where the spherical wave comes to a focus.

There are two possible modes of operation, depending on the detector geometry [5]. With a spatially integrating detector, the resulting data is a convolution of the object's intensity distribution with the desired complex PSF (namely, a spherical wave with a radius of curvature *z*
_{0} + *z*, and a radius *a* = *z*
_{0}sin*α*, in this case). This detection mode leads to a hologram from which the three-dimensional distribution of scattering intensity, absorption, and fluorescence intensity can be reconstructed. With a point detector at the center of a conjugate pupil plane, the resulting data is a convolution of the object's complex amplitude distribution with the same complex PSF. The reconstruction of this hologram gives the three-dimensional distribution of the specimen's complex amplitude transmittance. In particular, assuming that multiple scattering can be ignored, the phase of the reconstruction is a quantitative measure of the integrated optical path length through the specimen. These two modes of operation are similar to the usual coherent/incoherent imaging modes of a conventional system, which are obtained by using, respectively, a point source, or a large spatially incoherent source. In scanning holography, the detector size plays a similar role to that of the source size in conventional imaging.

For simplicity, let's assume an object with an amplitude transmittance *T*(\overrightarrow{r}
, *z*). Note that for an incoherent or fluorescent object, the phase of the transmitted field is a random variable, and only the intensity *I*(\overrightarrow{r}
, *z*) = |*T*(\overrightarrow{r}
, *z*)|^{2} is measurable. For a quasi transparent object, the phase is equal to the integrated optical thickness of the object: Φ(\overrightarrow{r}
) = (2*π*/*λ*)∫*dzn*(\overrightarrow{r}
, *z*), where *n*(\overrightarrow{r}
, *z*) is the 3D distribution of refractive index. The amplitude distribution after the object is written as

*A*(\overrightarrow{r}
, *t*) = ∫*dzS*(\overrightarrow{r}
, *z*)*T*[\overrightarrow{r}
-\overrightarrow{r}
_{
S
}(*t*), *z*], (3)

where \overrightarrow{r}
_{
S
}(*t*) is the instantaneous position of the 2D raster scan.

The incoherent imaging mode is obtained with a spatially integrating detector, leading to a temporal signal proportional to the integrated intensity, ∫*d*^{2}
*r*|*A*(\overrightarrow{r}
, *t*)|^{2}, which is stored in the computer. The data corresponding to each hologram line is cut from the signal, and band pass filtered to extract the term oscillating at Ω. The lines are then rearranged in a 2D format. The resulting hologram amplitude is found to be

*H*
_{
I
}(\overrightarrow{r}
) = ∫*dzI*(\overrightarrow{r}
, *z*) ⊕ [*p*
_{1}(\overrightarrow{r}
, *z*)*p**_{2}(\overrightarrow{r}
, *z*)], (4)

where ⊕ symbolizes a convolution integral, *I*(\overrightarrow{r}
, *z*) = |*T*(\overrightarrow{r}
, *z*)|^{2}, and *p*
_{1,2}(\overrightarrow{r}
, *z*) are the inverse Fourier transforms of the defocused pupil distributions (eq.1b). The superscript * stands for complex conjugate. The hologram is thus the convolution of the object intensity with a spherical wave, i.e. an in-line single-sideband Gabor hologram. In Fourier space, the hologram can be written as

{\tilde{H}}_{I}(\overrightarrow{\rho})={\displaystyle \int dz\tilde{I}}(\overrightarrow{\rho};z)\left[{\tilde{P}}_{1}(\overrightarrow{\rho};z)\otimes {\tilde{P}}_{2}(\overrightarrow{\rho};z)\right],\left(5\right)

where ⊕ symbolizes a correlation integral. With the pupils of eq.2, we find, in the paraxial approximation,

{\tilde{H}}_{I}(\overrightarrow{\rho})={\displaystyle \int dz\tilde{I}}(\overrightarrow{\rho};z)\mathrm{exp}\left[i\pi \lambda ({z}_{0}+z){\rho}^{2}\right]circ(\rho /{\rho}_{MAX}),\left(6\right)

which is the Fourier transform of the Fresnel hologram of the object's intensity distribution.

The coherent imaging mode is obtained by using a pinhole or a point detector at the center of a conjugate plane of the pupil of the objective. This leads to the hologram amplitude

*H*
_{
C
}(\overrightarrow{r}
) = ∫*dz*[*T*(\overrightarrow{r}
, *z*) ⊕ *p*
_{1}(\overrightarrow{r}
, *z*)][*T*(\overrightarrow{r}
, *z*) ⊕ *p*
_{2}(\overrightarrow{r}
, *z*)]*. (7)

Where again, ⊕ symbolizes a convolution product, and the superscript * stands for complex conjugate. In Fourier space,

{\tilde{H}}_{C}(\overrightarrow{\rho})={\displaystyle \int dz\tilde{T}}(\overrightarrow{\rho};z){\tilde{P}}_{1}(\overrightarrow{\rho};z)\otimes \tilde{T}(\overrightarrow{\rho};z){\tilde{P}}_{2}(\overrightarrow{\rho};z).\left(8\right)

With the pupils of eq.2, we find

{\tilde{H}}_{C}(\overrightarrow{\rho})={\displaystyle \int dz\tilde{T}\ast (0;z)\tilde{T}}(\overrightarrow{\rho};z)\mathrm{exp}\left[i\pi \lambda ({z}_{0}+z){\rho}^{2}\right]circ(\rho /{\rho}_{MAX}).\left(9\right)

Aside from an inconsequential complex constant (the first term under the integral), eq.9 is the Fourier transform of the Fresnel hologram of the object's complex amplitude distribution. Thus, the reconstruction of the hologram recorded in the coherent mode carries a quantitative measure of the object's phase distribution.