Modeling the depth-sectioning effect in reflection-mode dynamic speckle-field interferometric microscopy.

Unlike most optical coherence microscopy (OCM) systems, dynamic speckle-field interferometric microscopy (DSIM) achieves depth sectioning through the spatial-coherence gating effect. Under high numerical aperture (NA) speckle-field illumination, our previous experiments have demonstrated less than 1 μm depth resolution in reflection-mode DSIM, while doubling the diffraction limited resolution as under structured illumination. However, there has not been a physical model to rigorously describe the speckle imaging process, in particular explaining the sectioning effect under high illumination and imaging NA settings in DSIM. In this paper, we develop such a model based on the diffraction tomography theory and the speckle statistics. Using this model, we calculate the system response function, which is used to further obtain the depth resolution limit in reflection-mode DSIM. Theoretically calculated depth resolution limit is in an excellent agreement with experiment results. We envision that our physical model will not only help in understanding the imaging process in DSIM, but also enable better designing such systems for depth-resolved measurements in biological cells and tissues.

Interferometric microscopy offers extreme sensitivity in measuring sample deformation or absorption along the axial dimension without using fluorescence staining [18][19][20][21][22]. In an interferometric microscopy system, the sectioning effect can be realized via either the spatialor the temporal-coherence property of light under wide-field imaging mode. OCT, as an interferometric imaging technique, normally uses the temporal-coherence gating effect to achieve depth resolved measurements [13]. Its depth resolution is typically a few microns, which is mainly determined by the bandwidth or the temporal coherence of the light source used. Similarly, the spatial-coherence gating effect has also been utilized in interferometric microscopy to obtain depth-resolved measurements [23]. B. Redding et al. reported a fullfield interferometric confocal imaging method, where the spatial coherence was manipulated by using a multimode fiber [24]. The measured spatial resolution however was limited to a few microns. Soon after, Y. Choi et al. demonstrated a reflection-mode dynamic speckle-field quantitative phase microscopy system with ~500 nm lateral resolution and ~1 micron depth resolution [25]. This type of system is promising in studying the nanoscale dynamics of depth-resolved structures such plasma and nucleic membranes in complex eukaryotic cells. If applied to 3D imaging, this reflection phase imaging system can potentially solve the "missing cone" problem during image reconstruction, which otherwise requires priori constraints, such as the non-negativity and piecewise smoothness for convergence [26,27].
Despite of the recent experimental advances in depth-resolved interferometric imaging using dynamic speckle-fields, there has not been a full physical model to describe the sectioning effect in such systems [23]. Most of the previous theoretical analysis of depth resolution was based on small scattering angle approximations or paraxial approximations, including the SCM transfer function calculations [16,17], where the diffraction effects that potentially degrade the image reconstruction quality in high NA imaging are not fully accounted for. Later on, C. J. R. Sheppard's group has also calculated the 3D CTF for high NA imaging conditions for holographic tomography [28]. Recently, through solving the inverse scattering problem with the diffraction tomography theory, accurate 3D CTF has been obtained for low temporal-coherence interferometric tomography systems, which enabled more precise 3D reconstruction with improved spatial resolution in all dimensions for tissue [29,30] and cellular imaging [31][32][33]. This highlights the importance of including the diffraction effects in coherent imaging.
In this paper, we have extended the diffraction tomography theory to dynamic specklefield interferometric microscopy or DSIM. We have successfully developed a model to calculate the axial response function in reflection-mode DSIM systems, which can be used to determine the depth resolution. The theoretically calculated depth resolution agrees well with our previous experimental results [25]. In the following, a full description of the physical model is provided. First of all, we describe a typical reflection-mode DSIM system, including the light scattering process, the interference fields, and the detection measurement function. Then, we solve the scattered field from the inhomogeneous wave equation to calculate the cross-correlation function, which is directly related to the measurement quantity. Finally, we calculate the axial response function of a thin 2D slice to obtain the depth resolution. Our study shows that the depth resolution is proportional to the square of the NA of the illumination and imaging objective. In the discussion section, we also verify that transmission-mode DSIM systems do not have sectioning effects for flat objects.

Reflection-mode dynamic speckle-field interferometric microscopy
In this section, we first describe the working principle of a typical reflection-mode DSIM system. Then, we solve the backward scattered field for an arbitrary object to determine the measurement function on the detector plane. This lays the foundation for calculating the axial response function and the depth resolution. Typically, a Linnik-type interferometer is used in a reflection-mode DSIM system. Figure  1(a) shows the schematic of such a system (more details can be found in [23,25]). The field of interest starts from the diffuser plane, consisting of a disk shape ground glass, which is conjugated to the back focal planes of the reference arm objective (Obj1) as well as that of the imaging arm objective (Obj2). The sample surface, reference mirror, and the detector are also in conjugate planes through imaging optics. When the diffuser rotates at a high-speed (this allows for sufficient averaging of speckles during the camera integration time), the generated dynamic speckle-field forms a smooth distribution in the objective back aperture planes. According to our following theoretical model, it would be ideal for this field distribution to uniformly fill up the back aperture of Obj1 and Obj2 to achieve the optimum illumination with the best sectioning effect. Next, we describe the fields that are involved in the imaging process as described in Fig. 1(b). In our imaging system, the diffuser is in the Fourier plane where the speckle field is generated. Following the theory framework in [23], we assume that the speckle-field, immediately after the diffuser plane, has an angular spectrum distribution, ( )  , ,

System configuration and working principle
is the position vector. The plane wave illuminates the sample, described by the scattering potential ( ) ( ) 2 2 , , n x y z n , n x y z is the sample refractive index distribution. As a result, a backward scattered field, is generated. To obtain the depth-resolved measurements, the sample needs to be scanned along the axial direction around the focal plane. Assuming the sample focal displacement is , R z the backward scattered field in the sample and detector space is denoted by ). On the detector plane, there is also a plane wave component, coming from the reference arm, which has a form similar to that of the incident field, The backscattered sample and the reference fields interfere at the detector plane, creating an intensity distribution. From the measured intensity, we obtain which is the real part of the cross-correlation function for each R z . In this paper, we are interested in modeling the physical imaging process, thus, we need to fully describe the cross-correlation function, which requires solving the sample scattered field.

Solving the backward scattered field
The sample scattered field can be described by the inhomogeneous wave equation [14]: where ( ) U r is the total driving field which consists of both the incident and the scattered fields, Under the first-order Born approximation, we have r that allows us to solve the backward scattered field, denoted as , bs U in the z > 0 sample space, for different sample focal displacement as (refer to the Appendix for the derivation) is the scattered field axial projection ( x k and y k have units of m −1 ). Notice that for simplicity, we will use the same representation for a physical parameter in different spaces by carrying the variables throughout this paper. For example, in Eq. (4) χ is in the 3D Fourier transform space as evidenced from its variables. The imaging condition ensures that the field at z = 0 (defined at the sample surface) is conjugated with the camera detector plane, x y P k k has been introduced, which defines the spatial frequency bandwidth limited by the objective numerical aperture. Next, the scattered field solution will be used to calculate the cross-correlation function to obtain the system response function.

System response function
In this section, we will calculate the axial response function in reflection-mode DSIM. First, we calculate the scattered field from a thin step phase object. Then, we calculate the crosscorrelating function 12 Γ by considering the speckle statistics.

Thin step object response
A homogeneous thin object, as described in Fig. 2, is used as the sample to calculate the axial response function. The one-dimensional object has an infinite lateral dimensions and an axial width of z o , thus, its scattering potential can be described with a rectangle function in z as (the constant part of the scattering potential has been ignored, as it does not contribute to the axial response calculation). The 3D Fourier transform of this scattering potential is Substituting the above expression into Eq. (5), we obtain the backward scattered field in the sample space as, Next, we take a 2D inverse Fourier transform of Eq. (7) The dispersion relation of the incident field makes  ik z e signifies the double path of the field in the sample. Interestingly, in transmission-mode operation, the forward scattered field does not have a R z dependent phase term, indicating that it will not be able to provide the sectioning effect for flat objects (see more details in the discussion part).

Speckle-field statistics
Next, we calculate the correlation function while considering the speckle-field statistics. The speckle-field angular spectrum distribution ( ) where N 2 is the number of independent scattering areas. The distributions of ( ) In the above equation, we have changed the notation of ( )

The axial response function
In order to calculate the axial response function, the solution of ( ) 12 2 Re Γ in Eq. (16) The range of r k is limited by the objective numerical aperture through ( ) The above integral can be easily evaluated to give an analytical solution as sin c 2 cos sin c 2 cos , where the sinc function is defined as: The above framework establishes a mathematical model that describes the axial response of the reflection-mode DSIM system: if we know the objective numerical aperture, the illumination wavelength, we can compute ( ) 12 2 Re Γ as a function of R z to obtain the axial response function in DSIM. Finally, with the axial response function, the depth resolution can also be determined.

Depth resolution
In this section, the axial response function model is tested using the specifications from an experimental system, and this model is subsequently used to quantify the sectioning effect in terms of depth resolution. For this study, we use the parameters from our previous experiment [25], which also provides a way to validate our theoretical model. In that reflection-mode DSIM system, the laser wavelength is 0 0.8 λ = μm, the sample host medium is water with refractive index 1.33, n = and two water immersion objectives are used with 1.

obj NA =
Inserting these parameters into Eq. (21), we obtain the axial response function as shown in Fig. 3, where the solid black curve is the axial response function, i.e.,  achieved. It should be noted that in order to get the best depth resolution, a uniform speckle spectral distribution over the whole back aperture area of the high numerical aperture objective is necessary.

Discussion
We have developed a physical model to precisely describe the sectioning effect in a reflection-mode speckle-field illumination interferometric system. The sectioning effect comes from the spatially incoherent illumination. It is also possible to use a broadband source in such a system to further enhance the depth selectivity. However, it is not clear how much depth selectivity can be enhanced with this addition. In principle, this theoretical framework can be extended to incorporate temporal coherence to answer this interesting question. We note that frameworks considering temporal coherence have been reported for transmission case in earlier publications [31,32]. Another important question is whether transmission-mode DSIM can provide sectioning effect. In order to answer this, we calculate the forward scattered field using Eq. (33b) in the Appendix for the same step object described in Section 3. The field is given as Similarly, we can Fourier transform the field into the spatial domain representation, It turns out that the forward scattered field, as described in Eq. (25), is not a function of R z , thus giving no sectioning effect. Note that the above calculation assumes flat thin objects, with no lateral structures. However, for objects that have lateral features, there will be sectioning as was demonstrated in [36]. The missing axial frequency information in the low transverse region is called the "missing cone" problem in 3D optical imaging. Our following paper will discuss this issue in more details by calculating the 3D CTF in both reflection and transmission-mode DSIM systems.

Summary
In conclusion, we have developed a mathematical model to describe the axial response function in reflection-mode dynamic speckle-field interferometric microscopy. This model is based on the diffraction tomography theory and speckle statistics, and provides a spatial correlation function. Using this function, the axial response function is obtained and used to determine the depth resolution. The theoretically calculated depth resolution is in excellent agreement with our experimental results. Using this method, the connection between depth resolution and objective numerical aperture is also studied, which reveals an inverse square law relationship that is also expected. We envision that developed physical model will contribute to the understanding of sectioning effect in spatially incoherent illumination interferometry systems. It can also guide on the design of such systems for better performance in the future.

Appendix: optical diffraction tomography
We start with the inhomogeneous wave equation that describes the scattered field [