# Reflective Inverse Diffusion
## 2 Background
### Mathematical Model  
Scalar diffraction theory was used to develop a mathematical description of reflective inverse diffusion. The propagation distances of the experiment are all inside the Fresnel zone; thus, the Rayleigh-Sommerfeld diffraction formula is used for propagation. The transfer function for Rayleigh-Sommerfeld diffraction is [18]:  
\[H(f_{x},f_{y})=\exp\left(jkz\sqrt{1-(\lambda f_{x})^{2}-(\lambda f_{y})^{2}} \right), \tag{11}\]where \(z\) is the propagation distance, \(\lambda\) is the optical wavelength, \(k\) is the wavenumber \(2\pi/\lambda\) and \(f_{x}\) and \(f_{y}\) are the respective horizontal and vertical spatial frequencies of the source field. Using the transfer function to propagate the source field, \(U_{\text{src}}(x,y)\), a distance \(z\), the observed field is given by [19]:  
\[U_{obs}(x,y)=\frac{z}{j\lambda}\;\mathcal{F}^{-1}\{\mathcal{F}\{U_{src}(x,y)\} \;H(f_{x},f_{y})\}, \tag{12}\]  
where \((x,y)\) are the Cartesian coordinates orthogonal to \(z\) corresponding to the spatial frequencies \((f_{x},f_{y})\).  
The field at each pixel of the SLM when illuminated by a plane wave can be considered simply as the phase delay applied at that pixel. Using Equation (12), the field is propagated from the SLM to just prior to lens \(L2\) in Figure 1, normalized with constant phase terms removed,  
\[U_{L2}^{-}(x,y)=\mathcal{F}^{-1}\{\mathcal{F}\{U_{SLM}(x,y)\}\;H(f_{x},f_{y})| _{z=R_{0}}\}, \tag{13}\]  
where \(R_{0}\) is the distance from the SLM to lens \(L2\). The Fourier transform property of lens \(L2\) is then used to determine the field at the diffusely-reflecting sample located at the lens focus,  
\[\begin{split} U_{sample}^{-}(x,y)&=e^{j\frac{k}{2 }(u^{2}+v^{2})}\mathcal{F}\{U_{lens}^{-}(x,y)\}\\ &=e^{j\frac{k}{2}(u^{2}+v^{2})}\mathcal{F}\{\mathcal{F}^{-1}\{ \mathcal{F}\{U_{SLM}(x,y)\}\;H(f_{x},f_{y})|_{z=R_{0}}\}\}\\ &=e^{j\frac{k}{2}(u^{2}+v^{2})}\mathcal{F}\{U_{SLM}(x,y)\}\;H(f_ {x},f_{y})|_{z=R_{0}}.\end{split} \tag{14}\]  
Lens L2 causes a coordinate transformation to the \((u,v)\) plane, which is related to the spatial frequencies by \(u=\frac{\lambda f_{x}}{z}\) and \(v=\frac{\lambda f_{y}}{z}\), where \(z=R_{0}\)[18].  
The field at the sample given by Equation (14) is multiplied by \(e^{j\Theta(u,v)}\), which represents the phase scattering properties of the reflector. The result is then propagated to the charge-coupled device (CCD) in the observation plane using Equation (12):  
\[U_{CCD}(u,v)=\mathcal{F}^{-1}\{\mathcal{F}\{e^{j\frac{k}{2f}(u^{2}+v^{2})}e^{ j\Theta(u,v)}\mathcal{F}\{U_{SLM}(x,y)\}H(f_{x},f_{y})|_{z=R_{0}}\}\;H(f_{u},f_{y})| _{z=R_{2}}\}, \tag{15}\]  
where \((x,y)\) are the coordinate axes of the SLM and \((f_{x},f_{y})\) are the respective horizontal and vertical spatial frequencies of the field at the SLM. The coordinate axes of the reflective sample are \((u,v)\), and \((f_{u},f_{y})\) are the respective horizontal and vertical spatial frequencies of the field at the reflector. The focal length of lens \(L2\) is \(f\), and \(R_{0}\) and \(R_{2}\) are the distances from the SLM to the lens and the reflective sample to the CCD, respectively. This transform relationship is unique to the reflective inverse diffusion setup in Figure 1. Transmissive inverse diffusion uses microscope objectives on both sides of the scattering sample that re-image the light onto the CCD (see Figure 2 in [9]). The differences in the experimental setups create different transform relationships for reflective and transmissive inverse diffusion. However, since both processes are linear, Equation (1) is valid for both.  
Figure 1: Reflective inverse diffusion setup. A vertically-polarized HeNe laser is expanded, collimated and normally incident on the SLM. The phase modulated beam is then focused onto the reflecting sample with a positive lens, and the reflected speckle pattern is recorded by the CCD.