1118 Anatoly K. Khitrin et al. Two-dimensional (2D) generalization can be done as


follows. By analogy with Equation (4), the position vector (x',y') on the focal plane, where the continuation of refracted rays crosses the focal plane, can be written as:


ðÞ= x; y x0; y0 ðÞ+hx; y ðÞ∇hðx; yÞ; ðÞ n - 1 x0 =x +hx; y ðÞðÞ n - 1 ∂hx; y


ðÞ ∂x ; y0 =y +hx; y


ðÞ n - 1 ðÞ ∂hx; y (8)


where ∇ is the 2D gradient in the xy-plane. In terms of the components:


ðÞ ∂y : (9)


The line elements dx and dx' in Equation (6) have to be


replaced by the area elements dxdy and dx'dy', and the equivalent of Equation (6) takes the form:


I -1 x0; y0ðÞ= ∂ðx0; y0Þ : ∂ðx; yÞ


   


(9), one obtains: I - 1 x0; y0ðÞ= ∂x0


determinant ∂ðx0;y0Þ ∂ðx;yÞ


  


∂x


   


(10) Thus, the inverse intensity of the image is the Jacobian


  . By substituting x' and y' from Equation


∂y0 ∂y - ∂y0


∂x


∂x0 ∂y = 1 + ðn - 1Þ h ∂2h


"# 1 + ðn - 1Þ h ∂2h 2


() ∂y2 + ∂h


dy - n - 1


2 ðÞ h ∂2h


 dx


∂x∂y + ∂h


∂h dy


: ð11Þ This equation can be simplified if we assume that var-


iations of the image intensity are small compared with the average intensity. Then:


I -1 x0; y0ðÞ=1 + n - 1 ðÞ h ∂2h


=1 + n - 1 2 ∇2h2 x; y


() ∂x2 + ∂2h 2


+ ∂h dx


∂y2


=1 + n - 1 fgðÞ∇  hx; y ðÞ;


ðÞ∇hx; y ðÞ ð12Þ


where ∇2 denotes the 2D Laplacian in the xy-plane. This can be viewed as a general equation for image intensity. It can be readily rearranged as a Poisson’s equation for object height h in terms of known quantities (measured intensity and refractive index) as:


n - 1 I -1 x0; y0ðÞ- 1 =∇2h2 x; y


2 ðÞ; (13)


which can be solved uniquely for h2 provided boundary conditions on h2 are supplied. Equation (13) has a strikingly similar form to the transport-of-intensity equation (TIE), which will be discussed in the following section, in that the measured intensity for a pure-phase object is related to a function of the sample’s thickness through a Poisson’s equation. Therefore, many of the techniques developed for solving the TIE may be applicable to this model as


+ ∂h dy


2 "#


() ∂x2 + ∂h


2 dx





well (Paganin and Nugent, 1998; Volkov et al., 2002; Bardsley et al., 2011; Tian et al, 2012; Kostenko et al., 2013; Martinez-Carranza, et al., 2013). To create a more easily interpretable contrast, one can collect several images taken under slightly different conditions:


(1) Shift of the focal plane A shift of the focal plane by dz corresponds to the sub-


stitution h(x,y)→h(x,y)−dz in Equation (12) and has no effect on the derivatives. Therefore, one can construct the difference:


I -1 2 x0; y0ðÞ- I -1 1 x0; y0ðÞ= -dz n - 1 ðÞ∇2hðx; yÞ; (14)


where I1 and I2 are the image intensities at two different positions of the focal plane. Now Equation (14) has a simple interpretation: the contrast is proportional to the local curvature of the object boundary. This is a well-known fact, used, for example, in “defocusing” microscopy (Agero et al., 2004). It is interesting to compare this result with TIE


(Teague, 1983; Streibl, 1984) obtained fromthe paraxial wave equation. The TIE equation for the “logarithmic derivative” [Equation (7b) in Streibl, 1984 at uniform transmittance] is:


lnIx; y ; z=0 dz ðÞ= -∇2φðx; yÞ; d (15)


where φ(x,y) is the phase. If we use a low-contrast approxi- mation [as in Equation (12)] and realize that (n−1)h(x,y)is equivalent to the phase ϕ(x,y), then Equations (14) and (15) become identical. Both are the Poisson equations for the object profile h(x,y), where the left side represents experi- mental data. In the presence of image noise and for objects with complex shape, this equation is difficult to solve. Indeed, by applying the 2D Gauss theorem, one can see that pertur- bation from a noisy pixel does not decay, but grows logarith- mically with the distance from the pixel. Within the TIE approach, various computational methods have been developed to minimize artifacts in restored phase maps (Volkov et al., 2002; Waller et al., 2010; Bardsley et al., 2011; Bie et al., 2012; Tian et al., 2012; Zheng et al., 2012; Kostenko et al., 2013; Petruccelli et al., 2013; Zuo et al., 2013; Jingshan et al., 2014). Note that although both Equations (13) and (14) are


Poisson’s equations, they are based on two different mechanisms of contrast generation. In both cases, it is propagation of refracted rays that generates measurable intensity contrast, where refraction depends on variations in the object’s height h. In the case of Equation (13), propaga- tion is through the object itself, which is why the Poisson’s equation depends on the square of object height. In Equation (14), defocus by an amount dz is used to generate the contrast so the Poisson’s equation depends on the product of dz and the object height. In the practical realization of the vertical shift method,


two additional considerations apply. First, the vertical shift of the focal plane is not equivalent to the vertical shift of the objective (or of the stage), which is the only distance reported by the microscope hardware. Thus, to obtain the true dz to be


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