## Abstract

We comment on a recent paper by Kim et al. [Opt. Exp. **20**(6) 6737-6745 (2012)], in which the authors
claimed to present a new method for first-order differentiation of phase objects called gradient field microscopy
(GFM). We consider that the method does not substantially differ from well-known Fourier methods discussed in
textbooks. Also, we discuss some deficiencies of the paper.

© 2013 OSA

In a recent paper, Kim et al. [1] presented a method for first-order differentiation of phase objects called gradient field microscopy (GFM). Our purpose in the present Comment is not to discuss the applicability of the GFM for the study of biological cells, with what we agree.

The point we like to remark is the following: The authors of [1] claim to use
a sine-function in the *y*-direction of the spatial frequency domain, $H({k}_{x},{k}_{y})=A\left[1+\mathrm{sin}(a{k}_{y})\right]$. This filter function gives the convolution $\left[\delta (x,y)+\frac{\delta (x,y-a)-\delta (x,y+a)}{2i}\right]*U(x,y)$ across the image plane. The important issue is that the approximation used in Eq. (1)
of [1], $\left[\frac{\delta (x,y-a)-\delta (x,y+a)}{2i}\right]*U(x,y)\approx ia\frac{\partial U(x,y)}{\partial y}$, is only valid in the limit $a\to 0$.

The limit $a\to 0$ means that the period of $\mathrm{sin}(a{k}_{y})$ must be larger than the spatial frequencies involved in $U({k}_{x},{k}_{y})$. In other words, they are not using a sine function at all, but only a small portion of
the spatial filter around the origin of the *y*-coordinate in the Fourier plane, in which
$\mathrm{sin}(a{k}_{y})\approx a{k}_{y}$. Thus, actually they are using a linear function (i.e., a ramp) for performing the
spatial filtering instead of a sine-function as claimed. In fact, it could be absolutely equivalent to use any other
function ($F(a{k}_{y})$) with the behavior $F(a{k}_{y})\approx a{k}_{y}$ in the limit $a\to 0$, since the appearance of a first derivative is actually granted by the formula
$FT\left[{k}_{y}U({k}_{x},{k}_{y})\right]=i\frac{\partial U(x,y)}{\partial y}$.

We like to mention that the proposed method does not substantially differ from well-known Fourier methods discussed in several old papers and textbooks. For example, the “gradient field” as result of spatial filtering is discussed in references [2,3], and a review (also improvements to the method described by the authors of [1]) is presented in [4].

## References and links

**1. **T. Kim, S. Sridharan, and G. Popescu, “Gradient field microscopy of unstained specimens,” Opt.
Express **20**(6), 6737–6745 (2012). [CrossRef] [PubMed]

**2. **S. K. Yao and S. H. Lee, “Spatial differentiation and integration by coherent optical-correlation
method,” J. Opt. Soc. **61**(4), 474–477 (1971). [CrossRef]

**3. **J.K.T. Eu, C.Y.C. Liu, and A.W. Lohmann, “Spatial filters for differentiation,” Opt. Comm .
**9**(2), 168-171 (1973).

**4. **G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, eds.,
*The New Physical Optics Notebook: Tutorials in Fourier Optics* (SPIE, 1989), Chap.
32.