## Abstract

We correct several errors in the derivation of the MTF shown in our original manuscript.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In our original manuscript, [1] Eqs (1) through (9) provided a derivation of the MTF for a partial-coherence measurement model of a DIC microscope. Our first error was a typo appearing in Eq. (6), where one of the terms is missing a factor of $\frac{1}{2}$. The correct Eq. (6) is:

Our second error was in assuming that the polarization matrices that we originally obtained from Ref. 2 were correct. On reflection, it is obvious that they cannot be. Ref. 2 gives the Jones matrices for a 45° linear polarizer as

Likewise, the matrices representing the retardances induced by the Nomarski prisms, ${\mathbf{N}}_{1}$ and ${\mathbf{N}}_{2}$, were given by Ishiwata *et al.* as

These are not a valid retarder matrices, and should instead be

These when combined together form the matrix $\mathbf{A}=\mathbf{P}\left(\pm {45}^{\circ}\right){\mathbf{N}}_{2}{\mathbf{N}}_{1}\mathbf{P}\left({45}^{\circ}\right)$ for the two analyzed polarization directions.

When these revised forms for the polarization matrices are inserted into the equation for the transmission cross-coefficient *R*, we obtain

**R**must take the form of a 2 × 1 vector rather than the scalar represented in Ref. 2 and our original manuscript. Thus, the illumination model $\mathbf{Q}\left(\xi \right)$ too must be expanded to take the form of a 2 × 1 polarization vector rather than a scalar. Finally, the equation representing the intensity distribution at the detector array, Eq. (1) in our original manuscript, should be calculated as the sum of squares of the two elements of the polarization vector. Since

**A**has the simple form

In addition, in the statement below Eq. (9), we stated that the background component (a) of Eq. (8) consists of $M\left(0\right)$ plus an integral that evaluates to a constant. Due to the presence of the variable *x* inside the integral, however, it does not evaluate to a constant but instead to a function of *x*.

The remainder of the paper, including the experimental results, is unaffected by each of these corrections.

Finally, in the acknowledgements, Dr. Ishiwata’s first name was given incorrectly as Wataru, but should have been given as Hiroshi.

## Funding

Japan Society for the Promotion of Science (15H03941).

## References

**1. **S. Shibata, W. Takano, N. Hagen, M. Matsuda, and Y. Otani, “Video-rate quantitative phase analysis by a DIC microscope using a polarization camera,” Biomed. Opt. Express **10**, 1273–1281 (2019). [CrossRef] [PubMed]

**2. **H. Ishiwata, M. Itoh, and T. Yatagai, “A new method of three-dimensional measurement by differential interference contrast microscope,” Opt. Comm. **260**, 117–126 (2006). [CrossRef]