In a recent paper by Zhang and Ding [1], it is claimed that an error in the calculation of the Poynting vector was found in [2]. They claimed that they found a simple expression for the magnetic field calculation for highly focused radially-polarized light beams. With their formula, they found that their calculated Poynting vector had higher magnitude than that was reported in [2]. Thus they concluded that an error occurred in [2]. The authors further provided a “guess”, stating that a simple scaling of the radial component of the electrical focal field was used to calculate the magnetic field in [2].

Unfortunately, these claims are incorrect. First of all, the expression for magnetic field calculation is very straightforward and can be easily adapted from the azimuthal polarization focal field formula that is widely known (for example, Eq. (10) in [3]). The Poynting vector in [2] was indeed calculated with a formula that is exactly the same as Eq. (7) in [1]. Thus the “guess” offered by the authors of [1] is not right.

The reason for the difference is very likely due to how the electric field amplitude is calculated for a given input power. One should pay attention to a very important assumption in the well-known Richards-Wolf method [4], which is the index of refraction on the object and image space are the same. This assumption was used to relate the electric field amplitude in the object space to the electric field amplitude in the image space through the conservation of power flow [Eq. (2.10)] on page 362 of [4]). In order to use the Richards-Wolf formula to quantitatively estimate the focal field, the field amplitude in the object space must be evaluated in a medium that has the same index in the image space. Otherwise, errors will occur.

Let’s compare a uniform beam with cross section S, electrical field amplitude E₀ in air (index is 1) and E₁ in a medium (index of n) respectively. Assume the power of the beam is P for both media (equivalent to the assumption of ignoring the reflection and refraction losses in [4]), we have:

If the index of refraction in the image space is n and one wants to use the Richards-Wolf method [4] and its derivatives [3], it is the E₁ that should be used in the equations as the input pupil field amplitude for quantitative calculations. Obviously, the pupil electrical field amplitude to be used in Eqs. (3) and (4) in [1] will be ntimes lower than the amplitude E₀ calculated in air. If the authors calculated the focal electrical field distributions with E₀, instead of E₁, the focal electrical field will be increased by factor n and the same thing happens to the magnetic field in Eq. (7) in [1]. As a consequence, the Poynting vector will be increased by a factor of n. In my paper [2], the peak of the <S_z> shown in Fig. 3(b) is 9.08 × 10¹⁰ W/m². Multiplying this value with n = 1.33 (water) gives a peak value of 12.08 × 10¹⁰ W/m², which is exactly the peak value reported in Fig. 2(b) in [1].

In most previous publications dealing with the highly focused cylindrical vector (CV) beams, only the relative field distributions are concerned. In those cases, the Richards-Wolf method and its derivatives can be used without paying much attention to the absolute field amplitude. As a matter of fact, typically an arbitrary constant is used and it gets normalized. However, for quantitative estimation in applications such as optical tweezers, these details are important and should be paid attention to. For a given power of illumination, the field amplitude should be calculated within a medium with the same index of refraction as the image space. Another equivalent approach would be to re-derive Eq. (2.10) in [4] to account for the index of refraction difference between the object space and image space. And the results will be the same as I reported here and in [2].

In conclusion, the real reason for the discrepancies between the results reported in [1] and [2] is given. The magnetic field in [2] was calculated with the correct formula for an azimuthal magnetic pupil field. The discrepancy between [1] and [2] was caused by the wrong electric field amplitude E₀ used in the calculations in [1].