## Abstract

In this comment, problems associated with an oversimplified FDTD based model used for trapping force calculation in recent papers “Computation of the optical trapping force using an FDTD based technique” [Opt. Express 13, 3707 (2005)], and “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping” [Opt. Express 12, 2220 (2004)] are discussed. A more rigorous model using in Poynting vector is also presented.

© 2006 Optical Society of America

For trapping force calculation, there are several approaches, such as ray optics (RO) based methods [3] and the rigorous method using Maxwell’s stress tensor [4] or direct invocation of the Lorentz force law [5]. This letter is intended as a comment on recently published papers: “Computation of the optical trapping force using an FDTD based technique” [1] and “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping” [2]. Reference [1] discusses the calculation of the trapping force from differences in Poynting vector between “two step” simulations. The first step is determining the Poyning vector without a particle inside the computational domain, and the second step is carried out with a particle inside the computational domain. And Ref. [2] discusses the process of momentum exchange between light and the particle, basing on the “two step” simulations. It seems reasonable that the difference in Poynting vector is due to the particle influence and one can calculate the radiation pressure from this difference. However, the “two step” method, which only adding up the Poynting vector at specific grids, as Refs. [1] and [2] do, is an oversimplification in using the principle of conservation of momentum.

The amount of energy and momentum carried by a photon is *ħω* and *ħk⃗*, respectively, where *ħ* is the Planck’s constant, *ω* is the angular frequency and *k⃗* is the wave vector. For any given enclosed surface including the object under investigation, the electromagnetic energy passing through a surface element *da* during time *dt* can be express as *dE*=*S⃗*·*σ⃗dtda*, where *S⃗*=*E⃗*×*H⃗* is the Poynting vector, *σ⃗* is the unit vector of a surface element. The amount of optical momentum *dp⃗* passing through the surface element *da* in the period of *dt* can be expressed as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\frac{n}{c}\frac{\overrightarrow{k}}{\mid k\mid}(\overrightarrow{S}\xb7\overrightarrow{\sigma})dtda$$

where *n* is the refractive index and *c* is the speed of light. By applying the principle of conservation of momentum on the enclosed surface, the time averaged force is given by

where <> is the time average, *P⃗* is the total optical momentum in the enclosed surface. Equation 2 is derived from the rate of change of optical momentum, represented by the Poynting vector. The expressions given in Ref. [2] are, however, different:

The main difference between Eq. (1) and Eq. (3) lies in handling the directions of energy flow and surface normal. The Eq. (3) is oversimplified and valid only when the Poynting vector is normal to the surface element. Equation (4) is quite confusing, if the dot is treated as “dot product”, the left side is a vector and the right side is a scalar. In Ref. [2], it is stated that “*Δ S* represents the difference between the energy density flux through the unit area traveling into the object and coming out of the object,” which indicates that the “

*Δ*” in Eq. (4) should be the Poynting vector

**S***S⃗*. It is defined in Ref. [2] of a momentum transfer region (MTR), with a diameter of two grids spacing larger than that of the particle. It is stated in Ref. [2] that “We add up the light momentum passing through MTR without and with the particle, and then compare the momentum change between the two cases. Due to conservation of momentum, the difference of light momentum in the two cases should be transferred to the particle.” The optical momentum calculations in Ref. [2], by summing the Poynting vectors in the MTR, are meaningful, since it is not calculated using Eq. (3), but using the volume integration of the optical momentum density

*g⃗*=

*n*

^{2}

*S⃗*/

*c*

^{2}[6], which is proportional to the Poynting vector

*S⃗*. According to the principle of conservation of momentum, the difference of the optical momentum with and without the presence of the particle is the momentum transferred to the particle. However, such momentum conservation is only applicable when the volume integration is performed over the entire space, not just in the MTR defined in Ref. [2]. Due to the presence of the particle, the optical field outside the MTR is also affected, hence, causing a change of optical momentum outside the MTR. Under such circumstance, only considering the change inside the MTR region, which is only two grids spacing larger than the particle is not appropriate in using the principle of conservation of momentum.

Reference [1] discusses a 2D problem using the similar formula given in Ref. [2]. It is stated that “The reference and object Poynting vector values calculated can be used to obtain the optical forces on the object. … In a 2-D computation domain with the Δx and Δy grid space equal, the X and Y force components can be computer through:”

$${F}_{y}=\frac{{\Delta S}_{y}}{c}L,\phantom{\rule{.9em}{0ex}}\Delta {S}_{y}={S}_{yo}-{S}_{y}$$

“with L equal to the number of points over which the Poynting value is computed multiplied by the axial grid point spacing.”

Equation (5) is a simplified form of Eq. (4) for the two dimensional case, which is valid only when the directions of energy flow and surface normal are coincidence.

In summary, the direction of the Poynting vector can be significantly different from the normal of a surface element, e.g. under high numerical aperture objective cases, and therefore, the force calculation from Eqs. (3), (4) and (5) is improper in most optical trapping cases. In addition, the application of the “two step” method, which only takes into account the momentum changing within the MTR defined in Ref. [2], is inappropriate for the principle of the momentum conservation.

## References and links

**1. **R. C. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express **13**, 3707–3718 (2005). [CrossRef] [PubMed]

**2. **D. Zhang, X.-C. Yuan, S. C. Tjin, and S. Krishnan, “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping,” Opt. Express **12**, 2220–2230 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2220 [CrossRef] [PubMed]

**3. **A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. **61**, 569–582 (1992). [CrossRef] [PubMed]

**4. **W. L. Collett, C. A. Ventrice, and S. M. Mahajan, “Electromagnetic wave technique to determine radiation torque on micromachines driven by light,” Appl. Phys. Lett. **82**, 2730–2732 (2003). [CrossRef]

**5. **A. R. Zakharian, M. Mansuripur, and J. V. Moloney, “Radiation pressure and the distribution of electromagnetic force in dielectric media,” Opt. Express **13**, 2321–2336 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2321 [CrossRef] [PubMed]

**6. **J. D. Jackson, *Classical Electrodynamics* (Wiley, New York, 1962).