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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 25 — Dec. 11, 2006
  • pp: 12494–12496
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Comment on: Computation of the optical trapping force using an FDTD based technique

Fei Zhou, Xiaosong Gan, Wendong Xu, and Fuxi Gan  »View Author Affiliations


Optics Express, Vol. 14, Issue 25, pp. 12494-12496 (2006)
http://dx.doi.org/10.1364/OE.14.012494


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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

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]

] and the rigorous method using Maxwell’s stress tensor [4

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]

] or direct invocation of the Lorentz force law [5

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]

]. This letter is intended as a comment on recently published papers: “Computation of the optical trapping force using an FDTD based technique” [1

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

] and “Rigorous time domain simulation of momentum transfer between light and microscopic particles in optical trapping” [2

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]

]. Reference [1

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

] 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

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]

] 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

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

] and [2

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]

] 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

dp=(S·σ)dtdaћωћkkk,
=nckk(S·σ)dtda
(1)

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

<F>=<dPdt>=<nckk(S·σ)da>,
(2)

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

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]

] are, however, different:

dPda=(nc)S
(3)
F=ncΔS·dA
(4)

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

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]

], it is stated that “ΔS 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 “ΔS” in Eq. (4) should be the Poynting vector S⃗. It is defined in Ref. [2

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]

] 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

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]

] 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

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]

], 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

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

], 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

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]

]. 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

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

] discusses a 2D problem using the similar formula given in Ref. [2

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]

]. 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:”

Fx=ΔSxcL,ΔSx=SxoSx
Fy=ΔSycL,ΔSy=SyoSy
(5)

“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

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]

], 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).

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Trapping

History
Original Manuscript: January 27, 2006
Revised Manuscript: April 13, 2006
Manuscript Accepted: April 15, 2006
Published: December 11, 2006

Virtual Issues
Vol. 2, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Fei Zhou, Xiaosong Gan, Wendong Xu, and Fuxi Gan, "Comment on: Computation of the optical trapping force using an FDTD based technique," Opt. Express 14, 12494-12496 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12494


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