## Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle

Optics Express, Vol. 17, Issue 3, pp. 1753-1765 (2009)

http://dx.doi.org/10.1364/OE.17.001753

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### Abstract

Propagations of coherent and partially coherent flat-topped beams through a focusing optical system are formulated. The radiation force on a Rayleigh dielectric sphere induced by focused coherent and partially coherent flat-topped beams is investigated theoretically. It is found that we can increase the transverse trapping range at the planes near the focal plane by increasing the flatness (i.e., beam order) of the flat-topped beam, and increase the transverse and longitudinal trapping ranges at the focal plane by decreasing the initial coherence of the flat-topped beam. Moreover the trapping stiffness of flat-topped beam becomes lower as the beam order increases or the initial coherence decreases. The trapping stability is also analyzed.

© 2009 Optical Society of America

## 1. Introduction

9. M. S. Bowers, “Diffractive analysis of unstable optical resonator with super-Gaussian mirrors,” Opt. Lett . **19**, 1319–1321 (1992). [CrossRef]

4. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A: Pure Appl. Opt . **8**, 537–545 (2006). [CrossRef]

9. M. S. Bowers, “Diffractive analysis of unstable optical resonator with super-Gaussian mirrors,” Opt. Lett . **19**, 1319–1321 (1992). [CrossRef]

21. G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun . **260**, 687–690 (2006). [CrossRef]

22. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am . A **23**, 2623–2628 (2006). [CrossRef]

23. R. Borghi and M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett . **23**, 313–315 (1998). [CrossRef]

24. Y. Zhang, B. Zhang, and Q. Wen, “Changes in the spectrum of partially coherent flat-top light beam propagating in dispersive or gain media,” Opt. Commun . **266**, 407–412 (2006). [CrossRef]

25. M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng . **46**, 357–362 (2008). [CrossRef]

27. Y. Baykal and H. T. Eyyuboglu, “Scintillations of incoherent flat-topped Gaussian source field in turbulence,” Appl. Opt . **46**, 5044–5050 (2007). [CrossRef] [PubMed]

28. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett . **33**, 1795–1797 (2008). [CrossRef] [PubMed]

## 2. The characteristics of coherent and partially coherent flat-topped beams

11. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett . **27**, 1007–1009 (2002). [CrossRef]

11. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett . **27**, 1007–1009 (2002). [CrossRef]

*E*

_{0N}is a normalization factor,

*w*

_{0}is the waist size of the fundamental Gaussian mode,

*N*is called the beam order of the flat-topped beam, When

*N*=1, Eq. (1) reduces to the electric field of a Gaussian beam. Here we assume the power of the flat-topped beam to be

*P*. Following [32

32. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun . **124**, 529–541 (1996). [CrossRef]

*n*is the refractive index of the surrounding medium,

_{m}*ℰ*

_{0}is the dielectric constant, and

*c*is the speed of the light in vacuum. Substituting Eq. (1) into Eq. (2), we can obtain the expression for the normalization factor

*E*

_{0N}as follows

*P*=1W, which means every beam discussed in the following text carries the same power. Figure 1 shows the intensity distributions of a flat-topped beam for four different values of

*N*with

*w*

_{0}= 10

*mm*. One finds from Fig. 1 that the beam profile becomes flatter (i.e., beam width increases) but the maximum intensity decreases as

*N*increases, which means the potential wells of flatter beams (for example

*N*=4,

*N*=3) are

*lower*or

*shallower*than their counterparts (for example

*N*=2,

*N*=1). Fig. 1(b) is consistent with Fig. 1(a) of [11

11. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett . **27**, 1007–1009 (2002). [CrossRef]

18. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt . **6**, 390–395 (2004). [CrossRef]

**r**= (

^{T}*x y*),

*k*= 2

*π*/

*λ*is the wave number with

*λ*being the wavelength,

**Q**

_{1n}

^{-1}is a 2×2 matrix named complex curvature tensor, and is given by

18. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt . **6**, 390–395 (2004). [CrossRef]

*I*(

**r**,

*z*= 0) = Γ(

**r,r**,

*z*= 0) is the intensity distribution of the partially coherent beam, and

*g*(

**r**

_{1}-

**r**

_{2}) is the spectral degree of coherence given by

*σ*

_{0}being the transverse spatial coherence width.

21. G. Wu, H. Guo, and D. Deng, “Paraxial propagation of partially coherent flat-topped beam,” Opt. Commun . **260**, 687–690 (2006). [CrossRef]

22. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am . A **23**, 2623–2628 (2006). [CrossRef]

_{0}= 0), Eq. (10) reduces to

**r̃**

^{T}= (

**r**

_{1}

^{T}

**r**

_{2}

^{T}) = (

*x*

_{1}

*y*

_{1}

*x*

_{2}

*y*

_{2}) and

**M**

_{1nm}

^{-1}is a 4×4 complex matrix given by

**I**is a 2×2 unit matrix. Within the validity of the paraxial approximation, by applying the tensor ABCD law of partially coherent beam [50

50. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett . **27**, 216–218 (2002). [CrossRef]

ρ ˜

^{T}= (

**ρ**

_{1}

^{T}

**ρ**

_{2}

^{T}),

**Ā, B̄, C̄**and

**D̄**are defined as follows:

## 3. The focusing properties of coherent and partially coherent flat-topped beams

*f*is located at the input plane (z=0), and the output plane is located at

*z*=

*f*+

*z*

_{1}, where

*z*

_{1}is the axial distance from the focal plane to the output plane. Then the transfer matrix between the input and output planes can be expressed as follows

7. W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Z. Chen, Y. Gu, and S. J. Wang, “Field description and electron acceleration of focused flattened Gaussian laser beams,” Europhys. Lett . **73**, 211–217 (2006). [CrossRef]

*N*at several propagation distances with

*f*= 10

*mm*,

*w*

_{0}= 10

*mm*,

*λ*= 632.8

*nm*. One finds from Fig. 3 that the focusing properties of a flat-topped beam (

*N*>1) are very interesting and different from that of Gaussian beam (

*N*=1). At the focal plane (

*z*

_{1}=0), the intensity of the focused flat-topped beam is of quasi-Gaussian distribution, and maximum intensity decreases as the beam order

*N*increases. When the output plane is a little away from the focal plane, the intensity of focused flat-topped beam becomes flat-topped, and the beam profile becomes flatter but the maximum intensity decreases as

*N*increases. From Eq. (1), one can find that the flat-topped beam is not a pure mode, but a combination of different Gaussian modes, different Gaussian modes evolve differently upon propagation, what’s more, different modes overlap and interfere during propagation, thus leading to the interesting focusing properties of flat-topped beam. More information about the propagation properties of the flat-topped beam can be found in [10

10. F. Gori, “Flattened gaussian beams,” Opt. Commun . **107**, 335–341 (1994). [CrossRef]

20. X. Lü and Y. Cai, “Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system,” Opt. Commun . **269**, 39–46 (2007). [CrossRef]

## 4. Radiation force produced by focused coherent and partially coherent flat-topped beams on a Rayleigh particle

*a*≪

*λ*). The schematic for trapping a Rayleigh dielectric sphere with a focused flat-topped beam is shown in Fig. 2. A Rayleigh dielectric sphere with refractive index

*n*is placed near the focus. In this case, the particle is treated as a point dipole. It’s well known that there are two kinds of the radiation force: scattering force and gradient force. The scattering force

_{p}*F*

_{Scat}caused by the scattering of light by the sphere is proportional to light intensity and is along the direction of light propagation. The scattering force can be expressed as [32

32. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun . **124**, 529–541 (1996). [CrossRef]

*I*is the intensity of the focused beam at the output plane,

_{out}*e→*is a unity vector along the beam propagation,

_{z}*α*= (8/3)

*π*(

*ka*)

^{4}

*a*

^{2}[(

*γ*

^{2}-1)/(

*γ*

^{2}+2)]

^{2},

*γ*=

*n*/

_{p}*n*. The gradient force

_{m}*F*

_{Grad}produced by non-uniform electromagnetic fields is along the gradient of light intensity, and is expressed as [32

32. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun . **124**, 529–541 (1996). [CrossRef]

*θ*=

*a*

^{3}(

*γ*

^{2}- 1)/(

*γ*+ 2). By applying the Eqs. (6), (7), (14)–(16), (18) and (19), we can calculate the radiation force induced by focused coherent and partially coherent flat-topped beams on a Rayleigh dielectric sphere. We choose the radius of the particles to be

*a*= 50 nm, the refractive index of the particle to be

*n*= 1.59 (i.e., glass) and the refractive index of the ambient to be

_{p}*n*= 1.33 (i.e., water).

_{p}*N*at different positions

*z*

_{1}, and in Figs. 5(g) and 5(h) the longitudinal gradient force for two different transverse positions

*x*. The sign of radiation force means the direction of the force. Positive

*F*

_{Scat}means the direction of the scattering force is along +z direction. Positive

*F*

_{Grad-z}or

*F*

_{Grad-x}means the direction of the gradient force is along +z or +x direction. One finds from Fig. 5 that both scattering force and gradient force decrease as the initial beam order

*N*increases (i.e., beam profile becomes flatter), which means that the trapping stiffness (i.e., stability) of flatter beam is lower. What’s more, the forward scattering force always is much smaller than the longitudinal gradient force, so the scattering force in this case can be neglected. From Figs. 5(d) and 5(g), one finds that at the focal plane (

*z*

_{1}= 0), there is one stable equilibrium point, and we can use focused flat-topped beam to trap a Rayleigh particle whose refractive index is larger than the ambient at the focus. As the initial beam order

*N*increases, the trapping stability decreases due to the decrease of radiation force, and both transverse trapping range and longitudinal trapping range becomes smaller (i.e., the positions of peak values approach the focus as

*N*increases). So a coherent flat-topped beam (

*N*>1) at the focal plane does not offer any advantage for trapping a Rayleigh particle over a Gaussian beam. From Figs. 5(e) and 5(f), one finds that we can increase the transverse trapping range (i.e., increase the width of potential well of flatter beam) by increasing the initial beam order

*N*suitably near the focal plane (

*z*

_{1}=1

*μm*), even though the trapping stiffness decreases as

*N*increases. Table 1 shows the trapping range of Fig. 5(f) for four different values of

*N*. From Table 1 we can find that the trapping range increases remarkably as the beam order

*N*increases. The trapping range for the case of

*N*=4 is much large than the case of

*N*=1 (Gaussian beam). But the initial beam order

*N*can’t be arbitrary large, because the trapping stability decreases with increasing

*N*, as discussed in section 5. Rayleigh particles will diffuse instead of being trapped when

*N*is very large (i.e., the trapping stiffness is very low) because the radiation forces are comparable to the Brownian force as shown later (see Fig. 8(a)). We also note from Fig. 5(h), because there are two stable equilibrium points, the particle can’t be stably trapped along the z direction at off-axis points. To solve this problem, we can use two face to face focused flat-topped beams to construct a true optical potential well or “optical bottle” as shown in Fig. 6. To check for stability, we can interrupt one beam for a moment, which causes the particle to be accelerated rapidly in the remaining beam along its propagation direction. When another beam is turned on again, the particle is decelerated slowly and returns to its equilibrium region. So two face to face focused flat-topped beams can be used to trap a particle in a stable manner at on-axis and off-axis points.

*N*=3 for different values of σ

_{0}at different positions

*z*

_{1}, and in Fig. 7(g) and 7(h) the longitudinal gradient force for two different transverse positions

*x*. One finds from Fig. 7 that scattering force, transverse and longitudinal gradient forces decrease as the coherence of the initial flat-topped beam decreases. The forward scattering force also is much smaller than the longitudinal gradient force. From Figs. 7(d) and 7(g), we find that at the focal plane, as the initial coherence of flat-topped beam decreases, both transverse and longitudinal trapping ranges become larger (i.e., the positions of peak values deviate away from the focus as σ

_{0}decreases), while the trapping stiffness reduces due to the decrease of radiation force. Table 2 shows the trapping range of Fig. 7(d) for four different values of σ

_{0}. The trapping range also increases remarkably as initial coherence width σ

_{0}decreases from Table 2. The trapping range for the case of σ

_{0}= 2 mm is much larger than the case of σ

_{0}= 100 mm. From Figs. 7(e), 7(f) and 7(h), we find at the planes near the focal plane, both trapping range and stability are reduced as the initial coherence of flat-topped beam decreases. Similarly, we can use two face to face focused partially coherent flat-topped beams to trap a particle in a stable manner at on-axis and off-axis points.

*N*) of the flat-topped beam, we can increase the transverse trapping range at the planes near the focal plane. By decreasing the initial coherence of the flat-topped beam, we can increase the transverse and longitudinal trapping ranges at the focal plane. In both cases, the stability of trapping decreases, so it is necessary to choose suitable beam order

*N*and initial coherence (i.e., σ

_{0}) in order to extend the trapping range. This conclusion is the main result of present paper.

## 5. Analysis of the trapping stability

*F*| = (12

_{B}*πκak*)

_{B}T^{1/2}[49], where

*κ*is the viscosity of the ambient (in our case,

*κ*= 7.977 × 10

^{-4}Pas at the

*T*= 300K),

*a*is the radius of the particle and

*k*

_{B}is the Boltzmann constant. Then we obtain (after calculation) the magnitude of the Brownian force

*F*

_{B}= 2.5 × 10

^{-3}pN. Comparing the radiation forces in Figs. 5 and Fig. 7, we can find that both scattering force and gradient force in our numerical examples are all larger than the Brownian force. So Brownian motion does not influence the main conclusion of present paper. We illustrate in Fig. 8 the dependencies of the radiation forces

*F*

_{Scat}

^{Max},

*F*

_{Grad-x}

^{Max}and

*F*

_{Grad-z}

^{Max}induced by a flat-topped beam on initial beam order

*N*and initial coherence (i.e., σ

_{0}) at

*z*

_{1}=0. For comparison, Brownian force

*F*

_{B}is also shown in Fig. 8. From Fig. 8(a), one finds that both scattering force and gradient force decrease as

*N*increases, which is consistent with Fig. 5. When

*N*=15, the scattering force equals Brownian force

*F*

_{B}, but the gradient force is still much larger than the scattering force and Brownian force

*F*

_{B}, so a flat-topped beam with

*N*=15 can still be used to trap a particle. When

*N*=20, the gradient force nearly equals the Brownian force

*F*

_{B}, so a flat-topped beam with

*N*≥ 20 cannot be used for trapping a particle. From Fig. 8(b), one finds that both scattering force and gradient force decrease as the initial coherence of flat-topped beam decreases, which is consistent with Fig. 8. When σ

_{0}is smaller than 0.04, we cannot use a partially coherent flat-topped beam for trapping a particle because the Brownian force is larger than the radiation force. The line Q in Figs. 8 (a) and 8(b) can be regarded as the critical line. From above discussions, we come to conclusion that the trapping stability decreases as the initial order increases or initial coherence decreases, and we should choose suitable values of

*N*and σ

_{0}of flat-topped beam for particle trapping.

## 6. Conclusion

## Acknowledgments

## References and links

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**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(140.7010) Lasers and laser optics : Laser trapping

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: December 4, 2008

Revised Manuscript: January 19, 2009

Manuscript Accepted: January 25, 2009

Published: January 29, 2009

**Virtual Issues**

Vol. 4, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Chengliang Zhao, Yangjian Cai, Xuanhui Lu, and Halil T. Eyyuboğlu, "Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle," Opt. Express **17**, 1753-1765 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-3-1753

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### References

- D. C. Cowan, J. Recolons, L. C. Andrews, and C. Y. Young, "Propagation of flattened Gaussian beams in the atmosphere: a comparison of theory with a computer simulation model," in Atmospheric Propagation III, C. Y. Young and G. C. Gilbreath, eds., Proc. SPIE 6215, 62150B (2006).
- H. T. Eyyuboğlu, A. Arpali, and Y. Baykal, "Flat topped beams and their characteristics in turbulent media," Opt. Express 14, 4196-4207 (2006). [CrossRef] [PubMed]
- Y. Baykal and H. T. Eyyuboğlu, "Scintillation index of flat-topped Gaussian beams," Appl. Opt. 45, 3793-3797 (2006). [CrossRef] [PubMed]
- Y. Cai, "Propagation of various flat-topped beams in a turbulent atmosphere," J. Opt. A: Pure Appl. Opt. 8, 537-545 (2006). [CrossRef]
- N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, and M. Nakatsuka, "Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target," Opt. Rev. 7, 216-220 (2000). [CrossRef]
- D. W. Coutts, "Double-pass copper vapor laser master-oscillator power-amplifier systems: Generation of flat-top focused beams for fiber coupling and percussion drilling," IEEE J. Quantum Electron. 38, 1217-1224 (2002). [CrossRef]
- W. Wang, P. X. Wang, Y. K. Ho, Q. Kong, Z. Chen, Y. Gu, and S. J. Wang, "Field description and electron acceleration of focused flattened Gaussian laser beams," Europhys. Lett. 73, 211-217 (2006). [CrossRef]
- L. Wang and J. Xue, "Efficiency comparison analysis of second harmonic generation on flattened Gaussian and Gaussian beams through a crystal CsLiB6O10," Jpn. J. Appl. Phys. 41, 7373-7376 (2002). [CrossRef]
- M. S. Bowers, "Diffractive analysis of unstable optical resonator with super-Gaussian mirrors," Opt. Lett. 19, 1319-1321 (1992). [CrossRef]
- F. Gori, "Flattened gaussian beams," Opt. Commun. 107, 335-341 (1994). [CrossRef]
- Y. Li, "Light beam with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002). [CrossRef]
- A. A. Tovar, "Propagation of flat-topped multi-Gaussian laser beams," J. Opt. Soc. Am. A 18, 1897-1904 (2001). [CrossRef]
- S. A. Amarande, "Beam propagation factor and the kurtosis parameter of flattened Gaussian beams," Opt. Commun. 129, 311-317 (1996).
- R. Borghi, M. Santarsiero, and S. Vicalvi, "Focal shift of focused flat-topped beams," Opt. Commun. 154, 243-48 (1998). [CrossRef]
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