## Under-filling trapping objectives optimizes the use of the available laser power in optical tweezers |

Optics Express, Vol. 19, Issue 12, pp. 11759-11768 (2011)

http://dx.doi.org/10.1364/OE.19.011759

Acrobat PDF (987 KB)

### Abstract

For optical tweezers, especially when used in biological studies, optimizing the trapping efficiency reduces photo damage or enables the generation of larger trapping forces. One important, yet not-well understood, tuning parameter is how much the laser beam needs to be expanded before coupling it into the trapping objective. Here, we measured the trap stiffness for 0.5–2 *μ*m-diameter microspheres for various beam expansions. We show that the highest overall trapping efficiency is achieved by slightly under-filling a high-numerical aperture objective when using microspheres with a diameter corresponding to about the trapping-laser wavelength in the medium. The optimal filling ratio for the lateral direction depended on the microsphere size, whereas for the axial direction it was nearly independent. Our findings are in agreement with Mie theory calculations and suggest that apart from the choice of the optimal microsphere size, slightly under-filling the objective is key for the optimal performance of an optical trap.

© 2011 OSA

## 1. Introduction

1. K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. **23**, 247–285 (1994). [CrossRef] [PubMed]

2. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. **45**, 1943–1949 (1998). [CrossRef]

3. V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express **16**, 13831–13844 (2008). [CrossRef] [PubMed]

4. A. Jannasch, V. Bormuth, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Coated microspheres as enhanced probes for optical trapping,” *Proc. SPIE* p. 70382B (2008). [CrossRef]

1. K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. **23**, 247–285 (1994). [CrossRef] [PubMed]

5. 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]

*overfill*the objective. Kim

*et al.*[6] measured an optimal “laser beam radius-to-the aperture radius” of 1.2 for 5

*μ*m-diameter microspheres in agreement with the ray-optics predictions. In contrast, Bing-Huan

*et al.*[7

7. M. Bing-Huan, Z. Jin-Hua, Z. Min-Cheng, L. Yin-Mei, W. Jian-Guang, and R. Hong-Liang, “Improvement of transverse trapping efficiency of optical tweezers,” Chin. Phys. Lett. **25**, 2300–2302 (2008). [CrossRef]

*μ*m-diameter microspheres. Recently, Samadi and Reihani [8

8. A. Samadi and N. S. Reihani, “Optimal beam diameter for optical tweezers,” Opt. Lett. **35**, 1494–1496 (2010). [CrossRef] [PubMed]

*under-filling*. Discrepancies between the studies are due to (i) numerous definitions for the ratio of the beam diameter to some measure of the objective, (ii) different criteria for efficiency (escape force or trap stiffness), (iii) varying microsphere diameters, and (iv) a lack of rigorous Mie theory calculations necessary for particle sizes comparable to the wavelength of light.

*filling-ratio*parameter which is independent of the objective-back-opening size. The visible stops and apertures in the back of an objective maybe larger than the NA of the objective and furthermore depend on the manufacturer. Therefore, we base our filling-ratio parameter on Abbe’s sine condition. All high-quality microscope objectives are designed according to this condition to prevent spherical aberrations and coma. Essentially this means that the first principal plane of the objective is a hemisphere (H

_{1}in inset of Fig. 1(a)). Light rays emerging from the focus under an angle

*θ*relative to the optical axis leave the back of the objective (on the image side) at a radius

*r*=

*n f*sin

*θ*where

*n*is the refractive index of the lens and

*f*is the focal length of the objective obtained by dividing the manufacture’s tube length by the magnification of the objective. Note that for large values of

*θ*, common for high-NA objectives, this radius significantly differs from the one obtained with a straight principle plane (for a straight principal plane the radius would scale with tan

*θ*). Using the definition NA = nsin

*θ*, the marginal rays that corresponds to the NA of the objective thus span the diameter

*D*

_{NA}= 2

*f*NA. This diameter in microscopy is also called the exit pupil. Therefore, we define the filling ratio

*α*based on this diameter as where

*ω*

_{0}is the laser beam radius at which the intensity decreases to

*e*

^{−2}of the central value. If

*α*is multiplied by the NA of the objective, an effective NA can be assigned to the laser beam expansion. This effective numerical aperture in particular the optimal value of it, NA

_{opt}is a dimensionless parameter which can be used to compare the performance of different objectives. Here, we determined the optimal filling ratio and thus NA

_{opt}for microsphere sizes in the range of 0.5–2

*μ*m using an oil-immersion objective (NA = 1.3) and compared the results to Mie theory calculations. As criterion for

*α*, we measured the central trap stiffness since—contrary to the escape force—it is independent of the non-linear trapping force field [9

9. M. Jahnel, M. Behrndt, A. Jannasch, E. Schäffer, and S. Grill, “Measuring the complete force field of an optical trap,” Opt. Lett. **36**, 1260–1262 (2011). [CrossRef] [PubMed]

_{opt}≈ 1.25, resulted in the highest trapping efficiency for both lateral and axial directions.

## 2. Materials and methods

### 2.1. Optical tweezers setup and laser profile

10. M. Mahamdeh and E. Schäffer, “Optical tweezers with millikelvin precision of temperature-controlled objectives and base-pair resolution,” Opt. Express **17**, 17190–17199 (2009). [CrossRef] [PubMed]

_{4}, infrared laser,

*λ*= 1064 nm; Smart Laser Systems, Berlin, Germany). Briefly, the linearly polarized laser (in the

*y*-direction) was expanded to the desired width using two telescopes: a

*three-lens*Galilean telescope for continuously-adjustable expansion and a Kepler telescope with a discrete magnification depending on the choice of lenses. The Kepler telescope was used to increase the limited expansion range of the Galilean telescope. After a change in the expansion, we always realigned the complete laser path. The laser was then coupled into the trapping objective (CFI S Fluor 100×/0.7–1.3 NA oil,

*f*= 2 mm; Nikon, Japan). An identical condenser objective collected the laser light and its back focal plane was imaged onto a quadrant photo-diode (QPD: QP154-Q-HVSD enhanced for 1064 nm detection; Pacific Silicon Sensor, Westlake Village, USA) for microsphere-position detection in three dimensions [11

11. A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. **44**, 378–386 (1999). [CrossRef] [PubMed]

12. V. Bormuth, J. Howard, and E. Schäffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. **226**, 1–5 (2007). [CrossRef] [PubMed]

*ω*

_{0}directly before the objective in both the

*x*- and

*y*-direction using the knife-edge method [13

13. D. R. Skinner and R. E. Whitcher “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E: J. Sci. Instrum . **5**, 237–238 (1972). [CrossRef]

### 2.2. Sample preparation and calibration

^{2}and 22×22 mm

^{2}, No. 1.5; Corning, NY, USA) glued together by two pieces of double-sided tape leaving a channel of 18×3×0.1 mm

^{3}in size. The channel was filled with an aqueous solution of polystyrene (PS) microspheres (Bangs Laboratories, Fishers, USA) containing 1 mM KCl. The salt screens repulsive surfaces forces to less than 100 nm [14

14. E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir **23**, 3654–3665 (2007). [CrossRef] [PubMed]

14. E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir **23**, 3654–3665 (2007). [CrossRef] [PubMed]

15. S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. **77**, 103101 (2006). [CrossRef]

*a priori*knowledge of the viscosity, the microsphere diameter, or the distance to a surface. It provides—apart from the displacement sensitivity of the photo diode and the drag coefficient—the lateral and axial trap stiffness at the surface. The method is suitable for experiments close to a surface because it measures the drag coefficient independently from the other parameters. For every microsphere, we measured all parameters as a function of microsphere–cover-slip distance at ≈50 positions starting from a distance of about 3

*μ*m until the microsphere touched the surface. From the distance dependence of the drag coefficient, we determined the cover-slip surface position with nanometer precision. For this position, we state the determined trap stiffness [14

14. E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir **23**, 3654–3665 (2007). [CrossRef] [PubMed]

### 2.3. The optimal immersion oil and uniform objective transmission ensured diffraction-limited performance

#### Immersion oil

16. S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt. Lett. **32**, 1998–2000 (2007). [CrossRef] [PubMed]

*n*= 1.518, index-matching the cover slip, reduced their trapping efficiency. This was especially the case when working close to the cover glass surface using a NA = 1.32 oil-immersion objective. Using higher refractive-index oils increased their trapping efficiency because spherical aberrations were compensated. Therefore, we measured the trap stiffness as a function of immersion-oil refractive index (Fig. 1(b)). We varied the refractive index in the range of 1.5–1.54 in increments of 0.01 (Series A; Cat.-#. 18095; Cargille Laboratories, NJ, USA) including the standard oil we typically use (Immersol,

*n*= 1.518; ZEISS, Germany). For all directions, the trap stiffness peaked at the standard oil showing that it is indeed the optimal immersion oil for our setup.

#### Trapping objective transmission

17. N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, and P. A. M. Neto, “Characterization of objective transmittance for optical tweezers,” Appl. Opt. **45**, 4263–4269 (2006). [CrossRef] [PubMed]

17. N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, and P. A. M. Neto, “Characterization of objective transmittance for optical tweezers,” Appl. Opt. **45**, 4263–4269 (2006). [CrossRef] [PubMed]

*i*, we measured the power before,

*P*

_{before}, and after,

*P*

_{after}, the identical objectives. We then calculated the transmission of a single objective through an annulus of outer diameter

*D*according to

_{i}*D*

_{NA}= 5.2 mm. For larger diameters, the transmission dropped rapidly and was zero at ≈6.12 mm in agreement with the principal-plane diameter

*D*

_{H1}= 2

*n*

_{oil}

*f*= 6.07 mm. The visible stop in the back of the objective had a diameter of

*D*

_{0}≈ 6.5 mm (see schematic inset of Fig. 1(a)). Thus, the objective back opening was much larger (about 25 %) compared to the corresponding NA or pupil diameter. Note that significantly smaller filling ratios result when this stop is used as a reference.

#### Laser focus

*x*(

*y*)-direction, respectively. We expected this difference because the laser beam is not a perfect Gaussian beam as characterized by the so-called M

^{2}value. The diffraction-limited spot size is proportional to this M

^{2}value. According to the specifications of our laser,

*x*(

*y*)-direction, respectively. These values account for the increase in the measured spot size. As an independent visual control, we recorded an image of the laser focused on the cover slip surface. The size determined from cross-sections through the image were 15 % smaller (21 % larger) relative to the theoretical values for the

*x*(

*y*)-direction, respectively. We attribute these differences to polarization effects: before the light reaches the camera it passes a dichroic mirror and several other optics which may have a polarization-dependent transmission. In addition, the camera sensitivity may be polarization-dependent. There were no such uncertainties for the gold-particle scans. Therefore, we assume that the size obtained from the latter data reflect the true size of our laser focus. Taken together, the objective performed nearly diffraction-limited.

## 3. Results and discussion

### 3.1. Under-filling resulted in the highest trap stiffness

*μ*m) and seven filling ratios (

*α*≈ 0.54, 0.67, 0.82, 0.96, 1.15, 1.66 and 2.25) using a constant laser power before the trapping objective (Fig. 2). From the measurements we can draw the following conclusions: (i) The trap stiffness—for both the lateral and axial directions—was largest at a filling ratio below one, i.e. at under-filling conditions, for all microsphere sizes. (ii) For the lateral directions, the optimal under-filling ratio was size-dependent and decreased with increasing microsphere diameter. For example, the optimal filling ratios for the

*x*-direction were 0.94, 0.94, 0.8 and 0.66 for the four different microsphere sizes, respectively. (iii) For the axial direction, the optimal under-filling ratio was rather size-independent and about 0.95. (iv) As expected [3

3. V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express **16**, 13831–13844 (2008). [CrossRef] [PubMed]

*μ*m. This diameter corresponds to about the trapping laser wavelength in the medium,

*λ/n*

_{medium}≈ 800 nm, where a Mie resonance occurs [18]. During the measurements, we noticed that with filling ratios

*α*≲ 0.9, the trapping of microspheres became more difficult and the distance from the surface up to which microspheres could still be trapped decreased (to about 5

*μ*m from the cover-slip surface for

*α*= 0.67). This effect is due to the low effective NA and the increasing offset between the trap center and the focus (see below and inset Fig. 2(c)). With increasing distance to the surface, spherical aberrations increase. This increase weakens the trap until microspheres cannot be trapped anymore. At the lowest filling ratio of

*α*= 0.54 corresponding to an effective NA of 0.7, we were not able to trap any microspheres anymore because the gradient force in the axial direction was too small compared to the scattering force. In summary, the optimal laser expansion depended on the experimental choice of the microsphere size.

### 3.2. Mie theory calculations confirm the under-filling optimum

19. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt . **9**, S196–S203 (2007). [CrossRef]

^{®}scripts is provided in the Appendix.

*n*

_{PS}= 1.57 trapped in water,

*n*

_{medium}= 1.326, with 1064 nm-light polarized in the

*y*-direction truncated at an angle corresponding to an effective NA of 1.2. For this global fit, we varied only two parameters: the truncation angle, i.e. the effective NA of the objective, and a scaling factor for the effective power in the focus. We found that the effective NA was smaller compared to the oil-immersion objective’s specification due to spherical aberrations at the glass-water interface (inset Fig. 1(a)). The theoretical power was significantly lower (scaling factor of 0.62) than our measured power in the focus. To show how the NA affects the shape of the trap stiffness curves, we also calculated the trap stiffness for the objective NA of 1.3 using the same scaling factor of 0.62 (thin lines). The calculations for NA = 1.3 deviated significantly more than the ones for NA = 1.2 indicating that the effective NA of our objective was reduced by the spherical aberrations at the glass-water interface.

3. V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express **16**, 13831–13844 (2008). [CrossRef] [PubMed]

*α*= 2.5, [20

20. V. N. Mahajan, “Uniform versus Gaussian beams - a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A **3**, 470–485 (1986). [CrossRef]

*μ*m distance where we measured the trap stiffness), we do not expect that these aberrations fully account for the large difference between theory and experiment [21

21. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. **169**, 391–405 (1993). [CrossRef]

22. A. Rohrbach, “Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. **95**, 168102 (2005). [CrossRef] [PubMed]

23. Note that version 1.0 of the optical tweezers computational toolbox (Ref. [19]) contained a power scaling error in the code which was corrected in version 1.1.

21. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. **169**, 391–405 (1993). [CrossRef]

*α <*1 based on our calculations (inset Fig. 2(c)). This increase is exacerbated by the aberrations mentioned above with the consequence of a systematic weakening of the trap. Overall, our error bars on the trap stiffness measurements were smaller than the mean deviation from the theory, however, considering that with each filling ratio the complete laser path, and with each sample the condenser objective and detector had to be re-aligned, the overall agreement for the range of microsphere sizes and filling ratios is remarkable.

## 4. Conclusions

*α*[see Eq. (3)] and therefore the effective NA, while the diffraction-limited spot size scales only inversely with the effective NA. One direct consequence of optimizing trapping efficiencies is the reduction of heating and photo-damage in particular when working with biological samples. The least power in the focus to generate a trap stiffness of 1 pN/nm was needed for the 0.85

*μ*m-diameter microspheres for

*α*= 1 (≈360 mW based on our calculations, black solid line Fig. 2(d)). For the 0.46

*μ*m-diameter microspheres, the power to achieve the same trap stiffness was at least 50 % larger (thick magenta dotted line). The larger the over-filling ratio was for this size, the less power in the focus was necessary. This trend was contrary to the other microsphere sizes we used. For the two large microsphere sizes we tested, under-filling conditions lead to the least power for a 1 pN/nm-trap. When designing optical tweezers, Fig. 2(d) may also serve as a reference for how much output power the laser needs to have. For

*α*≳ 1, truncation leads to significant power loss (thin lines). The truncated power may lead to additional heating and thermal drift of the objective [10

10. M. Mahamdeh and E. Schäffer, “Optical tweezers with millikelvin precision of temperature-controlled objectives and base-pair resolution,” Opt. Express **17**, 17190–17199 (2009). [CrossRef] [PubMed]

*et al.*[6] measured a rather large optimal

*α*≈ 1.7 due to aberrations (usage of an oil-immersion objective and trapping ≈25

*μ*m away from the surface without refractive index compensation). The optimal value of

*α*= 0.83 (NA

_{opt}= 1.12) from Bing-Huan

*et al.*[7

7. M. Bing-Huan, Z. Jin-Hua, Z. Min-Cheng, L. Yin-Mei, W. Jian-Guang, and R. Hong-Liang, “Improvement of transverse trapping efficiency of optical tweezers,” Chin. Phys. Lett. **25**, 2300–2302 (2008). [CrossRef]

*μ*m-diameter-microsphere measurements. Samadi and Reihani’s [8

8. A. Samadi and N. S. Reihani, “Optimal beam diameter for optical tweezers,” Opt. Lett. **35**, 1494–1496 (2010). [CrossRef] [PubMed]

*μ*m-diameter microspheres corresponds to an optimal NA

_{opt}of 1.14 and

*α*= 0.88 in good agreement with our measurements. Thus, using a reference parameter for the filling ratio which is independent of the objective’s back opening size helps in comparing different studies.

*α*

_{opt}≈ 0.95 when using a 1.3 NA–oil-immersion objective. This value corresponds to an optimal filling numerical aperture of NA

_{opt}≈ 1.25. According to our calculations the optimal values for the lateral directions were lower (

*α*≈ 0.8, NA

_{opt}≈ 1.04). Therefore, we expect that optimal values for water-immersion objectives with minimized spherical aberrations are lower, in particular for objectives corrected in the near-infrared. As total-internal-reflection-fluorescence (TIRF) objectives are used with immersion oils, we expect that spherical aberrations reduce the effective NA to values comparable to our measurements. Any light rays at radial distances corresponding to NAs larger than the refractive index of the trapping medium cannot contribute to the trap because of total internal reflection. Thus, for TIRF objectives with NA ≥ 1.4, optimal under-filling ratios should be less than 0.9 based on NA

_{opt}≈ 1.25.

*α*≈ 1 is optimal independent of the microsphere size as shown by our data. Additionally it is independent of the NA of the objective based on our calculations (data not shown). In contrast for the lateral direction, under-filling conditions with respect to the definition in Eq. (1) are optimal. What the exact value for the optimal

*α*is, may depend on the microsphere size, whether an oil or a water immersion objective is used, and how well the objective is corrected for the trapping laser wavelength. Based on our experiments and calculations, an under-filling ratio of

*α*≈ 0.9±0.05 is a good compromise for a high-NA objective. The problem of maximizing the trap stiffness for optical tweezers is analogous to minimizing the extent of a laser focus while retaining the most power in case of confocal microscopy or other high-resolution scanning techniques using multi-photon excitation or stimulated emission depletion [24

24. J. Pawley, *Handbook of Biological Confocal Microscopy* (Springer, 2006). [CrossRef]

## Appendix

*E*for a radially symmetric profile is expressed in terms of an incoming angle

*θ*instead of a radial coordinate

*r*in accordance with Abbe’s condition (see Introduction). Using our definition of the filling ratio [Eq. (1)], we obtain where

*θ*is in the range of 0–

*π*. Note the usage of the medium refractive index to account for the glass-water interface. To implement laser power losses due to truncation by the objective, the trap stiffness is reduced by multiplying with a power normalization factor assuming a constant transmission

*T*

_{0}where

*D*

_{max}is the largest diameter up to which light is still transmitted. Since we measured the transmission of the objective, we multiplied the Gaussian beam with the fitted transmission curve and integrated the intensity up to the diameter of the principal plane. This changed

*P*

_{trunc}slightly compared to the above equation.

## Acknowledgments

## References and links

1. | K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. |

2. | N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. |

3. | V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express |

4. | A. Jannasch, V. Bormuth, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Coated microspheres as enhanced probes for optical trapping,” |

5. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

6. | H.-I. Kim, I.-J. Joo, S.-H. Song, P.-S. Kim, K.-B. IM, and C.-H. Oh, “Dependence of the optical trapping efficiency on the ratio of the beam radius-to-the aperture radius,” J. Korean Phys. Soc. |

7. | M. Bing-Huan, Z. Jin-Hua, Z. Min-Cheng, L. Yin-Mei, W. Jian-Guang, and R. Hong-Liang, “Improvement of transverse trapping efficiency of optical tweezers,” Chin. Phys. Lett. |

8. | A. Samadi and N. S. Reihani, “Optimal beam diameter for optical tweezers,” Opt. Lett. |

9. | M. Jahnel, M. Behrndt, A. Jannasch, E. Schäffer, and S. Grill, “Measuring the complete force field of an optical trap,” Opt. Lett. |

10. | M. Mahamdeh and E. Schäffer, “Optical tweezers with millikelvin precision of temperature-controlled objectives and base-pair resolution,” Opt. Express |

11. | A. Pralle, M. Prummer, E. L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. |

12. | V. Bormuth, J. Howard, and E. Schäffer, “LED illumination for video-enhanced DIC imaging of single microtubules,” J. Microsc. |

13. | D. R. Skinner and R. E. Whitcher “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E: J. Sci. Instrum . |

14. | E. Schäffer, S. F. Nørrelykke, and J. Howard, “Surface forces and drag coefficients of microspheres near a plane surface measured with optical tweezers,” Langmuir |

15. | S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. |

16. | S. N. S. Reihani and L. B. Oddershede, “Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations,” Opt. Lett. |

17. | N. B. Viana, M. S. Rocha, O. N. Mesquita, A. Mazolli, and P. A. M. Neto, “Characterization of objective transmittance for optical tweezers,” Appl. Opt. |

18. | H. van de Hulst |

19. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt . |

20. | V. N. Mahajan, “Uniform versus Gaussian beams - a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A |

21. | S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive-index,” J. Microsc. |

22. | A. Rohrbach, “Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. |

23. | Note that version 1.0 of the optical tweezers computational toolbox (Ref. [19]) contained a power scaling error in the code which was corrected in version 1.1. |

24. | J. Pawley, |

25. | T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf . 79–80, 1005–1017 (2003). |

**OCIS Codes**

(000.2170) General : Equipment and techniques

(140.7010) Lasers and laser optics : Laser trapping

(180.0180) Microscopy : Microscopy

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: March 4, 2011

Manuscript Accepted: March 20, 2011

Published: June 2, 2011

**Virtual Issues**

Vol. 6, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Mohammed Mahamdeh, Citlali Pérez Campos, and Erik Schäffer, "Under-filling trapping objectives optimizes the use of the available laser power in optical tweezers," Opt. Express **19**, 11759-11768 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-12-11759

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### References

- K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994). [CrossRef] [PubMed]
- N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943–1949 (1998). [CrossRef]
- V. Bormuth, A. Jannasch, M. Ander, C. M. van Kats, A. van Blaaderen, J. Howard, and E. Schäffer, “Optical trapping of coated microspheres,” Opt. Express 16, 13831–13844 (2008). [CrossRef] [PubMed]
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