## Shifting the spherical focus of a 4Pi focusing system |

Optics Express, Vol. 19, Issue 2, pp. 673-678 (2011)

http://dx.doi.org/10.1364/OE.19.000673

Acrobat PDF (933 KB)

### Abstract

In a 4Pi focusing system radially polarized laser beams can be focused to a spherical focal spot. For many applications, e.g., for moving trapped particles or for scanning a specimen, one would like to change the position of focal spot along the optical axis without moving lenses or laser beams. We demonstrate how this can be achieved by modulating the phase of the input field at the pupil plane of the lens. The required phase modulation function is determined by spherical wave expansion of the plane wave factors in the Richards–Wolf integral.

© 2011 OSA

## 1. Introduction

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [CrossRef] [PubMed]

2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

3. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A **73**(4), 043402 (2006). [CrossRef]

4. P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. **43**(2), 025401 (2010). [CrossRef]

5. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. **85**(25), 6239–6241 (2004). [CrossRef]

6. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express **12**(15), 3377–3382 (2004). [CrossRef] [PubMed]

10. M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. **48**(32), 6143–6151 (2009). [CrossRef] [PubMed]

11. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics **2**(8), 501–505 (2008). [CrossRef]

12. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. **35**(7), 965–967 (2010). [CrossRef] [PubMed]

13. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. **29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

14. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. **34**(16), 2444–2446 (2009). [CrossRef] [PubMed]

13. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. **29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

14. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. **34**(16), 2444–2446 (2009). [CrossRef] [PubMed]

## 2. Theory

**E**

_{input}=

*l*(

*θ*)

**e**

*, where*

_{ρ}**e**

*(*

_{ρ}*θ*) is the unit vector in radial direction and

*l*(

*θ*) represents the input field. We denote the input fields at the pupil planes by

*l*

_{left}(

*θ*) and

*l*

_{right}(

*θ*) for the left and right lens, respectively, where

*θ*covers the range from 0° to 90° for

*l*

_{left}and from 180° to 90° for

*l*

_{right}, while the single input field

*l*(

*θ*) at the pupil plane of the lenses covers the full range from 0° to 180°. When

*l*

_{left}and

*l*

_{right}have a certain phase relation, the

*z*-component of the electric fields near the focus becomes remarkably strong thanks to constructive interference, while the radial component experiences destructive interference and thus is very weak. Mathematical description of this interference effect can be established by using the well-known Richards–Wolf integral [2

2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**(2), 77–87 (2000). [CrossRef] [PubMed]

15. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A **253**(1274), 358–379 (1959). [CrossRef]

*E*and

_{ρ}*E*are the radial and axial components of the electric field at an observation point

_{z}**R**(

*ρ*,

*z*) near the focus,

*ρ*and

*z*are cylindrical coordinates, and

*X*(

*θ*) is the pupil apodization function where

*X*(

*θ*) = (cos

*θ*)

^{1/2}for an aplanatic lens and

*X*(

*θ*) = 1 for a Herschel-type lens (used in this paper),

*J*(

_{n}*x*) is the cylindrical Bessel function of first kind of order

*n*,

*h*(

*θ*) =

*k*cos

*θ*is the

*z-*component of

**k**, and

*k*is the wave number.

*l*(

*θ*) to be

*l*

_{0}(

*θ*) = sin

*θ*exp(−2sin

^{2}

*θ*) and substituting this into the integrals (1) and (2), one can see that the intensity of the focal field, dominated by the

*z*-component, exhibits an almost spherical distribution, i.e., a spherical focal spot, at

*z*= 0. Our aim is to determine

*l*(

*θ*) such that it corresponds to a spherical (focal) spot translated to another position

*z*=

*z*

_{0}along the optical axis. As mentioned, the by far dominating contribution to the spherical intensity distribution at the focus is coming from the

*z*-component. Since |

*E*|

_{ρ}^{2}is negligibly weak compared to |

*E*|

_{z}^{2}, [for example, if

*l*(

*θ*) =

*l*

_{0}(

*θ*), max(|

*E*|

_{ρ}^{2})/max(|

*E*|

_{z}^{2}) ≈0.1303 in the focal region] it is sufficient to consider only Eq. (2). As we know, the

*z*-component of the electric field obeys the scalar wave equation, whose spherically symmetrical solution is the first kind of spherical Bessel function of zero order:

*j*

_{0}(

*kR*) ≡ sin(

*kR*)/(

*kR*), where

*R*= |

**R**|. For a fixed vector

**z**=

**u**

_{3}

*z*

_{0}on the optical axis (here

**u**

_{3}is the unit vector in the

*z*direction), the desired solution is

*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*) with an intensity |

*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*)|

^{2}that forms a spherical spot at the shifted position

*z*

_{0}.

*l*(

*θ*) by which this can be achieved, we transform from cylindrical to spherical coordinates by expressing the factor

*J*

_{0}(

*kρ*sin

*θ*)

*e*

^{ih}^{(}

^{θ}^{)}

*in terms of the spherical wave functions [16]:where*

^{z}*P*(

_{n}*x*) denotes the Legendre polynomial of order

*n*,

*α*is the polar angle of the observation point

**R**, and

*j*(

_{n}*x*) is the spherical Bessel function of the first kind with order

*n.*When Eq. (3) is inserted into Eq. (2), one obtains [Note that we use here

*X*(

*θ*) = 1]withNow we have to find the coefficients

*A*by which the right hand side of Eq. (4) transforms into an expression of the form of

_{n}*j*

_{0}(

*k|*

**R**

*−*

**z**

*|*) that represents a spherical spot centered at

*z*

_{0}on the optical axis. Consider the addition theorem of the spherical Bessel functions [16]Noting that

*j*(

_{n}*x*) = (−1)

*(*

^{n}j_{n}*x*) and equating the right hand sides of Eqs. (4) and (6), we find thatPutting

*x*= cos

*θ*and

*g*(

*x*) =

*l*(

*θ*), Eq. (5) becomesWe recognize this as the coefficients of the Legendre series expansion of (1−

*x*

^{2})

^{1/2}

*l*(

*x*). So, we require the input field

*g*(

*x*) to beUnfortunately, (1−

*x*

^{2})

^{−1/2}has a divergence at

*x*= 1. To overcome this, we replace (1−

*x*

^{2})

^{−1/2}by

*g*

_{0}(

*x*). Here

*g*

_{0}(

*x*) = (1−

*x*

^{2})

^{1/2}exp[−2(1−

*x*

^{2})] or

*l*

_{0}(

*θ*) = sin

*θ*exp[−2sin

^{2}

*θ*], respectively, denotes the fundamental radial polarization mode. Since Bokor and Davidson use the input field

*l*

_{0}(

*θ*) = sin

*θ*exp(−2sin

^{2}

*θ*) to produce a spherical focal spot [13

13. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. **29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

*l*(

*θ*) can be thought of as a modulation of the fundamental mode

*l*

_{0}(

*θ*) by

*A*(

*θ*).

## 3. Numerical results

*l*(

*θ*). In this section, we substitute

*l*(

*θ*) into Eqs. (1) and (2) to check whether this gives the desired result. For

*A*(

*θ*) = 1, the situation is the same as in [13

**29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

*l*(

*θ*) =

*l*

_{0}(

*θ*) at the origin, but what we want is a spherical spot at another position

*z*

_{0}on the optical axis. For numerical demonstration we have chosen

*z*

_{0}= 1.5λ. We calculate the coefficients

*A*from Eq. (7), and the input field

_{n}*l*(

*θ*) from Eq. (10). In Fig. 2(a) the intensity |

*l*(

*θ*)|

^{2}of the input field is plotted. We find that

*l*

_{left}(

*θ*) and

*l*

_{right}(

*θ*) have exactly the sameintensity distributions and each distribution exhibits the intensity property of a fundamental radially polarized field: an annular intensity distribution with intensity minimum in the center, i.e., both are still fields with radial polarization. In fact, it can be verified that the modulation function has a constant modulus, suggesting that modulation does not change the intensity distribution of the input field. In Fig. 2(b), we present the phase of the input field

*l*(

*θ*). From the phase distribution, a two-belt phase structure is found for

*l*

_{left}(

*θ*). The first belt covers the range from 0 to about 70° and the second one goes from 70° to 90°. In the first belt, the phase increases almost linearly from 0 to 2

*π*. Then, after a transition to 0 at

*θ*= 70°, it increases linearly from 0 to

*π*in the second belt. The phase of

*l*

_{right}(

*θ*) has the same two-belt structure as

*l*

_{left}(

*θ*), but with a difference of sign in the corresponding belts. Figures 2(a) and 2(b) suggest that the input field

*l*(

*θ*) can be obtained from a fundamental radial field mode by a two-belt phase modulation, which can be achieved with a spatial light modulator. Having determined

*l*(

*θ*), we calculate from the integrals (1) and (2) the electric intensities in the focal region. Figure 3 shows line scans of the total intensity

*I*= |

*E*|

_{z}^{2}+ |

*E*|

_{ρ}^{2}in the axial direction (solid line) and in the transversal direction (dashed line), respectively. The maximum of the intensity has been normalized to unity. As can be seen, a nearly spherical intensity spot centered at

*z*

_{0}= 1.5λ can indeed be realized. Axial and transversal spot diameters are nearly identical and have a width of approximately 0.5λ, in agreement with the results in [13

**29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

14. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. **34**(16), 2444–2446 (2009). [CrossRef] [PubMed]

*z*

_{0}= 0.

*z*

_{0}continuously, it is evident that the spherical spot will move along the optical axis, i.e., the goal of a real time shifting of a spherical spot thus is solved. For illustration, we plot 2D (

*XZ*plane) color graphs of the intensity in the vicinity of the focus for different values of

*z*

_{0}in Fig. 4 and demonstrate by a movie (Media 1) the evolution of the intensity in the vicinity of the focus when the parameter

*z*

_{0}is changed continuously from −2λ to 2λ (The value of

*z*

_{0}increases by 0.05λ every 0.05s). Figures 4(a)-(e) are successive frames extracted from the movie at

*t*= 0s, 1s, 2s, 3s and 4s (or

*z*

_{0}= −2λ, −1λ, 0λ, 1λ and 2λ), respectively. As desired, we obtain a series of spots centered at

*z*

_{0}that are excellent approximations to the desired spherical spots of intensity and that all have a radius of approximately 0.5λ. The corresponding movie (Media 1) proves that the intensity distribution keeps its nearly spherical shape over the whole translation range. To quantify the extent of the shape of the spot to spherical shape, we introduce the size mismatch parameter ∆

*, measured by the difference between transverse and axial diameters. We find that the size mismatch ∆*

_{XZ}*(slightly elongated along the*

_{XZ}*z*axis) are all roughly −0.036λ for three

*z*

_{0}( = 0λ, 1λ and 2λ), implying quantitatively the almost spherical shape of the spot during the whole translation.

**29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

*X*(

*θ*) = 1, for which a fundamental radial polarization mode is focused into a spherical spot centered at the focus. For an aplanatic focusing system, the apodizer factor

*X*(

*θ*) = (cos

*θ*)

^{1/2}, the resulting focal spot is not of spherical shape but slightly transversely elongated (see Fig. 2 in [13

**29**(17), 1968–1970 (2004). [CrossRef] [PubMed]

*l*(

*θ*) of the incoming beam by (cos

*θ*)

^{1/2}, which means the introduction of an amplitude modulation, as done by Chen and Zhan in [14

**34**(16), 2444–2446 (2009). [CrossRef] [PubMed]

*θ*

_{max}to be 90° for the objective lens, which is absolutely impossible for practical objective lens. Meanwhile, when the aplanatic focusing system is used, the introduction of the amplitude modulation [

*l*(

*θ*)/(cos

*θ*)

^{1/2}, as discussed above] also requires

*θ*

_{max}less than 90°. For two high numerical aperture objective lenses with

*θ*

_{max}= 79.6° [NA = 1.49 oil (1.515) immersion objective, Nikon], we find that the focal spot can still be moved along the axis, but the spot is elongated along the transverse direction with the size mismatch ∆

*≈0.1480λ (*

_{XZ}*z*

_{0}= 2λ). However, we can reduce the size mismatch by properly choosing the minimum converging angle

*θ*

_{min}of the focusing system, which can be achieved by blocking the central region of the incoming beam with an opaque disk. For the case discussed here (

*θ*

_{max}= 79.6°), if we put

*θ*

_{min}= 25°, the size mismatch will be ∆

*≈0.082λ. The final comment refers to the translation range of the spot, i.e, the maximum value of*

_{XZ}*z*

_{0}. In our calculation, we find that for larger values of

*z*

_{0}our design still works. For example, when

*z*

_{0}= 10λ, we obtain a nearly spherical spot with ∆

*≈-0.037λ. But the corresponding phase distribution of the input field will become a ten-belt structure, while the phase structure for*

_{XZ}*z*

_{0}= 1.5

*λ*is of only two belt as shown in Fig. 2(b). With further increasing

*z*

_{0}, the phase structure becomes more complex. As a result, we conclude that our design can realize ± 10λ translation of the spherical spot of intensity without complex phase modulation.

## 4. Conclusion

*l*(

*θ*) that does the trick to produce a spherical intensity spot at any designated position

*z*

_{0}. As concerns the intensity,

*l*(

*θ*) has the same distribution as the fundamental radial polarization mode, but its phase is modulated. In other words,

*l*(

*θ*) turns out to be simply a phase modulated fundamental mode. We have shown that the position parameter

*z*

_{0}can be varied continuously such that the spherical intensity distribution of the focus is maintained during dynamical movement of the focal spot along the optical axis. In conclusion, we have pointed out a way how to move a trapped particle or to scan a specimen without moving objective lenses or laser beams.

## Acknowledgments

## References and links

1. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

2. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

3. | Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A |

4. | P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. |

5. | N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. |

6. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express |

7. | S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A |

8. | H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. |

9. | T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. |

10. | M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. |

11. | H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

12. | K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. |

13. | N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. |

14. | W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. |

15. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A |

16. | J. A. Stratton, |

**OCIS Codes**

(110.2990) Imaging systems : Image formation theory

(140.3300) Lasers and laser optics : Laser beam shaping

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 8, 2010

Revised Manuscript: November 19, 2010

Manuscript Accepted: December 7, 2010

Published: January 5, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Shaohui Yan, Baoli Yao, and Romano Rupp, "Shifting the spherical focus of a 4Pi focusing system," Opt. Express **19**, 673-678 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-673

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### References

- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]
- Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006). [CrossRef]
- P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010). [CrossRef]
- N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004). [CrossRef]
- Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]
- S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007). [CrossRef]
- H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007). [CrossRef] [PubMed]
- T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008). [CrossRef] [PubMed]
- M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009). [CrossRef] [PubMed]
- H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
- K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010). [CrossRef] [PubMed]
- N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef] [PubMed]
- W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009). [CrossRef] [PubMed]
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959). [CrossRef]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, 1941).

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