## On focused fields with maximum electric field components and images of electric dipoles |

Optics Express, Vol. 19, Issue 10, pp. 9157-9171 (2011)

http://dx.doi.org/10.1364/OE.19.009157

Acrobat PDF (1081 KB)

### Abstract

We study focused fields which, for a given total power and a given numerical aperture, have maximum electric field amplitude in some direction in the focal point and are linearly polarized along this direction. It is shown that the optimum field is identical to the image of an electric dipole with unit magnification. In particular, the field which is the image of an electric dipole whose dipole vector is parallel to the optical axis, is identical to the field whose longitudinal component is maximum at the image point.

© 2011 OSA

## 1. Introduction

3. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A **253**, 349–357 (1959). [CrossRef]

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

5. X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science **265**, 361–364 (1994). [CrossRef] [PubMed]

9. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. **86**, 329–334 (2007). [CrossRef]

10. Q. Zhan and J. Leger, “Focus Shaping using cylindrical vector beams,” Opt. Express **10**, 324–331 (2002). [PubMed]

11. J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express **18**, 21965–21972 (2010). [CrossRef] [PubMed]

12. N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. **30**, 1479–1481 (2005). [CrossRef] [PubMed]

15. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. **271**, 40–47 (2007). [CrossRef]

16. C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. **41**, 1495–1505 (1994). [CrossRef]

18. A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. **2**, 07008 (2007). [CrossRef]

19. V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. **6**, 317–345 (1997). [CrossRef]

25. H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A **79**, 013825 (2009). [CrossRef]

*not*linearly polarized in that direction but in another direction. Exceptions are the maximum longitudinal and maximum transverse components. In those cases the optimum fields are linearly polarized along the longitudinal and transverse directions, respectively.

25. H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A **79**, 013825 (2009). [CrossRef]

25. H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A **79**, 013825 (2009). [CrossRef]

6. L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. **86**, 5251–5254 (2001). [CrossRef] [PubMed]

19. V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. **6**, 317–345 (1997). [CrossRef]

27. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. **12**, 045707 (2010) [CrossRef]

**p**= (

*p*,

_{x}*p*,

_{y}*p*) is, for the given power, identical to the electric field in image space that has maximum amplitude in the direction (

_{z}*p*,

_{x}*p*, –

_{y}*p*), where the

_{z}*z*-axis is the optical axis. It follows in particular that the image of a dipole that is parallel to the optical axis is the field for which the longitudinal component is maximum.

24. H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. **100**, 1233904 (2008). [CrossRef]

**79**, 013825 (2009). [CrossRef]

## 2. The optimization problem

### 2.1. Formulation of the optimization problem

*n*, where

*n*is the refractive index in image space, and that the field has a prescribed power. Later we will discuss what fields in the entrance pupil of a lens with the chosen numerical aperture produce the optimum focused fields.

*x*,

*y*,

*z*) be a Cartesian coordinate system with the

*z*-axis parallel to the optical axis. Using the plane wave expansion, the electric field in a point

**r**in image space, which propagates in the direction of the positive

*z*-axis, can be written as where

**E**(

**r**) is the complex field,

**k**= (

*k*,

_{x}*k*,

_{y}*k*) the wave vector in the medium with refractive index

_{z}*n*,

*k*

_{0}= |

**k**|/

*n*= 2

*π*/

*λ*

_{0}with

*λ*

_{0}the wavelength in vacuum and

**A**(

*k*,

_{x}*k*) are the complex plane wave amplitudes. Since the field propagates in the positive

_{y}*z*-direction, we have

*k*,

_{x}*k*)-plane, which limits the integration area, will be denoted by Ω for brevity. Since the electric field is free of divergence, we have:

_{y}*z*= constant of the

*z*-component of the time averaged Poynting vector [25

**79**, 013825 (2009). [CrossRef]

**v̂**be a unit vector and consider the projection of the electric field at the origin

**r**=

**0**along this unit vector: The aim is to find the field for which the amplitude of

*ℰ*(

**0**,

*t*) ·

**v̂**is maximum for a given total power. Without restricting the generality we may assume that

**E**(

**0**) ·

**v**is real. In fact, if it is not real, applying a time shift can always make it real. Hence we will assume that Then the optimization problem is to find the plane wave amplitudes

**A**(

*k*,

_{x}*k*) for which the component of the electric field along

_{y}**v̂**has maximum amplitude for a given total power

*P*

_{0}: where

*ℋ*

_{0}is the space of plane wave amplitudes which have finite mean flow of power through a plane

*z*= constant and which satisfy Eq. (5). The problem is denoted as

*℘*

_{1}(

**v̂**) to indicate that we are optimizing in the direction of

**v̂**. It can be shown that optimization problem

*℘*

_{1}(

**v̂**) has a unique solution. The uniqueness is due to the imposed condition that

**E**(

**0**) ·

**v̂**is real. Without this condition the optimum solutions are unique except for a phase shift [25

**79**, 013825 (2009). [CrossRef]

### 2.2. Optimum plane wave amplitudes and field distributions

*℘*

_{1}(

**v̂**) can be computed by applying the Lagrange multiplier rule for inequality constraints. We remark that since we optimize a linear functional on a sphere in a Hilbert space (the sphere is the set of all fields of finite given power), the solution can alternatively be derived by using the Cauchy-Schwarz inequality. In any case, the solution can be computed in closed form and is given by [25

**79**, 013825 (2009). [CrossRef]

*α*

_{max}is such that It follows from Eq. (6) that the electric field vector of the plane wave with wave vector

**k**is parallel to the projection of

**v̂**on the plane perpendicular to

**k**. The electric field in an arbitrary point

**r**in image space can be obtained by substitution of the optimum plane wave amplitudes into Eq. (1). By changing the integration variables to respective polar and azimuthal angles, 0 <

*α*<

*α*

_{max}and 0 <

*β*< 2

*π*, given by (Fig. 1) and using cylindrical coordinates

*ρ*,

*φ*,

*z*for the point of observation:

**r**=

*x*

**x̂**+

*y*

**ŷ**+

*z*

**ẑ**=

*ρ*

**+**

ρ ^ *z*

**ẑ**=

*ρ*cos

*φ*

**x̂**+

*ρ*sin

*φ*

**ŷ**+

*z*

**ẑ**, one finds for the optimum electric field [25

**79**, 013825 (2009). [CrossRef]

**x̂**,

**ŷ**,

**ẑ**}.

## 3. Interpretation of optimum fields and second optimization

### 3.1. The electric field in focus

*l*≥ 1 it follows from Eq. (13) that with It is seen that the electric vector

**E**(

**0**) has real components, hence the electric field at

**r**=

**0**is linearly polarized. It is furthermore parallel to the plane through

**v̂**and the

*z*-axis. However, the direction of polarization is in general different from

**v̂**, i.e. from the direction along which the electric field amplitude is maximized. We shall write for convenience: Then Eq. (7) becomes: and Eq. (15) can be written as Let

**û**be the unit vector of the direction of polarization, i.e.

*v*= 0 (transverse polarization) and when

_{z}*v*=

_{x}*v*= 0 (longitudinal polarization). An exception is the case that the numerical aperture NA =

_{y}*n*. Then

*α*

_{max}=

*π*/2, so that

*γ*

_{+}=

*γ*

_{−}= 4/3 and hence

**û**and

**v̂**are always identical.

**v̂**lies in the (

*x*,

*z*)-plane, i.e. that

*v*= 0. Then

_{y}**û**also is in the (

*x*,

*z*)-plane. Let ϑ

*and ϑ*

_{v}*be the angle that*

_{u}**v̂**and

**û**, respectively, make with the

*z*-axis. Then and similarly, The relationship between both angles is displayed in Fig. 2 for several values of the numerical aperture. We see that always ϑ

*≥ ϑ*

_{u}*, i.e. the angle that*

_{v}**û**makes with the positive

*z*-axis is in general larger than that of

**v̂**. Furthermore, ϑ

*sweeps out all values between 0 and*

_{u}*π*/2 when ϑ

*varies between 0 and*

_{v}*π*/2. The difference between the two angles decreases when the numerical aperture is increased and the angles are the same in the limit NA =

*n*.

### 3.2. Optimum linearly polarized electric field

*℘*

_{1}(

**v̂**) have at the origin maximum electric field amplitude in the direction of

**v̂**, the electric field vector at the origin is, although linearly polarized, not parallel to

**v̂**. In fact it is pointing in the direction of the unit vector

**û**.

**v̂**

*and are in addition linearly polarized along*

**v̂**? The requirement that the electric field vector

*ℰ*(

**0**,

*t*) = Re[

**E**(

**0**)exp(–

*iωt*)] is parallel to

**v̂**at all times

*t*is equivalent to demanding that the vector

**E**(

**0**) is proportional to

**v̂**, which in turn is equivalent to: (When

**v̂**=

**ẑ**these conditions become:

**E**(

**0**) ·

**x̂**= 0 and

**E**(

**0**) ·

**ŷ**= 0). Hence, the second optimization problem that we want to consider is: where the space

*ℋ*

_{0}consists as before of all plane wave amplitudes

**A**(

*k*,

_{x}*k*) such that

_{y}**E**(

**0**) ·

**v̂**is real.

*℘*

_{2}(

**v̂**) can be determined analogously to that of problem

*℘*

_{1}(

**v̂**) by using the Lagrange multiplier rule or alternatively using the Cauchy-Schwarz inequality. In the latter case the constraint that the electric field should be linearly polarized along the direction of

**v̂**, should be incorporated into the Hilbert space of fields to which the Schwarz inequality is applied. Closed formulae are again obtained and the solution is again unique due to the fact that the phase is fixed by requiring that

**E**(

**0**) ·

**v̂**is real. It is then found that when

**v̂**is varied, more precisely, when ϑ

*is varied in the interval 0 ≤ ϑ*

_{v}*≤*

_{v}*π*/2, the two sets of optimum solutions of the two optimization problems are

*identical*when the proper directions are chosen. For every

**v̂**there is a unit vector

**û**such that the optimum field that is the solution of

*℘*

_{1}(

**v̂**) is also the solution of

*℘*

_{2}(

**û**), and conversely. In fact, the relationship between the unit vectors

**v̂**and

**û**is given by Eq. (23) defined in the previous section.

*℘*

_{2}(

**û**). Hence Since

**û**and since by Eq. (23)

**v̂**·

**û**> 0, it follows that but

*℘*

_{1}(

**v̂**) and since

*℘*

_{1}(

**v̂**) is unique, we thus conclude that Eq. (27) is indeed true.

### 3.3. Explicit expressions for the solution of problem ℘_{2}

*℘*

_{2}(

**û**) can be easily obtained. First we determine the unit vector

**v̂**by inverting Eq. (23): By substituting this into Eq. (6), we obtain the optimum plane wave amplitudes of problem

*℘*

_{2}(

**û**):

**r**becomes with Eq. (12): where

*ℳ̿*(

*ρ*,

*φ*,

*z*) is still given by Eq. (13).

### 3.4. Electric field strength in focus

*℘*

_{1}(

**v̂**) for some unit vector

**v̂**and let

**û**be the unit vector in the direction of

*v*=

_{y}*u*= 0. At the left-side of Fig. 3 the projections of

_{y}**v̂**and along

**û**are compared as function of ϑ

*for several values of the numerical aperture and for total power*

_{v}*P*

_{0}= 1W. The right-side of Fig. 3 shows a comparison between the projections of the solutions of

*℘*

_{1}(

**v̂**) and

*℘*

_{2}(

**v̂**) along

**v̂**, i.e.

*for several values of the numerical aperture and for*

_{v}*P*

_{0}= 1W). We see that for ϑ

*= 0*

_{v}*(longitudinal polarized optimum field) and ϑ*

^{o}*= 90*

_{v}*(transverse polarized optimum field), the projections are identical. As was pointed out before, this is also true for all angles 0 ≤ ϑ*

^{o}*≤*

_{v}*π*/2 when NA =

*n*.

## 4. Realization of the optimum fields

28. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. **39**, 205–210 (1981). [CrossRef]

### 4.1. Pupil fields that when focused yield the optimum field distributions.

**x̂**,

**ŷ**,

**ẑ**} be a Cartesian basis in image space such that the optical axis is parallel to

**ẑ**and the origin coincides with the focal point. Let {

**x̂**

*,*

_{p}**ŷ**

*} be the coordinate system in the entrance pupil of the lens, with*

_{p}**x̂**

*and*

_{p}**ŷ**

*parallel to*

_{p}**x̂**and

**ŷ**, respectively. Finally, let

*ρ*and

_{p}*φ*be polar coordinates in the entrance pupil, i.e. We have so that {

_{p}

ρ ^ *,*

_{p}

φ ^ *,*

_{p}**ẑ**} is a positively orientated orthonormal basis. The electric field in a point (

*ρ*,

_{p}*φ*) of the entrance pupil of the lens is decomposed on this basis: where the

_{p}**ẑ**-component of the electric field is neglected because the incident beam is predominantly propagating parallel to the optical

*z*-axis. This system is illustrated by Fig. 4.

*v*=

_{x}*v*=

_{y}*u*=

_{x}*u*= 0 (longitudinal case), the pupil field is radially polarized in all points of the pupil.

_{y}*℘*

_{1}(

**v̂**) (left figure) and

*℘*

_{2}(

**v̂**) (right figure), for unit vector

**v̂**with

*v*= 0 and ϑ

_{y}*= 45°. The numerical aperture is NA/*

_{v}*n*= 0.75 in both cases. The direction of the arrows is that of the linear polarization and their length is proportional to the electric field amplitude. The optimum pupil field at the right of Fig. 5 is similar to a radially polarized pupil field but with center shifted in the negative

*x*-direction.

_{p}## 5. Conclusion

*x*,

*y*)-plane as mirror.

## References and links

1. | V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd |

2. | V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , |

3. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A |

4. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A |

5. | X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science |

6. | L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. |

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

8. | L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. |

9. | M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. |

10. | Q. Zhan and J. Leger, “Focus Shaping using cylindrical vector beams,” Opt. Express |

11. | J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express |

12. | N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. |

13. | M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. |

14. | M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. |

15. | I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. |

16. | C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. |

17. | C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik |

18. | A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. |

19. | V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. |

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

21. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

22. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B |

23. | C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. |

24. | H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. |

25. | H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A |

26. | H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A |

27. | W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. |

28. | E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. |

29. | J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “ |

30. | R. Aarts, J. J. M. Braat, P. Dirksen, S. van Haver, C. van Heesch, and A. Janssen, “Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture,” J. Eur. Opt. Soc. Rapid Publ. |

31. | J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and S. F. Pereira, “Image formation in a multilayer using the extended Nijboer-Zernike theory,” J. Eur. Opt. Soc. Rapid Publ. |

32. | L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008). |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.6800) General : Theoretical physics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 5, 2011

Revised Manuscript: March 26, 2011

Manuscript Accepted: March 28, 2011

Published: April 27, 2011

**Citation**

R. de Bruin, H. P. Urbach, and S. F. Pereira, "On focused fields with maximum electric field components and images of electric dipoles," Opt. Express **19**, 9157-9171 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9157

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### References

- V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).
- V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).
- E. Wolf, “Electromagnetic diffraction in optical systems I. An integral repesentation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959). [CrossRef]
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959). [CrossRef]
- X. S. Xie and R. C. Dunn, “Probing single molecule dynamics,” Science 265, 361–364 (1994). [CrossRef] [PubMed]
- L. Novotny, M. R. Beverluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]
- Q. W. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef] [PubMed]
- L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002). [CrossRef]
- M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A: Mater. Sci. Process. 86, 329–334 (2007). [CrossRef]
- Q. Zhan and J. Leger, “Focus Shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002). [PubMed]
- J. Wang, W. Chen, and Q. Zhan, “Engineering of high purity ultra-long optical needle field through reversing the electric dipole array radiation,” Opt. Express 18, 21965–21972 (2010). [CrossRef] [PubMed]
- N. Sanner, N. Huot, E. Audouard, C. Larat, J.-P. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005). [CrossRef] [PubMed]
- M. A. A. Neil, F. Massoumian, R. Juskaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002). [CrossRef]
- M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948 (1996). [CrossRef] [PubMed]
- I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271, 40–47 (2007). [CrossRef]
- C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994). [CrossRef]
- C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).
- A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus of high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007). [CrossRef]
- V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light–theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
- C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]
- H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008). [CrossRef]
- H. P. Urbach and S. F. Pereira, “Focused fields of given power with maximum electric field components,” Phys. Rev. A 79, 013825 (2009). [CrossRef]
- H. P. Urbach and S. F. Pereira, “Erratum: focused fields of given power with maximum electric field components,” Phys. Rev. A 81, 059903 (2010). [CrossRef]
- W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12, 045707 (2010) [CrossRef]
- E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981). [CrossRef]
- J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468. [CrossRef]
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