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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9157–9171
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On focused fields with maximum electric field components and images of electric dipoles

R. de Bruin, H. P. Urbach, and S. F. Pereira  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9157-9171 (2011)
http://dx.doi.org/10.1364/OE.19.009157


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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

When light is focused by a lens with numerical aperture larger than 0.6, the rotation of polarization becomes important. The rotation of polarization of a diffraction limited lens of high numerical aperture is well described by the model of Ignatowsky [1

1. V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).

, 2

2. V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).

] and Wolf and Richards [3

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

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]

]. When the field in the lens aperture is that of a linearly polarized plane wave, the dominant electric field component near the focal point is parallel to the direction of polarization of the incident plane wave. But, as the numerical aperture increases, the maximum value of the longitudinal component of the electric field in the focal plane becomes quite substantial, although it vanishes at the focal point itself.

In many applications such as optical recording, photolithography and microscopy, it is important to be able to shape the focused spot. A field in focus with maximum electric component in a certain direction is of interest for manipulating and probing single molecules and particles, and in materials processing [5

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

9

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]

]. The focused wavefront can be tailored by setting a proper amplitude, phase and polarization distributions in the pupil of the focusing lens [10

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

,11

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]

]. Nowadays it is possible to realize almost any complex transmission function in the pupil plane, using for example liquid crystal-based devices [12

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

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

].

The optimization of focused fields has been studied by [16

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

18

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]

] for rotational symmetric systems. In the first two papers it was noted that the focused linearly polarized plane wave in the theory of Ignatowsky and Richard and Wolf, is in the Debye approximation identical to the sum of fields that are truncated by the lens aperture and converge to orthogonal electric and magnetic dipoles, with a particular ratio of dipole strengths. A nice treatment is given in [19

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

].

It was noted by several authors [16

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

, 20

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

23

23. C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]

] that when a radially polarized beam is focused, the distribution of the longitudinal component can be considerably narrower than the focused spot obtained by focusing a linearly polarized plane wave. In [24

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

] the pupil field was determined which after focusing gives the maximum longitudinal component in the focal point for a given power incident on the lens. The optimum pupil field is radially polarized and its amplitude increases in a particular way with radial distance to the center of the pupil. Furthermore, the distribution in the focal plane of the optimized longitudinal component is narrower than the Airy spot. If a recording medium would be used which is sensitive for only the longitudinal component of the electric field, this optimized longitudinal component can yield higher resolution.

In [25

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]

] this analysis was generalized to optimum pupil fields of which the electric field amplitude in an arbitrary, not necessarily longitudinal, direction in the focal point is maximum. It was in particular found that the pupil field which gives maximum transverse amplitude in focus is similar but not identical to that of a linearly polarized plane wave.

An interesting feature of the optimum focused fields is that, although the electric field amplitude at the focal points is maximum in a required direction, the electric field is in general 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.

For certain applications it is desirable to have maximum electric field amplitude in the focal point in a given direction under the additional constraint that the electric field is also linearly polarized in that direction. We shall therefore consider this optimization problem in this paper. A major result is that as the optimization direction is varied the set of solutions is identical to that of the problem without the constraint of parallel polarization studied in [25

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]

]. To be more precise: the electric field that has maximum amplitude in a given direction but which is not linearly polarized along that direction, is also the field that has maximum amplitude in the direction of its linear polarization.

As was already explained in [25

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]

], the optimum focused fields can be realized by focusing appropriate pupil fields. These pupil fields are linearly polarized with the direction of polarization and amplitude that are functions of the pupil coordinates. An alternative method to realize the optimum fields is by imaging an electric dipole by an optical system with unit magnification, e.g. two identical lenses. The image of an electric dipole was studied previously in [6

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]

, 17

17. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

, 19

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

] and, more recently, in [27

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

]. We will show that when the imaging system has unit magnification, the image of an electric dipole with dipole vector p = (px, py, pz) is, for the given power, identical to the electric field in image space that has maximum amplitude in the direction (px, py, – pz), where the 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.

The paper is organized as follows. In Section 2 the optimization problem and the optimum fields that have been derived in [24

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

] and [25

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]

] are summarized. The optimization problem is formulated in homogeneous space without considering the optical system by which the optimum fields are generated. The reason is that the results thus obtained are general and independent of any model used for the optical system. In Section 3 the results of this optimization are interpreted and the new optimization problem is studied where the electric field is required to be linearly polarized along the direction for which the amplitude of the electric field is maximum. It is then shown that when this direction is varied, the set of optimum fields is identical to the set of optimum fields that are the solutions of the first optimization problem. Finally in Section 4 two methods to realize the optimum fields are considered. The first method is by focusing an appropriate pupil field. We use here the vector diffraction theory of Ignatowksy and Wolf and Richards, which is based on the Debye approximation. The second method is by imaging an electric dipole with unit magnification.

2. The optimization problem

2.1. Formulation of the optimization problem

We first define a few quantities and introduce some notations. As was stated in the Introduction, in the optimization problem we first leave the optical system out of consideration. The optimized fields are derived from the rigorous Maxwell equations and are therefore generally valid. We only assume that the field in image space has a certain numerical aperture NA ≤ 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.

The plane wave vectors in image space are limited by the numerical aperture of the imaging system. Let (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
(r,t)=Re[E(r)eiωt]=14π2Re[kx2+ky2k0NAA(kx,ky)ei(k·rωt)dkxdky],
(1)
where E(r) is the complex field, k = (kx, ky, kz) the wave vector in the medium with refractive index n, k 0 = |k|/n = 2π/λ 0 with λ 0 the wavelength in vacuum and A(kx, ky) are the complex plane wave amplitudes. Since the field propagates in the positive z-direction, we have kz=k02n2kx2ky2>0. The circle kx2+ky2k0NA in the (kx, ky)-plane, which limits the integration area, will be denoted by Ω for brevity. Since the electric field is free of divergence, we have:
A(kx,ky)·k=0.
(2)

The time-averaged power flow is given by the integral over a plane z = constant of the z-component of the time averaged Poynting vector [25

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]

]:
P(A)=12Re[E(r)×H(r)*]·z^dxdy=18π2k0(ɛ0μ0)1/2Ω|A(kx,ky)|2kzdkxdky.
(3)
Let be a unit vector and consider the projection of the electric field at the origin r = 0 along this unit vector:
(0,t)·v^=Re[E(0)·v^eiωt].
(4)
The aim is to find the field for which the amplitude of (0,t) · 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
Im[E(0)·v^]=14π2ΩIm[A(kx,ky)·v^]dkxdky=0.
(5)
Then the optimization problem is to find the plane wave amplitudes A(kx, ky) for which the component of the electric field along has maximum amplitude for a given total power P 0:
1(v^):maxA0Ev(A)E(0)·v^,undertheconstraintP(A)P0,
where 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() to indicate that we are optimizing in the direction of . It can be shown that optimization problem 1() has a unique solution. The uniqueness is due to the imposed condition that E(0) · is real. Without this condition the optimum solutions are unique except for a phase shift [25

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]

].

2.2. Optimum plane wave amplitudes and field distributions

The solution of problem 1() 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

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]

]:
A(kx,ky)=(8πP0n)1/2(μ0ɛ0)1/41kz1Γ(αmax,vz)[v^(v^·k^)k^],
(6)
where
Γ(αmax,vz)={43cosαmax13cos3αmaxvz2cosαmaxsin2αmax}1/2,
(7)
for which α max is such that
sinαmax=NA/n.
(8)
It follows from Eq. (6) that the electric field vector of the plane wave with wave vector k is parallel to the projection of 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 < 2π, given by (Fig. 1)
Fig. 1 Spherical coordinates in image space with respect to wave unit vector .
kx=ksinαcosβ,
(9)
ky=ksinαsinβ,
(10)
kz=kcosα,
(11)
and using cylindrical coordinates ρ,φ,z for the point of observation: r = x + y ŷ + z = ρ ρ^ + z = ρ cos φ + ρ sin φ ŷ + z , one finds for the optimum electric field [25

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]

]:
E(ρ,φ,z)=2πnP0λ0(μ0ɛ0)1/41Γ(αmax,vz)¯¯(ρ,φ,z)(vxvyvz),
(12)
where
¯¯(ρ,φ,z)=(g00,1(ρ,z)+g02,1(ρ,z)000g00,1(ρ,z)+g02,1(ρ,z)0002g00,3(ρ,z))+(g20,3(ρ,z)cos(2φ)g20,3(ρ,z)sin(2φ)2ig11,2(ρ,z)cosφg20,3(ρ,z)sin(2φ)g20,3(ρ,z)cos(2φ)2ig11,2(ρ,z)sinφ2ig11,2(ρ,z)cosφ2ig11,2(ρ,z)sinφ0),
(13)
and where
glν,μ(ρ,z)=0αmaxeik0nzcosαcosναsinμαJl(k0nρsinα)dα.
(14)
Note that the electric field vector and the matrix in Eq. (12) are on the Cartesian basis {,ŷ,}.

3. Interpretation of optimum fields and second optimization

3.1. The electric field in focus

We consider the optimum electric field vector at the origin. Because glν,μ(0)=0 when l ≥ 1 it follows from Eq. (13) that
E(0)=[g00,1(0)+g02,1(0)]vxx^+[g00,1(0)+g02,1(0)]vyy^+2g00,3(ρ,z)vzz^,
(15)
with
g00,1(0)=0αmaxsinαdα=1cosαmax,
(16)
g02,1(0)=0αmaxcos2αsinαdα=1313cos3αmax,
(17)
g00,3(0)=0αmaxsin3αdα=23cosαmax+13cos3αmax.
(18)
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 and the z-axis. However, the direction of polarization is in general different from , i.e. from the direction along which the electric field amplitude is maximized. We shall write for convenience:
γ+g00,1(0)+g02,1(0)=43cosαmax13cos3αmax,
(19)
γ2[g00,1(0)g02,1(0)]=432cosαmax+23cos3αmax.
(20)
Then Eq. (7) becomes:
Γ(αmax,vz)={γ+(γ+γ)vz2}1/2,
(21)
and Eq. (15) can be written as
E(0)=γ+(vxx^+vyy^)+γvzz^.
(22)
Let û be the unit vector of the direction of polarization, i.e.
u^=E(0)|E(0)|=γ+(vxx^+vyy^)+γvzz^γ+2(vx2+vy2)+γ2vz2.
(23)

We conclude that the direction of polarization at the origin is only parallel to the direction of optimization when vz = 0 (transverse polarization) and when vx = vy = 0 (longitudinal polarization). An exception is the case that the numerical aperture NA = n. Then α max = π/2, so that γ + = γ = 4/3 and hence û and are always identical.

Without restricting the generality we may assume that lies in the (x,z)-plane, i.e. that vy = 0. Then û also is in the (x,z)-plane. Let ϑv and ϑu be the angle that and û, respectively, make with the z-axis. Then
vx=sinϑv,vz=cosϑv,
(24)
and similarly,
ux=sinϑu,uz=cosϑu,
(25)
The relationship between both angles is displayed in Fig. 2 for several values of the numerical aperture. We see that always ϑu ≥ ϑv, i.e. the angle that û makes with the positive z-axis is in general larger than that of . Furthermore, ϑu sweeps out all values between 0 and π/2 when ϑv varies between 0 and 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.

Fig. 2 The relationship between the optimization angle ϑv for which problem 1 is solved and angle ϑu between the direction of the optimum electric field vector E(0) and the z-axis, for several values of the numerical aperture NA.

3.2. Optimum linearly polarized electric field

Although the solutions of problem 1() have at the origin maximum electric field amplitude in the direction of , the electric field vector at the origin is, although linearly polarized, not parallel to . In fact it is pointing in the direction of the unit vector û.

The question then arises: what are the optimum fields that have maximum amplitude in a given direction and are in addition linearly polarized along ? The requirement that the electric field vector (0,t) = Re[E(0)exp(–iωt)] is parallel to at all times t is equivalent to demanding that the vector E(0) is proportional to , which in turn is equivalent to:
E(0)·(v^×z^)=0andE(0)·[v^×(v^×z^)]=0.
(26)
(When = these conditions become: E(0) · = 0 and E(0) · ŷ = 0). Hence, the second optimization problem that we want to consider is:
2(v^):maxA0Ev(A)=E(0)·v^,undertheconstraintsP(A)P0and(26),
where the space 0 consists as before of all plane wave amplitudes A (kx, ky) such that E(0) · is real.

It is clear that E1v^ satisfies all constraints of problem 2(û). Hence
0<E1v^(0)·u^E2u^(0)·u^.
(28)
Since E1v^(0) and E2u^(0) are both pointing in the direction of û and since by Eq. (23) · û > 0, it follows that
E1v^(0)·v^E2u^(0)·v^.
(29)
but E2u^ satisfies all constraints of problem 1() and since E1v^ is the solution, it follows that in Eq. (29) equality holds. Because the solution of problem 1() is unique, we thus conclude that Eq. (27) is indeed true.

3.3. Explicit expressions for the solution of problem ℘2

With the results obtained above, explicit formulae for the optimum plane wave amplitudes that are solutions of optimization problem 2(û) can be easily obtained. First we determine the unit vector by inverting Eq. (23):
v^=γ(uxx^+uyy^)+γ+uzz^γ2(ux2+uy2)+γ+2uz2=γu^+(γ+γ)uzz^γ2(ux2+uy2)+γ+2uz2.
(30)
By substituting this into Eq. (6), we obtain the optimum plane wave amplitudes of problem 2(û):
A(kx,ky)=(8πP0n)1/2(μ0ɛ0)1/41kz1Γ(αmax,vz)[v^(v^·k^)k^]=(8πP0n)1/2(μ0ɛ0)1/41kz1Γ˜(αmax,uz)×{γ[u^(u^·k^)k^]+(γ+γ)uz[z^(z^·k^)k^]},
(31)
where
Γ˜(αmax,uz)Γ(αmax,vz)γ2(ux2+uy2)+γ+2uz2=γ+γ{γ(γ+γ)uz2}1/2,
(32)
Finally, the expression for the optimum electric field in an arbitrary point r becomes with Eq. (12):
E(ρ,φ,z)=2πnP0λ0(μ0ɛ0)1/41Γ˜(αmax,uz)¯¯(ρ,φ,z)(γuxγuyγ+uz),
(33)
where ℳ̿ (ρ, φ, z) is still given by Eq. (13).

3.4. Electric field strength in focus

Let E1v^ be the solution of optimization problem 1() for some unit vector and let û be the unit vector in the direction of E1v^(0) as defined by Eq. (23). Without restricting the generality, we assume again that vy = uy = 0. At the left-side of Fig. 3 the projections of E1v^(0) along the direction of optimization and along û are compared as function of ϑv for several values of the numerical aperture and for total power P 0 = 1W. The right-side of Fig. 3 shows a comparison between the projections of the solutions of 1() and 2() along , i.e. E1v^(0)·v^ and E2v^(0)·v^ (again as functions of ϑv for several values of the numerical aperture and for P 0 = 1W). We see that for ϑv = 0o (longitudinal polarized optimum field) and ϑv = 90o (transverse polarized optimum field), the projections are identical. As was pointed out before, this is also true for all angles 0 ≤ ϑvπ/2 when NA = n.

Fig. 3 Left: The electric field E1v^(0) projected along and û as functions of ϑv for several values of the numerical aperture. : The projections of the optimum electric fields E1v^(0) and E2v^(0) along the direction , as function of ϑv for several values of the numerical aperture.

4. Realization of the optimum fields

In this section we discuss two methods for realizing the optimum fields. In the first method an appropriate pupil distribution is focused by a lens of given numerical aperture. The optimum field is the field in image space of the lens with the focal point as point were the electric field is maximum. In the second method, an electric dipole with appropriate dipole moment is imaged by a lens system with magnification 1. The field component is then maximum at the image point of the dipole. It should be noted that both realizations of the image field are approximative because they are based on a model of the focusing and imaging of a high NA lens which are valid under certain, be it rather mild, assumptions [28

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.

We consider a diffraction limited lens with (high) numerical aperture NA and a light beam which propagates parallel to the optical axis and is focused by this lens. Let {,ŷ,} be a Cartesian basis in image space such that the optical axis is parallel to and the origin coincides with the focal point. Let { p,ŷ p} be the coordinate system in the entrance pupil of the lens, with p and ŷ p parallel to and ŷ, respectively. Finally, let ρp and φp be polar coordinates in the entrance pupil, i.e.
xp=ρpcosφp,yp=ρpsinφp.
(34)
We have
ρ^p=cosφpx^p+sinφpy^p,
(35)
φ^p=sinφpx^p+cosφp·y^p,
(36)
so that {ρ^ p, φ^ p,} is a positively orientated orthonormal basis. The electric field in a point (ρp, φp) of the entrance pupil of the lens is decomposed on this basis:
Ep(ρp,φp)=Eρp(ρp,φp)ρ^p+Eφp(ρp,φp)φ^p,
(37)
where the -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.

Fig. 4 The focusing lens with numerical aperture NA.

It follows from the formulas that the optimum pupil fields are linearly polarized in all points of the pupil and that in all points they are in phase. Furthermore, for the case that vx = vy = ux = uy = 0 (longitudinal case), the pupil field is radially polarized in all points of the pupil.

In Fig. 5 snapshots are shown of the optimum pupil distributions which after focusing give the solution of 1() (left figure) and 2() (right figure), for unit vector with vy = 0 and ϑv = 45°. The numerical aperture is NA/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 xp-direction.

Fig. 5 Left: Snapshots of the electric field in the entrance pupil, E p, which after focusing yields the field with maximum component along obtained by solving 1() (left) and the optimum pupil field that is solution of 2() (right), where is such that vy = 0 and ϑv = 45°. In both cases NA/n = 0.75.

4.2. Imaging of an electric dipole

Fig. 7 Schematics of the imaging of a dipole using the two lens system with (paraxial) magnification equal to 1. The dipole produces the electric field at the focal point which has maximum amplitude in the direction of and which has maximum amplitude and is linearly polarized in the direction of û.

5. Conclusion

References and links

1.

V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).

2.

V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).

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]

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]

7.

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

8.

L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002). [CrossRef]

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]

13.

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]

14.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948 (1996). [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]

17.

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).

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]

20.

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

21.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

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 72, 109–113 (2001).

23.

C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]

24.

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

26.

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]

27.

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

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]

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 “Progress in Optics,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468. [CrossRef]

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. 3, 08039 (2008). [CrossRef]

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. 4, 09048 (2009). [CrossRef]

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

  1. V. S. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd I, paper IV (1919), (in Russian).
  2. V. S. Ignatowsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd , I, paper V (1920), (in Russian).
  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]
  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]
  7. Q. W. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004). [CrossRef] [PubMed]
  8. L. E. Helseth, “Focusing of atoms with strongly confined light potentials,” Opt. Commun. 212, 343–352 (2002). [CrossRef]
  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]
  13. 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]
  14. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948 (1996). [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]
  17. C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177 (1997).
  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]
  20. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  21. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
  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 72, 109–113 (2001).
  23. C. J. R. Sheppard and A. Choudhurry, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]
  24. H. P. Urbach and S. F. Pereira, “The field in focus with maximum longitudinal electric component,” Phys. Rev. Lett. 100, 1233904 (2008). [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]
  26. 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]
  27. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12, 045707 (2010) [CrossRef]
  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]
  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 “Progress in Optics ,” Vol. 51, E. Wolf, ed. (Elsevier B.V., 2008), chap. 6, pp. 349–468. [CrossRef]
  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. 3, 08039 (2008). [CrossRef]
  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. 4, 09048 (2009). [CrossRef]
  32. L. Novotny and B. Hecht, “Principles of nano-optics,” Section 2.10.2, (Cambridge University Press, 2008).

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