Focused fields of given power with maximum electric field components

Focused fields of given power with maximum electric field components

H. P. Urbach

*

and S. F. Pereira

Optics Research Group, Department of Imaging Science and Technology, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands

共Received 27 August 2008; revised manuscript received 24 October 2008; published 28 January 2009兲 Closed formulas are derived for the field in the focal region of a diffraction limited lens, such that the electric field component in a given direction at the focal point is larger than that of all other focused fields with the same power in the entrance pupil of the lens. Furthermore, closed formulas are derived for the correspond-ing optimum field distribution in the lens pupil. Focused fields with maximum longitudinal or maximum transverse are considered in detail. The latter field is similar, but not identical, to the focused linearly polarized plane wave.

DOI:10.1103/PhysRevA.79.013825 PACS number共s兲: 42.30.Va, 42.25.Fx, 42.79.Hp, 42.79.Bh

I. INTRODUCTION

When a linearly polarized plane wave is focused by a diffraction-limited lens, the intensity distribution in the focal plane is in the scalar theory the well-known Airy pattern. However, when the lens has high numerical aperture, the rotation of polarization must be accounted for and the vector diffraction theory of Ignatowsky关1,2兴 and Richards and Wolf

关3,4兴 has to be applied to obtain the field distribution in the

focal region. We then obtain three electric and three mag-netic field components in the focal region. When the beam in the lens aperture is uniformly linearly polarized plane wave, the dominant electric field component in the focal region is found to be parallel to the polarization direction of the inci-dent 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.

An appropriately shaped focused spot is essential in many applications such as optical recording, photolithography, and microscopy. Furthermore, a field in focus with maximum electric component in a specific direction is important for manipulating single molecules and particles, and in materials processing关5–9兴. The focused wave front can be tailored by

setting a proper amplitude, phase, and polarization distribu-tions in the pupil of the focusing lens. Nowadays it is pos-sible to realize almost any complex transmission function in the pupil plane, using, for example, liquid-crystal-based de-vices关10–14兴.

In this paper, we maximize a specified component of the electric field in the focal point of a diffraction-limited lens. First we consider fields in free space or in homogeneous matter, without taking into account the way these fields are realized, in particular without considering the lens. We merely suppose that, with respect to a Cartesian coordinate system共x,y,z兲, the fields considered consist of plane waves propagating in the positive z direction and have wave vectors with angles with the positive z axis that do not exceed a specific maximum angle, i.e., the numerical aperture of the plane waves is restricted. The optimization problem is then to find the complex plane wave amplitudes such that, for a

given direction in space and for given mean flow of electro-magnetic power through a plane z = const, the amplitude at some point 共chosen at the origin兲 of the electric field com-ponent that is parallel to the chosen direction is larger than that of any other field for the same numerical aperture and the same mean total power flow. We shall derive closed for-mulas for the plane wave amplitudes of the optimum field.

The solutions of the optimization problems for the field propagating in homogeneous space is rigorous results since they are derived from Maxwell’s equations without any fur-ther assumptions. Next we will consider the realization of the optimum field using a diffraction-limited lens with the origin as the focal point. By using the vector diffraction theory of Ignatowksy and Richards and Wolf, closed expressions for the optimum pupil distributions will be derived, which after focusing give the optimum field component in the focal point of the lens. In contrast with the solution in terms of the plane wave expansion, the formulas for the optimum pupil fields are approximate since they are based on the vector diffrac-tion theory which is an approximate theory that is valid for lenses of which the focal distance and the pupil radius are many wavelengths.

When one considers the focusing by a lens it is obvious that the plane wave expansion in image space has finite nu-merical aperture of that of the lens, NA= n sin␣max, where n is the refractive index in image space and␣maxis half the top angle of the cone with the top the focal point and the base the pupil. But also when one would consider only waves in free space without a focusing lens, there is a good reason to re-strict the fields to finite numerical aperture, in that case in particular to NA= n. In fact, when NA⬎n a part of the eva-nescent waves are taken into account in the expansion. One can then construct fields with a given power which have arbitrary large components. The evanescent waves do not contribute to the total power and hence one can increase their amplitudes by any desired amount to achieve arbitrarily high local fields. Stated differently, by constructing suitable time-harmonic source distributions that emit singular fields with finite power flow, one can achieve arbitrary large field com-ponents by approaching these source distributions. At small distances to the sources the evanescent waves of course play a major role.

Among the directions for the optimized electric field com-ponent, two are of particular interest, namely, the directions parallel and perpendicular to the optical z axis. These direc-*h.p.urbach@tudelft.nl

tions are also called the longitudinal and transverse direc-tions. The field with maximum longitudinal component has been discussed in 关15兴, but without derivation. In this paper

details of the derivation are provided and the optimization problem is generalized to arbitrary directions of the electric field vector.

As was announced in关15兴, the pupil field that when

fo-cused gives maximum longitudinal component, is radially polarized. This means that in all points of the lens pupil, the electric field is linearly polarized with the electric field point-ing in the radial direction. Furthermore, the electric fields in all points of the pupil are in phase and the electric field amplitudes are rotationally symmetric. The amplitude of the electric pupil field vanishes at the center of the pupil and is a monotonically increasing function of the radial coordinate. The shape of this function depends on the numerical aper-ture.

It was noted by several authors关16–19兴 that when a

radi-ally polarized beam is focused, the distribution of the longi-tudinal component can be considerably narrower than the focused spot obtained by focusing a linearly polarized plane wave. With the development of a new generation of photo-resists关20兴, it is possible to control the photosensitive

mate-rial in such a way that it will react to only one of the polar-ization components of the electric field. Materials with molecules having fixed absorption dipole moments have been applied in 关6兴 to be able to probe field components

individually. When this component is the longitudinal com-ponent, a tighter spot can thus be obtained than with the classical Airy pattern.

Often the amplitude distribution of the radially polarized beam in the pupil plane is chosen to be a doughnut shape or a ring mask function关17兴. But these distributions do not give

the maximum possible longitudinal electric field component in focus for the given power and its amplitude as a function of the radial pupil coordinate differs from the optimum func-tion derived in the present paper.

The other case of particular interest is the optimization of the transverse electric field vector. Since the optical system is assumed to be rotationally symmetric around the optical axis, we may choose this direction parallel to the x axis. The so-lution of the optimization problem is then the field for which the amplitude of the x component of the electric field in the focal point is maximum for the given numerical aperture and the given total power flow. We will show that the corre-sponding pupil field is linearly polarized with direction of polarization predominantly, although not exactly, parallel to the x axis. Therefore, the focused optimum field is similar to the vectorial Airy pattern of a focused linearly polarized plane wave, although it is not identical to it.

The paper is organized as follows. In Sec. II, we will formulate the optimization problem and we will prove that the optimization problem has one and only one solution. In Sec.III, we will apply the Lagrange multiplier rule to obtain closed formulas for the plane wave amplitudes of the opti-mum field and for the optiopti-mum field distributions near focus. In Sec. IV, we will study the optimum fields in the focal region, in particular their mean energy flow. Then, in Sec.V, we apply the vector diffraction theory of Ignatowsky and Richards and Wolf to derive the electric field distribution in

the pupil of the lens that, when focused, yields the maximum field component in the focal region.

II. FORMULATION OF THE OPTIMIZATION PROBLEM FOR ARBITRARY ELECTRIC FIELD COMPONENT We begin with some notations. Consider a time-harmonic electromagnetic field in a homogeneous unbounded medium with real refractive index n共i.e., the material does not absorb electromagnetic radiation of the given frequency兲,

E共r,t兲 = Re关E共r兲e−i␻t_{兴, 共1兲} H共r,t兲 = Re关H共r兲e−i␻t_{兴, 共2兲} where␻⬎0. As stated in the Introduction, the lens is first not considered in the optimization problem. It is merely assumed that, with respect to the Cartesian coordinate system共x,y,z兲 with unit vectors xˆ, yˆ, and zˆ, the electromagnetic field 共1兲

and 共2兲 has numerical aperture NA艋n and that the plane

wave vectors have positive z component,

E共r兲 = 1 4␲2

冕冕

冑

k_x2+k_y2艋k0n sin␣max A共kx,ky兲eik·rdkxdky, 共3兲 H共r兲 = 1 4␲2 1 ␻␮0

冕冕

冑

k_x2+k_y2艋k0n sin␣max k⫻ A共kx,ky兲 ⫻eik·r_dk xdky, 共4兲 where k =共kx, ky, kz兲 with kz=共k02n2− kx 2 − ky 2_兲1/2_{, 共5兲} with k0=␻

冑

⑀0␮0= 2␲/␭0, where ␭0 is the wavelength in vacuum, and where NA= n sin␣maxwith␣maxthe maximum angle that the wave vectors make with the positive z direc-tion. If NA= n we have␣max=␲/2 and the plane wave spec-trum then consists of all homogeneous plane waves that propagate in the non-negative z direction 共there are no eva-nescent waves in the expansion兲. When NA⬍n, the cone of allowed wave vectors has top angle ␣max= arcsin共NA/n兲 ⬍90°. Because the electric field is free of divergence we have that

A · k = 0. 共6兲

We shall use spherical coordinates in reciprocal k space, kˆ = sin␣cos␤xˆ + sin␣sin␤yˆ + cos␣zˆ, 共7兲 ␣ˆ = cos␣cos␤xˆ + cos␣sin␤yˆ − sin␣zˆ, 共8兲 ␤ˆ = − sin␤xˆ + cos␤yˆ, 共9兲 where 0艋␣艋␣max and 0艋␤⬍2␲ are the polar and azi-muthal angles, respectively. Conversely, we have

xˆ = sin␣cos␤kˆ + cos␣cos␤␣ˆ − sin␤␤ˆ , 共10兲 yˆ = sin␣sin␤kˆ + cos␣sin␤␣ˆ + cos␤␤ˆ , 共11兲

zˆ = cos␣kˆ − sin␣␣ˆ . 共12兲 Note that兵kˆ,␣ˆ ,␤ˆ 其 is a positively oriented orthonormal basis, kˆ⫻␣ˆ =␤ˆ , ␣ˆ ⫻␤ˆ = kˆ, ␤ˆ ⫻ kˆ =␣ˆ . 共13兲 Furthermore, k = k₀nkˆ and the Jacobian of the transformation 共␣,␤兲哫共kx, ky兲 is

冢

⳵kx ⳵␣ ⳵kx ⳵␤ ⳵ky ⳵␣ ⳵ky ⳵␤

冣

= k0n

冉

cos␣cos␤ − sin␣sin␤

cos␣sin␤ sin␣cos␤

冊

, 共14兲 so that

dkxdky= k0 2

n2cos␣sin␣d␣d␤. 共15兲 Because of Eq.共6兲 we have

A共␣,␤兲 = A_␣共␣,␤兲␣ˆ + A_␤共␣,␤兲␤ˆ , 共16兲 for some functions A_␣ and A_␤. Then, using Eq.共13兲,

k⫻ A = k0n共− A_␤␣ˆ + A_␣␤ˆ 兲. 共17兲 The plane wave expansion can thus be written as

E共r兲 =n 2 ␭0 2

冕

0 ␣max

冕

0 2␲

共A␣␣ˆ + A␤␤ˆ 兲cos␣sin␣eik·rd␣d␤, 共18兲 H共r兲 =n 3 ␭02

冉

⑀0 ␮0

冊

1/2

冕

0 ␣max

冕

0 2␲ 共− A␤␣ˆ

+ A␣␤ˆ 兲cos␣sin␣eik·rd␣d␤. 共19兲 The ␣ˆ component is parallel to the plane through the wave

vector and the z axis, whereas the ␤ˆ component is perpen-dicular to this plane.

Let vˆ =vxxˆ +vyyˆ +vzzˆ be a real unit vector. We consider

the projection of the electric field at the origin at time t = 0 on the direction of vˆ, E共0兲 · vˆ =n 2 ␭0 2

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲v␣ + A_␤共␣,␤兲v_␤兴cos␣sin␣d␣d␤, 共20兲 where

v_␣= vˆ ·␣ˆ = vxcos␣cos␤+vycos␣sin␤−vzsin␣,

共21兲 v_␤= vˆ ·␤ˆ = − vxsin␤+vycos␤. 共22兲

We will consider E共0兲·vˆ as a 共linear兲 functional of A=A␣␣ˆ + A␤␤ˆ , which for brevity we will denote by F共A兲. Hence,

F共A兲 = def._n2 ␭02

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲v␣ + A_␤共␣,␤兲v_␤兴cos␣sin␣d␣d␤. 共23兲 Next, we calculate the total mean flow of power through a plane z = const. The total mean power flow is obtained by integrating the normal component of the vector 共1/2兲Re S over the plane z = const, where S = E⫻H* is the complex Poynting vector. By using Plancherel’s formula, the integral of Re S over this plane can be written as an integral over kx

and ky,

冕

−⬁ ⬁

冕

−⬁ ⬁ ₁ 2 Re关S共r兲兴dxdy = 1 2 Re

冕

_−⬁ ⬁

冕

−⬁ ⬁ E共r兲 ⫻ H共r兲*dxdy = 1 8␲2Re

冕冕

冑

k_x2+k_y2艋NAk0 A共kx,ky兲eikzz⫻

冋

k ␻␮0⫻ A共kx,ky兲*

册

e−ikzzdkxdky = 1 8␲2 1 ␻␮0

冕冕

_冑

k_x2+k_y2艋NAk0 兩A共kx,ky兲兩2kdkxdky, 共24兲

where we used that k is real and A共kx, ky兲·k=0. The total time-averaged flow of energy in the positive z direction through the

plane z = const is given by the z component of Eq. 共24兲,

冕

−⬁ ⬁

冕

−⬁ ⬁ ₁ 2 Re关Sz共r兲兴dxdy = 1 8␲2 1 ␻␮0

冕冕

_冑

k_x2+k_y2艋NAk0 兩A共kx,ky兲兩2kzdkxdky = n 3 2␭0 2

冉

⑀0 ␮0

冊

1/2

冕

0 ␣max

冕

0 2␲

This is independent of the plane z = const, as should be in a medium without losses.

The quantity F共A兲=E共0兲·vˆ is the complex electric field component in the direction of vˆ at time t = 0. Without restrict-ing the generality we may assume that F共A兲 is real. If it were not real, a time shift could be applied to make it real. Hence we may assume that

Im关E共0兲 · vˆ兴 =

冕

0 ␣max

冕

0 2␲ Im关A␣共␣,␤兲v␣ + A_␤共␣,␤兲v_␤兴cos␣sin␣d␣d␤= 0. 共26兲 The optimization problem is to find the plane wave am-plitudes A = A_␣␣ˆ + A_␤␤ˆ for which the electric field compo-nent at the origin 0 that is parallel to the direction of vˆ is larger than for any other field with the same mean power flow through a plane z = const and the same numerical aper-ture. To formulate this problem mathematically, we introduce the spaceH of plane wave amplitudes A=A_␣␣ˆ + A_␤␤ˆ which have finite mean flow of power through the planes z = const,

H =

再

A = A_␣␣ˆ + A_␤␤ˆ ;

冕

0 ␣max

冕

0 2␲ 关兩A␣共␣,␤兲兩2 +兩A_␤共␣,␤兲兩2兴cos2␣sin␣d␣d␤⬍ ⬁

冎

, 共27兲 and we define H0 as the subspace of H consisting of all A which satisfy Eq.共26兲. Then H is a Hilbert space with scalar

product 具A,B典_H=

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲B␣共␣,␤兲* + A_␤共␣,␤兲B_␤共␣,␤兲*兴cos2_{␣sin␣d␣d␤, 共28兲} andH₀ is a closed subspace of H. Note that the constraint 共26兲 means that Im共A兲 is perpendicular to the vector field

vˆ/cos␣ in the space H, i.e., Im共A兲 is perpendicular to vˆ/cos␣in the sense of scalar product共28兲.

Define the quadratic functional P共A兲 = def. _n3 2␭0 2

冉

⑀0 ␮0

冊

1/2

冕

0 ␣max

冕

0 2␲ 关兩A␣共␣,␤兲兩2

+兩A_␤共␣,␤兲兩2兴cos2 ␣sin␣d␣d␤, 共29兲 which is the mean power flowing through a plane z = const for fields with plane wave amplitudes A = A_␣␣ˆ + A_␤␤ˆ . Then the optimization problem is to find, for given P0⬎0, the solution of

共ⴱ兲 max

A苸H₀

F共A兲, under the constraint P共A兲 艋 P0. For any solution of problem 共*兲 the equality P共A兲= P0 holds, because otherwise A could be multiplied by the num-ber关P0/ P共A兲兴1/2⬎1 and this would increase the value of F without violating the constraint on the energy. Hence it does not matter whether we impose the equality constraint P共A兲 = P0or the inequality constraint P共A兲艋 P0on the mean flow of energy.

It is not completely obvious that problem共ⴱ兲 has a unique solution since it is posed in a linear space H0 of infinite dimension. However, there is a functional analytic theorem which states that a continuous real linear functional attains its supremum on a sphere in a Hilbert space and that the solution is unique关21兴. Since the functional F is linear, real

and continuous with respect to the norm onH, and since the feasible set of problem 共ⴱ兲 is a sphere in H0, this theorem applies to our problem. Hence the optimization problem has a unique solution. In the next section we shall compute the solution.

III. OPTIMUM PLANE WAVE AMPLITUDES Since F is a linear functional, the Fréchet derivative of F at A in the direction of B is simply F共B兲, i.e.,

␦F共A兲共B兲 = F共B兲 =n 2 ␭02

冕

0 ␣max

冕

0 2␲ 关B␣共␣,␤兲v␣ + B_␤共␣,␤兲v_␤兴cos␣sin␣d␣d␤. 共30兲 The Fréchet derivative of the quadratic functional P共A兲 is

␦P共A_␣兲共B_␣兲 =n 3 ␭0 2

冉

⑀0 ␮0

冊

1/2 Re

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲B␣共␣,␤兲* + A_␤共␣兲B_␤共␣,␤兲*兴cos2_{␣sin␣d␣d␤. 共31兲} According to the Lagrange multiplier rule for inequality con-straints 共also known as Kuhn-Tucker’s theorem兲 关22兴, there

exists a Lagrange multiplier ⌳艌0 such that, if A is the op-timum field, we have

␦F共A兲共B兲 − ⌳␦P共A兲共B兲 = 0 for all B in H0, 共32兲 and

⌳关P共A兲 − P0兴 = 0. 共33兲

In the previous section we have shown that P共A兲= P0, there-fore the last equation does not give new information. By substituting 共30兲 and 共31兲 into the Lagrange multiplier rule,

we obtain n2 ␭02

冕

0 ␣max

冕

0 2␲ 关B␣共␣,␤兲v␣+ B␤共␣,␤兲v␤兴cos␣sin␣d␣d␤ −⌳n 3 ␭0 2

冉

⑀0 ␮0

冊

1/2 Re

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲B␣共␣,␤兲* + A_␤共␣兲B_␤共␣,␤兲*兴cos2_{␣sin␣d␣d␤} = 0, for all B inH0. 共34兲

Because B satisfies Eq.共26兲, it follows that in the first

inte-gral we may replace B_␣ and B_␤ by B_␣*and B_␤*, respectively. Hence,

Re

冕

0 ␣max

冕

0 2␲

再

冋

v_␣ cos␣−⌳n

冉

⑀0 ␮0

冊

1/2 A_␣

册

B_␣* +

冋

v␤ cos␣−⌳n

冉

⑀0 ␮0

冊

1/2 A_␤

册

B_␤*

冎

cos2␣sin␣d␣d␤ = 0, for all B inH0. 共35兲 This is equivalent to

冕

0 ␣max

冕

0 2␲

再

冋

v_␣ cos␣−⌳n

冉

⑀0 ␮0

冊

1/2 Re共A_␣兲

册

Re共B_␣兲 −⌳n

冉

⑀0 ␮0

冊

1/2 Im共A_␣兲Im共B_␣兲 +

冋

v␤ cos␣−⌳n

冉

⑀0 ␮0

冊

1/2 Re共A␤兲

册

Re共B␤兲 −⌳n

冉

⑀0 ␮0

冊

1/2

Im共A_␤兲Im共B_␤兲

冎

cos2_{␣sin␣d␣d␤= 0,} 共36兲 for all B in H0, i.e., for all B for which

冕

0 ␣max

冕

0 2␲

关Im共B␣兲v␣+ Im共B␤兲v␤兴cos2␣sin␣d␣d␤= 0. 共37兲 Choose first B_␣and B_␤ real. Then Eq.共37兲 is obviously

sat-isfied and Eq. 共36兲 implies

Re共A␣兲 =_⌳n1

冉

␮0_⑀0

冊

1/2 _v ␣ cos␣, 共38兲 Re共A_␤兲 = 1 ⌳n

冉

␮0 ⑀0

冊

1/2 _v ␤ cos␣. 共39兲

By substituting this in Eq.共36兲 it follows that

冕

0 ␣max

冕

0 2␲

关Im共A␣兲Im共B␣兲

+ Im共A␤兲Im共B␤兲兴cos2␣sin␣d␣d␤= 0, 共40兲 for all B_␣, B_␤ that satisfy Eq. 共37兲. This can be stated

alter-natively by saying that if B = B_␣␣ˆ + B_␤␤ˆ is perpendicular to 共v␣/cos␣兲␣ˆ +共v␤/cos␣兲␤ˆ , then B is perpendicular to Im共A␣兲␣ˆ + Im共A␤兲␤ˆ 关perpendicular means here of course with respect to scalar product 共28兲兴. We conclude that

Im共A␣兲␣ˆ + Im共A␤兲␤ˆ is proportional to 共v␣/cos␣兲␣ˆ +共v_␤/cos␣兲␤ˆ ,

Im共A␣兲 = C v␣

cos␣, 共41兲

Im共A␤兲 = C v␤

cos␣, 共42兲

for some constant C. We shall now show that C = 0. By sub-stitution of Eqs.共41兲 and 共42兲 into Eq. 共26兲 we obtain

C

冕

0 ␣max

冕

0 2␲ 共v_␣2 +v_␤2兲cos␣sin␣d␣d␤= 0. 共43兲 If C⫽0, then we must have

v_␣=v_␤= 0, for all␣,␤with 0艋␣艋␣max, 0艋␤艋 2␲. 共44兲 Use the expressions 共21兲 and 共22兲 for v_␣andv_␤in terms of the Cartesian componentsvx,vy, andvz. It is then easily seen

that Eq. 共44兲 implies vx=vy=vz= 0. This contradicts the

as-sumption that v is a unit vector. Hence C = 0.

We thus conclude that the plane wave amplitudes of the optimum field are given by

A_␣= 1 ⌳n

冉

␮0 ⑀0

冊

1/2 _v ␣ cos␣, 共45兲 A_␤= 1 ⌳n

冉

␮0 ⑀0

冊

1/2 _v ␤ cos␣. 共46兲

The Lagrange multiplier⌳ can be determined by substituting Eqs. 共45兲 and 共46兲 into P共A兲= P0 and then using Eqs. 共21兲

and共22兲. We find P共A兲 = n 2⌳2␭02

冉

␮0 ⑀0

冊

1/2

冕

0 ␣max

冕

0 2␲ 共v_␣2 +v_␤2兲sin␣d␣d␤ 共47兲 = n 2⌳2␭02

冉

␮0 ⑀0

冊

1/2

再

vx 2

冕

0 ␣max

冕

0 2␲

共cos2_␣cos2_{␤+ sin}2_{␤兲sin␣d␣d␤} +vy 2

冕

0 ␣max

冕

0 2␲

共cos2_␣sin2_{␤+ cos}2_{␤兲sin␣d␣d␤+v}

z 2

冕

0 ␣max

冕

0 2␲ sin3␣d␣d␤ − 2vxvy

冕

0 ␣max

冕

0 2␲

sin3_{␣cos␤sin␤d␣d␤− 2v}

xvz

冕

0 ␣max

冕

0 2␲

− 2vyvz

冕

0 ␣max

冕

0 2␲

cos␣sin2_{␣sin␤d␣d␤}

冎

= ␲n 2⌳2_␭ 0 2

冉

␮0 ⑀0

冊

1/2

冋

冉

4 3− cos␣max− 1 3 cos 3_␣max

冊

_共v x 2_+v y 2_{兲 +}

冉

4 3− 2 cos␣max+ 2 3cos 3_␣max

冊

_v z 2

册

= ␲n 2⌳2␭02

冉

␮0 ⑀0

冊

1/2

冋

4 3− cos␣max− 1 3 cos

3_{␣max− sin}2_{␣maxcos␣maxv}

z

2

册

. 共48兲

It follows from P共A兲= P0and⌳艌0 that ⌳ =

冑

␲ 2 n1/2 P₀1/2␭0

冉

␮0 ⑀0

冊

1/4

冋

₄ 3 − cos␣max− 1 3 cos 3_␣max

− sin2␣maxcos␣maxvz2

册

1/2

. 共49兲

Herewith the derivation of the plane waves amplitudes of the optimum field is complete.

The maximum of the field component at the origin, i.e., of F, is F_max= F共A兲 =n 2 ␭0 2

冕

0 ␣max

冕

0 2␲ 关A␣共␣,␤兲v␣ + A_␤共␣,␤兲v_␤兴cos␣sin␣d␣d␤ = 1 ⌳ n ␭02

冉

␮0 ⑀0

冊

1/2

冕

0 ␣max

冕

0 2␲ 共v_␣2 +v_␤2兲sin␣d␣d␤ = 2⌳P0 =

冑

2␲P₀1/2n 1/2 ␭0

冉

␮0 ⑀0

冊

1/4

冋

₄ 3− cos␣max −1 3 cos

3_{␣max− sin}2_{␣maxcos␣maxv}

z

2

册

1/2 ,

共50兲

where we used Eqs.共47兲 and 共49兲.

IV. THE OPTIMUM ELECTROMAGNETIC FIELD

The electric field amplitudes of the plane waves of the optimum field are given by

A共␣,␤兲 = A_␣共␣,␤兲␣ˆ + A_␤共␣,␤兲␤ˆ = 1 ⌳n

冉

␮0 ⑀0

冊

1/2 共v␣␣ˆ + v␤␤ˆ 兲 1 cos␣= 1 ⌳n

冉

␮0 ⑀0

冊

1/2

冋

冉

cos␤␣ˆ − sin␤ cos␣␤ ˆ

冊

_v x+

冉

sin␤␣ˆ + cos␤ cos␣␤ ˆ

冊

_v y− tan␣␣ˆ vz

册

= 1 ⌳n

冉

␮0 ⑀0

冊

1/2

再

冋

冉

cos␣cos2␤+ sin

2_␤ cos␣

冊

xˆ −

sin2␣

cos␣ cos␤sin␤yˆ − sin␣cos␤zˆ

册

vx +

冋

−sin

2_␣

cos␣ cos␤sin␤xˆ +

冉

cos␣sin

2_␤+cos 2_␤

cos␣

冊

yˆ − sin␣sin␤zˆ

册

vy +

冋

− sin␣cos␤xˆ − sin␣sin␤yˆ +sin

2_␣

cos␣zˆ

册

vz

冎

. 共51兲

If we write the right-hand side of Eq.共51兲 as the product of a matrix and the vector v on the Cartesian basis xˆ, yˆ, zˆ, we obtain

A共␣,␤兲 = 1 ⌳n

冉

␮0 ⑀0

冊

1/2

冢

cos␣cos2␤+sin 2_␤ cos␣ −

sin2␣

cos␣ cos␤sin␤ − sin␣cos␤ −sin

2_␣

cos␣ cos␤sin␤ cos␣sin

2_␤+cos 2_␤

cos␣ − sin␣sin␤

− sin␣cos␤ − sin␣sin␤ sin

2_␣ cos␣

冣

冢

vx vy vz

冣

.

= r sin_␸, z = r cos␸, 共52兲 and the unit vectors ␸ˆ ,
ˆ are defined by

ˆ = cos_␸xˆ + sin␸yˆ, ␸ˆ = − sin␸xˆ + cos␸yˆ. 共53兲 Then

k · r = k0n共x sin␣cos␤+ y sin␣sin␤+ z cos␣兲 = k0n共
cos␸sin␣cos␤+
sin␸sin␣sin␤+ z cos␣兲

= k0n关
sin␣cos共␸−␤兲 + z cos␣兴 = k0n
sin␣cos共␸−␤兲 + k0nz cos␣. 共54兲 The optimum electric field in a point r with cylindrical coordinates
,␸, z is then

E共
,␸,z兲 =n 2 ␭02

冕

0 ␣max

冕

0 2␲

关A␣共␣,␤兲␣ˆ + A␤共␣,␤兲␤ˆ 兴eik·rsin␣cos␣d␣d␤

= n ⌳␭02

冉

␮0 ⑀0

冊

1/2

冕

0 ␣max sin␣eik0nz cos␣_d␣

冕

0 2␲

冢

1 − sin2␣cos2␤ − sin2␣cos␤sin␤ − cos␣sin␣cos␤ − sin2_{␣cos␤sin␤ 1 − sin}2_␣sin2_{␤ − cos␣sin␣sin␤} − cos␣sin␣cos␤ − cos␣sin␣sin␤ sin2␣

冣

⫻

冢

vx

vy

vz

冣

eik0n
sin␣ cos共␤−␸兲_d␤. 共55兲

Furthermore, the magnetic field amplitudes of the plane waves are共19兲

− A_␤共␣,␤兲␣ˆ + A_␣共␣,␤兲␤ˆ = 1 ⌳n

冉

␮0 ⑀0

冊

1/2 共− v␤␣ˆ + v␣␤ˆ 兲 1 cos␣ = 1 ⌳n

冉

␮0 ⑀0

冊

1/2

冋

冉

− sin␤ cos␣␣ˆ + cos␤␤ ˆ

冊

_v x+

冉

cos␤ cos␣␣ˆ + sin␤␤ ˆ

冊

_v y− tan␣␤ˆ vz

册

= 1 ⌳n

冉

␮0 ⑀0

冊

1/2

兵关− 2 cos␤sin␤xˆ +共cos2_{␤− sin}2_{␤兲yˆ + tan␣sin␤zˆ兴v}

x

+关共cos2␤− sin2␤兲xˆ + 2 cos␤sin␤yˆ − tan␣cos␤zˆ兴vy+共sin␤xˆ − cos␤yˆ兲tan␣vz其. 共56兲

Hence, on the Cartesian basis

− A_␤共␣,␤兲␣ˆ + A_␣共␣,␤兲␤ˆ = 1 ⌳n

冉

␮0 ⑀0

冊

1/2

冢

− 2 cos␤sin␤ cos2␤− sin2␤ tan␣sin␤ cos2_{␤− sin}2_{␤ 2 cos␤sin␤ − tan␣cos␤}

tan␣sin␤ − tan␣cos␤ 0

冣冢

vx

vy

vz

冣

. 共57兲

The optimum magnetic field is thus

H共
,␸,z兲 = n 3 ⌳␭0 2

冉

⑀0 ␮0

冊

1/2

冕

0 ␣max

冕

0 2␲

关− A␤共␣,␤兲␣ˆ + A␣共␣,␤兲␤ˆ 兴eik·rsin␣cos␣d␣d␤

= n 2 ⌳␭0 2

冕

0 ␣max sin␣eik0nz cos␣_d␣

冕

0 2␲

冢

− cos␣sin共2␤兲 cos␣cos共2␤兲 sin␣sin␤ cos␣cos共2␤兲 cos␣sin共2␤兲 − sin␣cos␤

sin␣sin␤ − sin␣cos␤ 0

冣冢

vx

vy

vz

冣

eik0n
sin␣ cos共␤−␸兲_d␤_.

共58兲 The integrals over ␤can be computed with the following formulas关23兴:

冕

0 2␲

ei␨ cos共␤−␸兲cos共m␤兲d␤= 2␲imJm共␨兲cos共m␸兲, 共59兲

冕

0 2␲

ei␨ cos共␤−␸兲sin共m␤兲d␤= 2␲imJm共␨兲sin共m␸兲, 共60兲

冕

0 2␲

ei␨ cos共␤−␸兲_cos2_{␤d␤=␲关J0共␨兲 − J2共␨兲cos共2␸兲兴, 共61兲}

冕

0 2␲

ei␨ cos共␤−␸兲sin2␤d␤=␲关J0共␨兲 + J2共␨兲cos共2␸兲兴, 共62兲

冕

0 2␲

ei␨ cos共␤−␸兲cos␤sin␤d␤= −␲J2共␨兲sin共2␸兲. 共63兲 By using the notation

gl␯,␮共
,z兲 =

冕

0 ␣max

eik0nz cos␣_cos␯␣_sin␮␣_J

l共k0n
sin␣兲d␣, 共64兲

the electric and magnetic fields can be expressed on the Cartesian basis as

E共
,␸,z兲 =␲ n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2

冦冢

g₀0,1共
,z兲 + g02,1共
,z兲 0 0 0 g₀0,1共
,z兲 + g₀2,1共
,z兲 0 0 0 2g₀0,3共
,z兲

冣

+

冢

g₂0,3共
,z兲cos共2␸兲 g₂0,3共
,z兲sin共2␸兲 − 2ig₁1,2共
,z兲cos␸ g₂0,3共
,z兲sin共2␸兲 − g20,3共
,z兲cos共2␸兲 − 2ig11,2共
,z兲sin␸ − 2ig₁1,2共
,z兲cos␸ − 2ig₁1,2共
,z兲sin␸ 0

冣冧

冢

vx vy vz

冣

, 共65兲 H共
,␸,z兲 =2␲n 2 ⌳␭0 2

冢

g₂1,1共
,z兲sin共2␸兲 − g21,1共
,z兲cos共2␸兲 ig10,2共
,z兲sin␸ − g₂1,1共
,z兲cos共2␸兲 − g₂1,1共
,z兲sin共2␸兲 − ig₁0,2共
,z兲cos␸

ig₁0,2共
,z兲sin␸ − ig₁0,2共
,z兲cos␸ 0

冣冢

vx

vy

vz

冣

. 共66兲

Alternative concise expressions are obtained when the elec-tric field, the magnetic field and the vector v are all written on the local cylindrical unit basis 兵
ˆ,␸ˆ , zˆ其 共attached to the point of observation r =

ˆ + zzˆ兲, E共
,␸,z兲 = ␲n ⌳␭02

冉

␮0 ⑀0

冊

1/2 „兵关g00,1共
,z兲 + g0 2,1_{共
,z兲 + g} 2 0,3_共
,z兲兴 ⫻共vxcos␸+vysin␸兲 − 2ig1

1,2_共
,z兲v

z其
ˆ

+关g00,1共
,z兲 + g02,1共
,z兲 − g20,3共
,z兲兴共− vxsin␸

+vycos␸兲␸ˆ +关− 2ig11,2共
,z兲共vxcos␸+vysin␸兲

+ 2g₀0,3共
,z兲vz兴zˆ… 共67兲 = ␲n ⌳␭02

冉

␮0 ⑀0

冊

1/2 „兵关g00,1共
,z兲 + g0 2,1_共
,z兲 + g₂0,3共
,z兲兴v
− 2ig1 1,2_共
,z兲v z其
ˆ + 关g0 0,1_共
,z兲 + g₀2,1共
,z兲 − g20,3共
,z兲兴v_␸␸ˆ +关− 2ig11,2共
,z兲v
+ 2g₀0,3共
,z兲vz兴zˆ…, 共68兲 H共
,␸,z兲 = −2␲n 2 ⌳␭02兵g2 1,1_{共
,z兲共− v} xsin␸+vycos␸兲
ˆ +关g₂1,1共
,z兲共vxcos␸+vysin␸兲 + ig1 0,2_共
,z兲v z兴␸ˆ + ig₁0,2共
,z兲共− vxsin␸+vycos␸兲zˆ其 共69兲 =−2␲n 2 ⌳␭0 2兵g21,1共
,z兲v␸
ˆ +关g21,1共
,z兲v
+ ig₁0,2共
,z兲vz兴␸ˆ + ig10,2共
,z兲v␸zˆ其. 共70兲 The time averaged Poynting vector of the optimum field is Re S共
,␸,z兲 =1 2Re关E共
,␸,z兲 ⫻ H共
,␸,z兲*兴 =1 2Re兵关E␸共
,␸,z兲Hz共
,␸,z兲* − Ez共
,␸,z兲H␸共
,␸,z兲*兴
ˆ +关Ez共
,␸,z兲H
共
,␸,z兲* − E
共
,␸,z兲Hz共
,␸,z兲*兴␸ˆ +关E
共
,␸,z兲H␸共
,␸,z兲*

− E_␸共
,␸,z兲H
共
,␸,z兲*兴zˆ其, 共71兲 with Re S
共
,␸,z兲 = 1 2Re关E␸共
,␸,z兲Hz共
,␸,z兲* − Ez共
,␸,z兲H_␸共
,␸,z兲*兴 =␲2 n 3 ⌳2_␭04

冉

␮0 ⑀0

冊

1/2 兵2 Im关g1 1,2_共g 2 1,1_兲*兴v
2 − Im关共g00,1+ g02,1− g0,32 兲共g10,2兲*兴v␸2 + 2 Im关g00,3共g10,2兲*兴v_z2− 2 Re关g11,2共g10,2兲* − g₀0,3共g₂1,1兲*兴v
_vz其, 共72兲 Re S_␸共
,␸,z兲 =1 2 Re关Ez共
,␸,z兲H
共
,␸,z兲* − E
共
,␸,z兲Hz共
,␸,z兲*兴 =␲2 n 3 ⌳2_␭04

冉

␮0 ⑀0

冊

1/2 兵Im关− 2g1 1,2_共g 2 1,1_兲* +共g₀0,1 + g₀2,1+ g₂0,3兲共g10,2兲*兴v
v_␸− 2 Re关g11,2共g10,2兲* + g₀0,3共g₂1,1兲*兴v_␸vz其, 共73兲 Re Sz共
,␸,z兲 = 1 2Re关E
共
,␸,z兲H␸共
,␸,z兲* − E_␸共
,␸,z兲H
共
,␸,z兲*兴 =␲2 n 3 ⌳2_␭ 0 4

冉

␮0 ⑀0

冊

1/2 兵− Re关共g00,1+ g02,1 + g₁0,3兲共g₂1,1兲*兴v
2 + Re关共g0 0,1 + g₀2,1− g₂0,3兲共g₂1,1兲*兴v_␸2 + 2 Re关g11,2共g10,2兲*兴vz2 − Im关共g00,1+ g₀2,1+ g₂0,3兲共g10,2兲* + g11,2共g21,1兲*兴v
vz其. 共74兲 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 1. Left: the normalized distribution of 兩E_z兩2_{in the z = 0 plane for the field with maximum longitudinal component when NA/n}

= 0.5 and P0= 1 共␭=␭0/n is the wavelength in the material with refractive index n兲. For comparison, the normalized electric energy

distribution共78兲 of the x-polarized focused plane wave is shown at the right for the same numerical aperture. The total flow of power in the z direction is the same for both fields.

−1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

V. OPTIMUM FIELD DISTRIBUTIONS

We will consider now in more detail subsequently fields obtained by optimizing the longitudinal, the transverse, and an intermediate component.

A. Optimum longitudinal component In this case

vˆ = zˆ. 共75兲

Then Eqs.共68兲 and 共70兲 become

E共
,␸,z兲 =2␲n ⌳␭02

冉

␮0 ⑀0

冊

1/2 关− ig1 1,2_{共
,z兲
ˆ + g} 0 0,3_{共
,z兲zˆ兴,} 共76兲 and H共
,␸,z兲 = −2␲in 2 ⌳␭0 2 g10,2共
,z兲␸ˆ . 共77兲 Since g₁1,2共0,0兲=0, the electric field in the origin is parallel to the z axis, hence it is purely longitudinal in the origin. In the z = 0 plane the functions g_ᐉ␯,␮ are real and therefore Eq. 共76兲 implies that the polarization ellipse of the electric field

in that plane has minor and major axis parallel to the
ˆ and zˆ axis关which is the major and which the minor axis depends on the relative values of g₀0,3共
,0兲 and g₁1,2共
,0兲兴. In the z = 0 plane, the phase of Ez is⫾␲ whereas the other electric

field components have phase ⫾␲/2. The magnetic field is everywhere parallel to ␸ˆ , i.e., it is azimuthal. In Fig. 1 the normalized distribution 兩Ez共x,y,0兲兩2 of the optimum field

with maximum longitudinal component for NA/n=0.5 in the z = 0 plane is compared to the normalized total electric en-ergy density, −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y/ λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 3. The squared modulus 兩E
兩2 _{of the radial component and electric energy density 兩E兩}2 _{of the electric field with maximum}

longitudinal component for NA/n=0.9. The maximum values of 兩E
兩2_{are approximately 25% of the maximum of the squared modulus兩E}

z兩2

of the longitudinal component.

−1.50 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 x/λ or y/λ |E| 2 /|E(0)| 2 Maximum longitudinal Airy spot along x−axis Airy spot along y−axis

(b) −1.50 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 x/λ or y/λ |E| 2 /|E(0)| 2 Maximum longitudinal Airy spot along x−axis Airy spot along y−axis

(a)

FIG. 4. Cross section of the rotational symmetric 兩E_z兩2 _{共solid curves兲 of the field with maximum longitudinal component and cross}

sections of兩E共x,0,0兲兩2_{共dashed兲 and 兩E共0,y,0兲兩}2_{共dotted兲 of the Airy spot for the x-polarized focused plane wave, for NA/n=0.5 共left兲 and}

兩Ex共x,y,0兲兩2+兩Ey共x,y,0兲兩2+兩Ez共x,y,0兲兩2, 共78兲

in the focal plane of the focused x-polarized plane wave. The coordinates x, y are expressed in units of the wavelength ␭ =␭0/n in the material with refractive index n. The formulas for the electric field of a focused plane wave are given in the Appendix 关see Eqs. 共A14兲–共A16兲兴. The power flow in the z

direction of the optimum longitudinal field and the focused plane wave are the same. The distribution of the optimum 兩Ez兩2 is rotationally symmetric while that of the electric

en-ergy density, Eq.共78兲, of the focused linearly polarized plane

wave is elliptical, with the short axis parallel to the y direc-tion共i.e., perpendicular to the direction of polarization of the focused plane wave兲. In Fig.2the distributions are compared for NA/n=0.9. In this case the short axis of the elliptic dis-tribution is considerably shorter than the long axis. In Fig.3

the squared modulus 兩E
兩2 and 兩E兩2 are shown. Due to the

broad doughnut shaped distribution of the radial component, 兩E兩2_{is broader than the Airy spot.}

Cross sections along the short and long axes are shown in Fig.4 for both NA/n=0.5 and NA/n=0.9. To compare the

shapes the maxima of all cross sections are rescaled to 1. It is seen that the longitudinal component has smaller full width at half maximum 共FWHM兲, but also higher secondary maxima.

The FWHM of the optimum longitudinal component is for NA/n=1 almost identical to that of the longitudinal com-ponent in 关18兴, obtained by focusing a radially polarized

beam using a ring mask function 共with radius 90% of the total pupil兲. However, the side lobes are higher at the cost of the central maximum compared to our longitudinal compo-nent. This is of course not surprising because the longitudinal component in 关18兴 was not optimized for a high maximum

on the optical axis.

In Fig.5the FWHM of兩Ez共x,y,0兲兩2of the optimum field

is compared to the FWHM in the x and y directions of the electric energy density兩E共x,y,z=0兲兩2of the focal spot of the

x-polarized plane wave. At the left, the FWHM is shown as a function of the numerical aperture NA/n, at the right as a function of n/NA, both in units of wavelength. It is seen that the FWHM of the longitudinal component is smaller than the FWHM in both the x and y directions of the focused spot. The FWHM as a function of n/NA is almost linear, but the slope is smaller for the longitudinal component. This means that the spot size of the optimum longitudinal 兩Ez兩2 is

rela-tively smaller than the Airy spot when the numerical aperture is larger. Nevertheless, for all values of the numerical aper-ture, the longitudinal spot is narrower than the Airy spot. As shown in Fig. 6, the maximum amplitude兩Ez共0兲兩 of the

op-timum longitudinal component is for most values of the nu-merical aperture smaller than the maximum amplitude 兩Ex共0兲兩 of the focused x-polarized plane wave. But for

NA/n⬎0.65 this ratio is already more than 0.5 and for NA/n⬎0.994 the maximum longitudinal component is even

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 NA/n FWHM [units of λ ] FWHM of optimized |E_z|2

FWHM of |E|2in x−dir. of Airy spot FWHM of |E|2in y−dir. of Airy spot

(a) 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 n/NA FWHM [units of λ ] FWHM of optimized |E_z|2

FWHM of |E|2in x−dir. of Airy spot FWHM of |E|2in y−dir. of Airy spot

(b)

FIG. 5. FWHM of 兩Ez共x,y,0兲兩2 of the field with maximum longitudinal component and FWHM of 兩E共x,y,0兲兩2 along the x and y

directions of the focused x-polarized plane wave. At the left the FWHM is shown as function of NA/n, at the right as function of n/NA in units of␭=␭0/n. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 NA/n |E max z (0)|/|E Ai ry x (0)|

FIG. 6. Ratio of maximum longitudinal amplitude兩E_z兩 and the amplitude兩Ex兩 of the focused x-polarized plane wave for the same

larger than the maximum amplitude 兩Ex共0兲兩 of the focused

x-polarized plane wave.

Along the optical axis we have
= 0 and

g₀0,3共0,z兲 =

冕

0 ␣max eik0nz cos␣_sin3␣_d␣ = i k₀nz

冉

1 + 2 k₀2n2z2

冊

关e ik₀nz cos␣max_{− e}ik0nz兴 + 2cos␣maxe ik₀nz cos␣max_{− e}ik0nz k₀2n2z2 − icos

2_␣maxeik₀nz cos␣max_{− e}ik0nz

k0nz

. 共79兲

Because J₁共0兲=0,

g₁1,2共0,z兲 = g₁1,2共0,z兲 = 0. 共80兲 It thus follows from Eq. 共76兲 that along the optical axis the

electric field is parallel to the z axis and that the modulus of the field is symmetric with respect to the focal plane. Fur-thermore, 兩Ez共x = 0,y = 0,z兲兩2 兩Ez共0,0,0兲兩2 = 兩g0 0,3_共0,z兲兩2 兩g00,3_共0,0兲兩2, 共81兲 where g₀0,3共0,0兲 = 2 3 − cos␣max+ 1 3 cos 3_{␣max. 共82兲} We define the focal depth of the optimum longitudinal com-ponent as the distance ⌬z to the focal plane for which the ratio

兩g00,3共0,⌬z兲兩2 兩g00,3共0,0兲兩2

= 0.8. 共83兲

For the focused linearly polarized plane wave, the electric energy density on the optical axis is given by Eq. 共A40兲,

兩E共x = 0,y = 0,z兲兩2_=兩E

x共x = 0,y = 0,z兲兩2+兩Ey共x = 0,y = 0,z兲兩2

+兩Ez共x = 0,y = 0,z兲兩2

=␲ 2_n2_f2

␭0

2 兩g01/2,1共0,z兲 + g03/2,1共0,z兲兩2. 共84兲 In the origin we have共A41兲

兩E共0,0,0兲兩2₌␲ 2_n2_f2 ␭0 2

冋

2 3共1 − cos 3/2_␣max兲 +2 5共1 − cos 5/2_␣max兲

册

2_{. 共85兲} The energy density on the optical axis is again symmetric around the focal plane and the focal depth is the distance⌬zA

such that 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 NA/n focal depth/ λ

∆ z optimum longitudinal component ∆ z_Aof focused linearly polarized plane wave 0.5n2/NA2

FIG. 7. Focal depth in unit of␭=␭0/n as function of NA/n for

the optimum longitudinal component and for the energy density of the focused linear polarized plane wave. The approximative scalar paraxial focal depth is also shown in units of␭.

−1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5

z/

λ

x/

λ

(b) −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5

z/

λ

x/

λ

(a)

FIG. 8. Poynting vector S = S

ˆ + Szzˆ for the fields with maximum longitudinal component, in the共z,x兲 plane 共i.e., the␸=␲/2 plane兲 for

兩E共x = 0,y = 0,⌬zA兲兩2

兩E共0,0,0兲兩2 = 0.8. 共86兲

Both⌬z and ⌬zAare shown as functions of NA/n in Fig.7.

The approximative focal depth 0.5␭0n/NA2_{, which is valid} in the scalar paraxial theory, is also shown. It is seen that for NA/n⬍0.4 the focal depths of the focused linearly polarized plane wave calculated in the vectorial and the scalar paraxial theory are almost identical. The focal depth of the optimized longitudinal spot is for NA/n⬍0.8 larger than that of the Airy spot, while for NA/n⬎0.8 it is smaller.

Equations共72兲–共74兲 imply S共
,␸,z兲 =2␲ 2_n3 ⌳2_␭ 0 4

冉

␮0 ⑀0

冊

1/2 „Im兵g00,3_{共
,z兲关g1}0,2_{共
,z兲兴*其
ˆ} + Re兵g11,2共
,z兲关g10,2共
,z兲兴*其zˆ…. 共87兲 The ␸ component of the Poynting vector thus vanishes and the Poynting vector is independent of the angle␸. In the共z = 0兲 plane the energy flows in the z direction. In Fig. 8 the time-averaged Poynting vector of the field with maximum longitudinal component is shown in the 共z,
兲 plane for the cases NA/n=0.5 and NA/n=0.9. The ␸ component of the Poynting vector vanishes and the other components S
and Sz

are independent of rotation angle ␸. The Poynting vector vanishes on the z axis.

B. Optimum transverse component

We choose the transverse component parallel to the x axis,

vˆ = xˆ, 共88兲

i.e., the x component of the electric field in the origin is optimized. On the local basis共attached to the point of obser-vation兲 of cylindrical coordinates
, ␸, z we have v

= cos␸, v_␸= −sin␸,vz= 0. Hence Eqs. 共65兲, 共68兲, 共66兲, and

共70兲 imply that the optimum electromagnetic field becomes

E共
,␸,z兲 = ␲n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2 兵关g00,1_{共
,z兲 + g0}2,1_共
,z兲 + g20,3共
,z兲cos共2␸兲兴xˆ + g20,3共
,z兲sin共2␸兲yˆ − 2ig₁0,2共
,z兲cos␸zˆ其 共89兲 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 9. The normalized distribution of兩E_x兩2_{共left兲 and 兩E}

y兩2共right兲 in the z=0 plane for the field with optimized x component when

NA/n=0.9. −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 10. Normalized distribution of兩E_z兩2_{and of the electric energy density兩E兩}2_{for the field with maximum E}

= ␲n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2 兵关g00,1_{共
,z兲 + g0}2,1_共
,z兲 + g₂0,3共
,z兲兴cos␸
ˆ −关g00,1_{共
,z兲 + g0}2,1_共
,z兲 − g₂0,3共
,z兲兴sin␸␸ˆ − 2ig₁1,2_{共
,z兲cos␸zˆ其, 共90兲} H共
,␸,z兲 = −2␲n

2

⌳␭0

2关g21,1共
,z兲sin 2␸xˆ − g21,1共
,z兲cos 2␸yˆ − ig₁0,2共
,z兲sin␸zˆ兴 共91兲 =2␲n 2 ⌳␭0 2关g2 1,1_{共
,z兲sin␸
ˆ − g} 2 1,1_{共
,z兲cos␸␸ˆ} + ig₁0,2共
,z兲sin␸zˆ兴. 共92兲 The squared amplitudes of the electric field components and

the electric energy density in the z = 0 plane are shown in Figs.9and10when NA/n=0.9. These field components are very similar to the components of the focused x polarized plane wave for the same NA/n. In Fig. 11 cross sections along the x and y axes of 兩Ex兩2 for the maximum transverse

field are the focused linearly polarized plane wave pattern are compared for NA/n=0.9. The maxima of all functions are normalized to 1. Along the y axis, the Ex spot of the

optimized transverse field is slightly narrower than for the Airy spot, but has also higher secondary maxima. It can be shown that, just as for the focused x polarized plane wave, Ex

of the optimized field is widest along the x axis. In Fig. 12

the FWHM as function of NA/n of 兩Ex兩2 of the optimized

field is compared to the FWHM of 兩Ex兩2 of the x-polarized

plane wave. The field distributions have elliptical shape. The FWHM is therefore defined with respect to the radial distri-bution obtained by averaging兩Ex共
,␸, z = 0兲兩2over the angle

0⬍␸⬍2␲, as explained in Eq.共A26兲 in the Appendix. Note

that, in contrast to Fig. 5, where the optimum Ezcomponent

was compared with the intensity of the Airy spot, we here compare the squared moduli of the x components of the elec-tric fields. It is seen that the FWHM of the optimized and the focused plane wave are almost identical.

The optimized electric field at the focal point is pointing in the x direction and is given by

Ex共0,0,0兲 =

冑

2␲P0

冑

n ␭0

冉

␮0 ⑀0

冊

1/4

冉

4 3− cos␣max −1 3 cos 3_␣max

冊

1/2 . 共93兲

Formulas for the field distributions of the focused x polarized plane wave are derived in the Appendix. For a power given by Eq.共A30兲, the x component of the electric field in focus is

given by Eq.共A41兲. Hence if the power is P0we have for the focused x-polarized plane wave

−1.50 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 x/λ |E x | 2 /|E x (0)| 2 Optimized E x

x−polarized Airy spot

−1.50 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 y/λ |E x | 2 /|E x (0)| 2 Optimized E x

x−polarized Airy spot

(b) (a)

FIG. 11. Cross sections of 兩Ex兩2 along the x 共left兲 and y axis 共right兲 for the field with maximum x component and for the focused

x-polarized plane wave with the same power. The numerical aperture is NA/n=0.9. The maxima of all cross sections are rescaled to 1.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 NA/n FWHM [unit of λ ] FWHM of optimized |E_x|2 FWHM of |E_x|2of Airy spot

FIG. 12. The FWHM of兩E_x兩2_{along the x axis of the optimized}

transverse field and the FWHM of the averaged 兩Ex兩2 of the

Ex plw_{共0,0,0兲 =}

冑

2␲P₀

冑

n ␭0

冉

␮0 ⑀0

冊

1/4 ₁ sin␣max ⫻

冉

16 15− 2 3cos 3/2_␣max−2 5 cos 5/2_␣max

冊

_. 共94兲 The ratio of Eqs. 共93兲 and 共94兲,

sin␣max

冉

4 3− cos␣max− 1 3cos 3_␣max

冊

1/2 16 15− 2 3cos 3/2_␣max−2 5 cos 5/2_␣max , 共95兲

is shown in Fig. 13 as function of NA/n=sin␣max. For ␣max→0 the ratio becomes 1, while for ␣max→␲/2 it be-comes 15/共8

冑

3兲=1.0825. Hence, the optimized field can

have more than 8% higher amplitude Exthan the x-polarized

Airy spot of the same power.

C. Intermediate case

Without restricting the generality, we may assume that the vector v is in the共x,z兲 plane, i.e., vy= 0. It is then convenient

to write the optimum focused electric and magnetic field on the orthonormal basis with unit vectors vˆ, yˆ and vˆ⫻yˆ. We have

vˆ =vxxˆ +vzzˆ, vˆ⫻ yˆ = − vzxˆ +vxzˆ, 共96兲

and therefore

xˆ =vxvˆ −vz共vˆ ⫻ yˆ兲, zˆ = vzvˆ +vx共vˆ ⫻ yˆ兲. 共97兲

Hence, Eqs.共65兲 and 共66兲 become

E共
,␸,z兲 =␲ n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2 „兵关g00,1+ g₀2,1+ g₂0,3cos共2␸兲兴vx − 2ig₁1,2cos␸vz其xˆ + 关g2 0,3_{sin共2␸兲v} x

− 2ig₁1,2sin␸vz兴yˆ + 关− 2ig11,2cos␸vx+ 2g00,3vz兴zˆ…

=␲ n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2 „兵关g00,1_{+ g} 0 2,1_{+ g} 2 0,3_{cos共2␸兲兴v} x 2 − 4ig₁1,2cos␸vxvz+ 2g00,3vz 2_{其vˆ + 关g2}0,3_{sin共2␸兲v} x

− 2ig₁1,2sin␸vz兴yˆ + 兵− 2ig11,2cos␸vx2

−关g₀0,1+ g₀2,1− 2g₀0,3+ g₂0,3cos共2␸兲兴vxvz + 2ig₁1,2cos␸vz 2_{其vˆ ⫻ yˆ…, 共98兲} and H共
,␸,z兲 =2␲n 2 ⌳␭0 2兵关g2 1,1_{sin共2␸兲v} x+ ig10,2sin␸vz兴xˆ

−关g21,1cos共2␸兲vx+ ig10,2cos␸vz兴yˆ

+ ig₁0,2sin␸vxzˆ其 =2␲n 2 ⌳␭0 2兵关g2 1,1_{sin共2␸兲v} x 2_{+ 2ig} 1 0,2_sin␸v xvz兴vˆ 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.02 1.04 1.06 1.08 1.1 NA/n |E max x (0)|/|E Ai ry x (0)|

FIG. 13. Ratio of the amplitudes of E_x共0,0,0兲 at the focal point of the field with maximum 兩Ex共0,0,0兲兩 and of the focused

x-polarized plane wave with the same power. The ratio is for NA = n maximum with value 15/共8

冑

3兲.

−1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 14. The normalized distribution of兩Ev兩2共left兲 and 兩Ey兩2共right兲 in the z=0 plane for the field with optimum field with␽v= 30° when

−关g21,1cos共2␸兲vx+ ig10,1cos␸vz兴yˆ

+关ig₁0,2sin␸vx2− g21,1sin共2␸兲vxvz

− ig₁0,2sin␸vz

2_{兴vˆ ⫻ yˆ其. 共99兲}

Because g_ᐉ␯,␮共0,z兲=0 when ᐉ艌1, it follows that on the op-tical axis,
= 0, we have

E共0,␸,z兲 =␲ n ⌳␭0 2

冉

␮0 ⑀0

冊

1/2 „兵关g00,1_{共0,z兲 + g0}2,1_{共0,z兲兴v} x 2 + 2g00,3共0,z兲vz 2_{其vˆ − 关g0}0,1_{共0,z兲 + g0}2,1_共0,z兲 − 2g₀0,3共0,z兲兴vxvzvˆ⫻ yˆ…, 共100兲

while the magnetic field vanishes there. It follows in particu-lar that the electric field in the origin
= z = 0 is parallel to vˆ only when v is perpendicular to the optical axis共vz= 0,

trans-verse case兲 or parallel to it 共vx= 0 longitudinal case兲. In all

other cases the projection of the electric field at the origin on the plane perpendicular to vˆ is nonzero.

Thev component of the electric field is

E共
,␸,z兲 · vˆ =␲ n ⌳␭02

冉

␮0 ⑀0

冊

1/2 兵关g0 0,1 + g₀2,1+ g₂0,3cos共2␸兲兴vx 2 − 4ig₁1,2cos␸vxvz+ 2g0 0,3 vz 2_{其. 共101兲}

In particular, since the functions g_ᐉ␯,␮are all real for z = 0, we find for the squared modulus of the v component in the z = 0 plane 兩E共
,␸,0兲 · vˆ兩2_=␲2 n 2 ⌳2_␭ 0 4

冉

␮0 ⑀0

冊

„关共g00,1+ g0 2,1 − g₂0,3兲vx 2 + 2g₀0,3vz 2_兴2_{+ 4兵共g} 0 0,1 + g₀2,1− g₂0,3兲g₂0,3vx 4 + 2关g0 0,3 g₂0,3+ 2共g₁1,2兲2兴vx 2 vz 2_其cos2_␸ + 4共g20,3兲2_v x 4_cos4_{␸…, 共102兲}

where all g_l␯,␮are evaluated at
and z = 0. If␽_v is the angle between the vector vˆ and the positive z axis, we have

vx= sin␽v, vz= cos␽v. 共103兲

In Figs.14and15,兩E·v兩2_,兩E

y兩2,兩E·共v⫻yˆ兲兩2and the electric

energy density 兩E兩2_{are shown in the z = 0 plane for the} opti-mum field with ␽v= 30° and NA/n=0.9. Figures16and17

−1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 15. Normalized distribution of 兩E·共vˆ⫻yˆ兲兩2 _{and of the electric energy density兩E兩}2 _{of the field with maximumv component for}

␽v= 30° and NA/n=0.9. −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 16. The normalized distribution of兩Ev兩2共left兲 and 兩Ey兩2共right兲 in the z=0 plane for the field with optimum field with␽v= 60° when

correspond to␽v= 60°.

In TableI the maxima of the indicated field components in the focal plane are listed for the optimum fields corre-sponding to NA/n=0.9 for several choices of␽v. In all cases

the total flow of power in the pupil of the lens is 1 W. Equation共102兲 is a quadratic function of cos2_{␸and since} the coefficient of cos4_{␸is non-negative, it follows that, for} every
, the maximum is attained when cos2␸= 1, i.e., when ␸= 0 and when␸=␲. To obtain a useful measure of the spot size, we average Eq. 共102兲 over 0⬍␸⬍2␲,

1 2␲

冕

₀ 2␲ 兩E共
,␸,0兲 · vˆ兩2_d␸=␲2 n 2 ⌳2_␭ 0 4

冉

␮0 ⑀0

冊

(关共g00,1+ g02,1 − g₂0,3兲vx 2 + 2g₀0,3vz 2_兴2_{+ 2兵共g} 0 0,1 + g₀2,1− g₂0,3兲g20,3vx4+ 2关g00,3g20,3 + 2共g1 1,2_兲2_兴v x 2 vz 2_{其 +}3 2共g2 0,3_兲2_v x 4 ), 共104兲 where we used 1 2␲

冕

0 2␲ cos2␸d␸=1 2, 1 2␲

冕

0 2␲ cos4␸d␸=3 8. 共105兲 The FWHM in the z = 0 plane is then defined as the value 2
0such that 1 2␲

冕

₀ 2␲ 兩E共
0,␸,0兲 · vˆ兩2d␸=1 2兩E共0,0,0兲 · vˆ兩 2_{. 共106兲} In Figs. 18 and 19 the FWHM of 兩E_v兩 and the maximum 兩Ev共0兲兩 are shown as a function of␽v for several values of

the numerical aperture and for P0= 1 W. In Fig. 19the am-plitude of 兩Ex共0兲兩 of the focused x-polarized plane wave is

shown for the same power are indicated by the dots. In con-trast to the FWHM, the maximum E_v共0兲 is a monotonically increasing function of␽v.

VI. THE OPTIMUM FIELD IN THE LENS PUPIL The optimum fields in the z = 0 plane can be obtained by focusing appropriate pupil distributions. We derive these pu-pil distributions in this section, using the vector diffraction theory of Ignatowksy and Richards and Wolf. This theory is based on 共1兲 Debye’s approximation which expresses the plane wave amplitudes in image space to the field vectors in

TABLE I. Maxima of the optimum electric field components 共not of their squares兲 in the focal plane when NA/n=0.9 and for several choices of the angle␽v. The total power in the lens pupil is

in all cases 1 W. Note that the maximum of兩Ey兩 and 兩E·共vˆ⫻yˆ兲兩 are

not attained at the focal point.

␽v共degrees兲 兩Ev兩 共V/m兲 兩Ey兩 共V/m兲 兩E·共vˆ⫻yˆ兲兩 共V/m兲

90° 45.28 6.03 13.67 60° 42.93 7.33 13.81 30° 37.76 14.17 14.04 0° 34.87 17.72 17.72 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x/λ y /λ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (b) (a)

FIG. 17. Normalized distribution of 兩E·共vˆ⫻yˆ兲兩2 _{and of the electric energy density兩E兩}2 _{of the field with maximumv component for}

␽v= 60° and NA/n=0.9. 0 20 40 60 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ϑ_v[degrees] FWHM [unit o fλ ] NA/n=0.45 NA/n=0.6 NA/n=0.75 NA/n=0.9

FIG. 18. FWHM expressed in units of␭=␭0/n of the optimum