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High-aperture focusing systems:Control of light concentration infocal region by pupil filteringPeep Adamson aa Institute of Physics, University of Tartu , Riia 142, 51014,Tartu, Estonia E-mail:Published online: 03 Jul 2009.

To cite this article: Peep Adamson (2004) High-aperture focusing systems: Control of lightconcentration in focal region by pupil filtering, Journal of Modern Optics, 51:1, 65-74

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JOURNAL OF MODERN OPTICS, 10 JANUARY 2004 VOL. 51, NO. 1, 65-74

+ Taylor & Francis 0 Tlylor LFIMclS Group

High-aperture focusing systems: control of light concentration in focal region by pupil filtering

PEEP ADAMSON

Institute of Physics, University of Tartu, Riia 142, 51014 Tartu, Estonia e-mail: peep@fi.tartu.ee

(Received 13 November 2002; revised 24 February 2003)

Abstract. The influence of various pupil filters on the field distribution in the neighbourhood of focus within the context of the vector diffraction theory is reviewed. The principal issue is with the focal depth via the calculations of integral parameters: encircled electric energy and encircled longitudinal electro- magnetic energy flow along the optical axis. It is shown that in the case of filters which change the phase, the integral parameters for small radii attain their own maximum values not in the geometrical focal plane, but in the plane which is displaced from the latter by a certain distance. The encircled electric energy and the encircled longitudinal flux have maximum values in the focal plane only for large radii. For the optimum enhancement of focal depth the most suitable filter, from those under consideration, is the truncated Bessel filter.

1. Introduction It is common knowledge that the structure of the focal region can be modified

by pupil plane filtering [l-81, which has been studied in the framework of scalar diffraction theory (paraxial optics) for some time. At present, however, focusing systems of high numerical aperture are increasingly important, for example in microscopy [9], optical information storage [lo], and lithographic projection lenses [ll]. Much of the initial theory of the diffraction of high-aperture focusing by aplanatic aberration-free systems was laid down by Ignatowsky [12], Wolf [13], Richards and Wolf [14], and later expanded by other researches [15-171. Recently, the effects of aberrations in high-aperture systems have also been investigated and a considerable variation with respect to the paraxial approximation has been demonstrated [18, 191.

T h e purpose of this paper is to study the effect of various pupil filters on the field distribution in the neighbourhood of focus, particularly on the focal depth, in the case of high-aperture focusing. Our principal concern will be with calculations of encircled electric energy and encircled longitudinal electromagnetic energy flow along the optical axis, which, for high-aperture systems, have not yet been considered.

2. Basic formulas We shall examine the properties of pupil filtering within the framework of the

vector diffraction theory [13, 14, 20-221. We shall also assume that the Fresnel

Journal of Modern Optics ISSN 09504340 print/ISSN 1362-3044 online 0 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals

DOI: 10.1080/095003403 10001 17907

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66 P. Adamson

number of the optical system NF = a 2 / l f (where a is the radius of the exit pupil, 1 is the vacuum (image space) wavelength, and f is the focal distance) is significantly greater than unity (as a rule, NF>> 1 in typical large-aperture focusing systems). The classical Debye approximation is then sufficiently precise [23], but allowance for the vector character of the field leads to a vector diffraction integral.

According to Richards and Wolf [14], when a monochromatic wave, linearly polarized in the x direction, is focused by a rotationally symmetric aberration-free aplanatic (axially stigmatic and obeying the sine condition) optical system, the electric and m_agnetic vectors in the focal region are given i,n cylindrical cooLdinates r , 4, z by E(r, rp, z, t) = Re(;(r, rp, z) exp(-iwt)] H(r , rp, z, t) = Re(h(r, rp, z) exp(-iwt)], respectively, where Re denotes the real part, and

and

10 = 1; t(0)(cos 0)1/2 sin 0(l + cos0)J0(2n(r/l) sin 0) exp(i2n(z/l) cos 0) do,

e m

Il = I0 t(B)(cos 0)1/2( sin 6~)~J,(2n(r/1) sin 0) exp(i2n(z/l) cos 0) do,

I2 = 6 t(B)(cos 0)1/2 sin 0( 1 - cos 0)J2(2n(r/l) sin 0) exp(i2n(z/l) cos 0) do,

(3)

in which Jo, J I and J z are the Bessel functions of the first kind, lo is the amplitude of the electric vector at the optical axis (incident beam center ( r = 0)) in the object space, 0, is half of the aperture angle in the image space; t(0) defines the relative angular dependence of the amplitude on the exit pupil (exit sphere) formed by the pupil amplitude-phase filter of the system. Instead of the two true spatial coordinates r and z, we shall henceforth use the traditional normalized quantities: so-called optical coordinates z, = 2n(r/1) sin 0, and u = 2n(z/l) sin2 0,.

Thus, for optical systems of high numerical aperture the focal field in the Debye approximation can be expressed in terms of three diffraction integrals Io, 11, and 1 2 . With these three integrals the relative distributions of the time-averaged longitudinal electromagnetic energy flux (axial component of the Poynting vector) p and the time-averaged electric energy density w are evaluated by means of the equations

where P, is+ the z-component of time-averaged Poynting vector (P) = (c/8n)Re(e' x (h)*) and (We) = (1/16n)(;. (v) is the time-averaged electric energy density (the asterisk denotes the complex conjugate). As shown in the paper by

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High-aperture focusing systems 67

Richards and Wolf [14], for systems with small angular aperture, diffraction integrals II and I2 are of lower order in 0, than Io, so that these integrals may be neglected in comparison with 10. Consequently, in the paraxial case, w z p is the light intensity in relative units.

T o control light concentration in the focal region a variety of exit pupil distribution can be produced when coating the exit aperture with absorbing and refracting materials suitably varies the amplitude and phase transmissions of the exit aperture of the focusing system. Here we shall discuss some typical distribu- tions [l, 2, 5, 7, 241. First, absorbing amplitude filters without phase changing which establish the angular dependences of t(0)

t(0) = 1 - sinO/ sine,,,,

t(0) = exp[- sin2 0/(2 sin2 &)I, t(e) = [ 2 ~ 1 (B sin O ) / P sin el2,

where 0~ is the effective half-aperture for the Gaussian distribution and B is a certain real constant. These functions actually show how many times the ampli- tude on the exit pupil at a distance h = f s ine from optical axis differs from the amplitude on the optical axis. For example, in the case of distribution (6) this dependence on h is in the form of a linear function, but function (7) describes the well-known Gaussian distribution of the amplitude over the aperture (the latter case can also be treated as the focusing of a Gaussian beam with a high-aperture aplanatic focusing system without pupil filtering). The filters (6)-(8) anywhere on the exit pupil do not reverse the amplitude. But a subject of particular interest is such pupil filter, which contain phase-shifting element [3] so that the amplitude is of opposite sign on some part of the exit aperture. In this paper we consider the following real-valued filter functions (contain 0 and 180" phase values only):

where a1,2,3 = f l , a 2 # a3,81 I 62 < Om, 1 < y < 2,a >> 2.4.

we introduce the integral parameters In order to characterize the degree of spatial localization of the focused light,

Kp = p(r , z ) r dr p(r , z )r dr, J-: / r K , = 1; 1; w(r, ip, z)r dr dgo/ I'" w(r, go, z)r dr dgo (13)

0 0

which describe quantitatively the transverse concentration of the radiation in the focal region. Note that calculation of the factor of encircled electric energy and of the factor of encircled longitudinal energy flow is not straightforward, and the integrated quantities have been considered previously only in some instances within the scalar approximation [25-271.

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68 P. Adamson

3. Results and discussion First, it should be pointed out that high-aperture focusing of linearly polarized

light not only takes out the rotational symmetry of electric-energy density w in the geometric-optical focal plane ( z = 0), but also appreciable dependence of w on rp (figure l(a), w(rp = 0") # w(q = 90")) and noticeable difference between quantities p and w clearly appear at points removed sufficiently far from geometrical focal plane (distribution of p has the property of rotation symmetry at any aperture angles [14]). The action of different amplitude filters (6)-(8) principally has the same character as in the paraxial domain. These filters suppressed the sidelobes and increased the width of the central maximum both in transverse and longi- tudinal (axial) directions: the stronger the suppression of sidelobes, the wider the central maximum (figure l(6)). In other words, by way of amplitude filters the electromagnetic energy of sidelobes can be transferred to the central maximum.

Incorporating phase-shifting masks into the focusing system permits us to generate the pupil filters (9)-( 11) in which the effect is reversed in comparison with the purely absorption filters of Equations (6) to (8), i.e. it decreases the width of the

20 40 60 ao u p=w

lo1

20 40 60 ao u 1u

p=w

20 40 60 80 u p=w

20 40 60 ao u

Figure 1 . Dependences of the quantities p and w on u for an aplanatic focusing system with 8,=70" (a) at v = 2 for t(0)El (p ( - - - - - ) # w for #=90° (-) and 0" (... . . . ) and (b, c, d) at v = 0 for different pupil filters: (b) filter (7) {-, the figures are values of OG}, filter (8) {-.-.-.- , the figures are values of b}, and filter (6) {- - - - -}; (c) filter (10) { - . - . - a - y = 1.3 and - y = 1 . 8 } and filter (9) { - - - - - a1 = -1,q = a3 = +1,& = 02 = 30"); (d) filter (11) for a = 5.52 ( l ) , 14.931 (2), 30.635 (3), 62.048 (4), 200 (5), 2000 (6).

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High-aperture focusing systems 69

central peak and enhances the sidelobes, in effect, it distributes the electromagnetic energy of the central peak into the sidelobes. For example, a filter (10) with little adverse amplitude (quantity y approaches the unity) affects the diffraction pattern exactly in such a manner in the focal plane. Such filters are known as resolution- improving filters [l]. We have found that these filters have a similar effect on the distribution of radiation along the axial direction (figure l ( c ) , chain curve for y = 1.3 ). However, if y approaches 2 where the contribution of adverse amplitude is of crucial importance, then a more powerful modification of the field distribution appears. Peaks of p and w do not appear at the geometrical focus, but are displaced from the latter by a certain distance along the optical axis. The physical reason for this lies in the fact that component plane waves in the Debye superposition integral have a different phase at the geometrical focus. It is known (can be readily seen from formulas (1)-(3)) that in the Debye approximation the diffraction field in the focal region always possesses specular reflection symmetry relative to the geometric-optical focal plane. Consequently, there also exists a second maximum, which is moved in the pupil direction. The observed behaviour that diffraction focus may not coincide with geometrical focus in the case Np>> 1 is recently discovered in the theory of aberrations [17-191. But it is worth stressing that in the instance NF>> 1 the diffraction focus is always located at the point of geometrical focus when phase shift is absent (because all the constituents in the superposition of plane waves should be equal in phase).

On the other hand, it has been known for a long time that with the constraint N F s 1 , when the Debye approximation is not valid, the asymmetric focal shift (only in the direction of the exit pupil) also occurs in the absence of pupil filtering [28, 291. However, in aberration-free focusing systems with large Fresnel numbers focal shift takes place in the case when the phase-shifting mask is entered into the focusing system. In this sense, particularly appreciable is the pupil filter ( l l ) , i.e. the truncated Bessel filter. As shown (figure l(6)) with increasing a! the maximum of the field is significantly removed away from the geometrical focus as a result of the phase modification, which is precisely the physical reason for the enhancement of focal depth.

The dependence of the quantities p and w on the coordinates yields a satisfactory understanding of the field extrema, but the efficiency of the spatial localization of the radiation in the focal region cannot be inferred from them. This is due to the fact that the numerous sidelobes may carry a significant part of the total power of focused beams (only a very small part of the latter can be concentrated in the main maximum). The closer examination of the efficiency of focusing and diffraction broadening of radiation is attainable with the integral quantities Kp and K,. Note that since w(r, q, z) is not rotationally symmetric, it is also instructive to examine the integral quantity K, which depends on more than one parameter; in other words, the cross-section of the ‘optical tube’ is not circular but, for example, elliptical or rectangular. T o ease the integration procedure we shall restrict consideration, to the simple type of K , defined by equation (1 3) .

As shown in the paraxial case [30, 311 the role of apodization for the localization of electromagnetic energy in the focal plane is essential if the value of z, is more than one optical unit. This is also the case for high-aperture focusing. Amplitude filters (6)-(8) elevate the degree of transverse concentration of total axial energy flow in the focal region. But calculations show that the influence of apodization on

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70 P. Adamson

1.0

0.8

0.6

0.4

0.2

(b)

10 20 30 40 v

K,& 1.0

0.8

0.6

0.4

0.2

(c)

10 20 30 40 v

KF loo

10-1

(d) lo-'

n-4

10 20 30 40 v

U

10 20 30 40 v I U

Figure 2. Dependences of the quantities K p (- - - - -) and K , (the other curves) on v for ern = 70" at fixed values of u (the figures at the curves) for (a ) t (0) = 1 and (b , c, d) different pupil filters: ( b ) filter (7) {- and - - - - - forec = 20°}, filter (8) I-.-.-.- for = 6 } , and filter ( 6 ) {......}; (c) filter (10) {- and - - - - - for y = 1.4}, filter (9) {-.-.-.- for a l = - l , a 2 = a 3 = + 1 , 0 1 =02=30° and ...... for al = - 1 , q = + l , q = -1,01 = 30°,02 =SO"}; (d) filter (11) {- a = 5.52, -.-.-.- for 30.635, and - - - - - . . . . . . for 200).

the diffraction pattern in these transverse planes, which are located away from the geometrical focal plane, for instance, in planes u = 32 or u = 64 optical units, is more powerful than in the Gaussian focal plane. In addition, as we can see in figure 2(b) in the Gaussian focal plane the pupil filter (8) is more ineffectual for focusing than the filters (6) and (7), but in the transverse planes u = 32 and u = 64 the situation is quite the reverse. Note that filters (6)-(8) reduce the difference between the two quantities Kp and K,< Kp as compared with the case of a uniformly illuminated exit pupil (figure 2(a) ) . At the same time, amplitude-phase filters (9)-(1 l ) , conversely, decrease the degree of transverse localization of the field in the focal plane as well as in the all transverse planes u = const (figure 2(c)). When diffraction focus is located away from the Gaussian focus, as takes place in the case of a Bessel filter (1 1) at high values of constant a, then expansion of the field in the Gaussian focal plane and nearby planes is crucial (figure 2(d)). To be more specific, in this case the diffraction image in the focal plane is shaped into a single light ring, as was demonstrated in the scalar theory [32] (in principle, this follows from the fact that the Fourier transform of the function Jo(x) is a circle).

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High-aperture focusing systems 71

KD,KW

10" 64

10-1 32 16 8 @)

4 2 1

lo-2

1 n-4 20 40 60 80 u LU

KD,KW

loo

lo-l

i n-3

64 32 16

8

4

2

20 40 60 80 u I U

20 40 60 80 u I U

loo

10-l

(d)

i n-3 1u 20 40 60 80 u

Figure 3. Dependences of the quantities Kp (- - - - -) and K , (-) on u for Om = 60" at fixed values of 21 (the figures at the curves) for (a) t (O) = 1 and (b, c, d) different amplitude filters: (7) at Oc = 15" (b), (8) at B = 6 ( c ) , and (6) (d).

The depth of focus (the resistance of the focused beam to diffraction broad- ening along the optical axis) is characterized best through the dependence of Kp,w on u for given values of v , which shows changes in the degree of concentration of radiation in a certain 'optical tube' with certain radius w centred about the optical axis. It is evident from the curves (figure 3(a)) that in the case of uniformly illuminated exit pupil the quantities KP,, periodically oscillate for small values of v . While the latter increases, the oscillations gradually decrease and for large values of v practically disappear. If the filters (6)-(8) are used, then the dependences of on u are identical for any of there filters, the curves are free from oscillations and K, x K, (figures 3(b)-3(4). We emphasize that the filters (6)-(8) suppress diffrac- tion broadening of the focused field beyond the geometrical focal plane as compared with the case where t ( 0 ) = 1. For example, if z, = 8 then at u = 100 we have KwX 5 x l op3 for t (0) = 1 (figure 3(a)) and Kw*7 x lo-* for filter (7) (figure 3(b) ) . The best in this regard is the pupil filter (8) (figure 3 ( c ) ) . It can be seen from figure 3 that in the case of filters (6)-(8) or in the absence of filters ( t (0 ) = l ) , for every value of v the highest concentration of radiation is always achieved in the geometric-optical focal plane.

The all-important effect for focusing, which appears in use of the amplitude- phase filters (9)-(ll), lies in the fact that for small values of z1 (for the central part of the focused beam) the maximum of Kp,w does not fall in the geometric-optical

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72 P. Adamson

loo

10-l

lo-a (a)

lo"

20 40 60 80 u Kp,KV?

loo 64

lo-l 32

8 2

0.5

lo-2

10"

10"'

20 40 60 80 u

loo' lo-l

lo-z @)

10"'

64 32

8

2

0.5 20 40 60 80 u

20 40 60 80 u

Figure 4. Dependences of the quantities Kp (- - - - - and . . . . . .) and K, (-) on u for Om = 60" at fixed values of v {the figures at the curves} (a) for filter (10) at y= 1.4; (b) for filter (9) at a1 = -l,a2 = a3 = +1 ,& = 02 = 30"; (c) for filter (9) at al = +1,a2 = -l,a3 = + 1 , & = 20O.82 = 40'; (d) for filter (11 ) at (Y = 30.635 (- and - - - - - ) and 2000 (...... ) .

focal plane (figure 4), but is displaced from the latter by a certain distance. In addition, in this case the dependence of Kp,w on u possesses a periodical oscillatory shape for filters (10) (figure 4(a)) and a difficult nonperiodical oscillatory character for filters (9) (figures 4(b) and 4(c)). However, for larger values of w when KP,, 2 0.5 the strongest transverse compression still appears in the geometric-optical focal plane and oscillations will be suppressed. Here, when this is the case, particular attention is given to the truncated Bessel filters ( l l) , which prevent the oscillations of integral parameters Kp,w (figure 4(d)). Furthermore, this filter is intriguing because it creates beyond the geometric-optical focal plane a region with a rather large focal depth, but it is important to bear in mind that for large values of (r in the centre of the focused beam we have Kp,w(u) Kp,w(u = 0) = const << 1 when Iu( 5 a! (figure 4(d)).We may also note that by optimization of the parameters of filters (9) and (10) we can create a region where Kp,w(u) GZ Kp,w(u = 0) in the near vicinity of the geometric-optical focal plane (lul 5 10). The value of Kp,w in this case amounts to several per cent. It must be emphasized that physically enhancement of the focal depth is always achieved by reduction of the degree of transverse concentration of radiation, or, in other words, by suppression of the central maximum and increase of the side maxima.

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High-aperture focusing systems 73

4. Conclusions In the case of filters (9)-(ll), which change the phase, integral parameters

(encircled electric energy and encircled longitudinal flux) for small radii do not attain their own maximum values in the geometric-optical focal plane, but in a plane which is displaced from the latter by a certain distance, and diffraction focus does not coincide with the geometric-optical focus. T h e encircled electric energy and the encircled longitudinal flux have maximum values in the focal plane only for large radii of the focused beam. For optimal enhancement of the focal depth the most suitable pupil filter of the filters discussed in this paper is the truncated Bessel filter. But essential enhancement of the focal depth always leads to strong suppression of the central maximum and notable increase in the side maxima.

For conventional absorbing filters (6)-(8) which d o not alter the phase, the integral parameters possess maximum values for any one of the given focused beam radii in the geometric-optical focal plane (as in the case when pupil filtering is absent). These filters significantly increase the values of the integral parameters (the degree of transverse concentration), particularly in planes which are located away from the geometric-optical focal plane, in comparison with the diffraction pattern of the focusing system with uniformly illuminated exit pupil.

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