Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 512971 8 pageshttpdxdoiorg1011552013512971

Research ArticleSharper Focal Spot for a Radially Polarized BeamUsing Ring Aperture with Phase Jump

Svetlana N Khonina and Andrei V Ustinov

Image Processing Systems Institute of the Russian Academy of Sciences Molodogvardeiskaya Street 151 Samara 443001 Russia

Correspondence should be addressed to Svetlana N Khonina khoninasmrru

Received 28 November 2012 Revised 4 March 2013 Accepted 10 March 2013

Academic Editor You-Rong Li

Copyright copy 2013 S N Khonina and A V Ustinov This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We study analytically and numerically in which way the width of ring aperture containing a phase jump affects the size and intensityof the focal spot generated with a radially polarized beam It is shown that by means of destructive interference of beams comingfrom the different-phase rings it becomes possible to overcome the scalar diffraction limit corresponding to the first zero of thezero-order Bessel functionThe minimal focal spot size (FWHM = 033120582) is found to be attained when the annular aperture widthamounts to 20 of the full-aperture radius In this case the side-lobe intensity is not larger than 30 of the central peak A widerannular aperture with the phase jump introduced is also shown to form a focal spot not exceeding the diffraction limit for a narrowannular aperture simultaneously providing a nearly six times higher intensity In this case the side lobes amount to 35 of thecentral peak

1 Introduction

Anarrow annular pupil that blocks the light frompropagatingpractically through the entire central part of the lens [1ndash3] isa simple albeit low-efficiency technique to generate narrow-extended beams in the focal plane

More complicated techniques for the full-apertureapodization of the pupilrsquos function that employ both purelyphase and amplitude-phase distributions have also beenreported [4ndash9] In this case a tighter focal spot is normallyobtained at a sacrifice of the energy redistribution fromthe central peak to the side lobes This situation is in fullcompliance with the Toraldo di Francia theory [10] whichstates that the central focal spot can be made as small asone likes but at a sacrifice of increasing side lobes whichsometimes become several times [11] or even orders [8 9]larger than the central peak

The presence of the substantial side lobes limits the useof ldquosuperresolutionrdquo elements in the imaging and optical datarecording systems in which the acceptable side-lobe level is30 of the central peak [12]

The optimization procedures aimed at controlling thegrowth of the side lobes inevitably lead to a widened central

spot [12 13] With the side lobesrsquo intensity being less than30 of the central peak intensity they can be filtered out [14]or leveled as a result of nonlinear light interaction with therecording medium [15]

The focal spot can be made tighter without an essentialincrease in the side lobes by introducing a radial phase jumpwhich leads to the destructive interference between two lightbeams generated by each of the pupilrsquos rings [15 16] Thedestructive action of the 120587-phase jump is well known in thecoding methods also [17 18]

Note that although the full-aperture apodization of thelens pupil with binary radial elements is usually considered[4 6ndash8 16 19ndash21] it has been numerically demonstrated [22]that by diaphragming the lens central part (sim75 in terms ofradius) the focus can be made considerably tighter

In this paper we study analytically and numerically inwhich way the size of an annular pupil with a radial phasejump affects the focal spot size generated by a radiallypolarized beam It has been shown that the destructiveinterference between the beams passing different-phase ringsenables going beyond a scalar limit corresponding to thezero-order Bessel function (FWHM = 036120582) A tighter focalspot (FWHM = 033120582) is attained at a sacrifice of a lower

2 Journal of Engineering

focal intensity and higher side lobes that corresponds to theToraldo di Francia theory However due to increase of widthof the ring aperture we can increase energy in the central spotand keep a level of side lobes below 30 of the central peak

The analysis of the sharp focusing based on the Richards-Wolf integrals [23] has shown that the dependence of the focalspot size on the width of the narrow annular pupil is differentfor the scalar and vector models In the former case thediffraction-limited focal spot size is attained when the widthof the ring that transmits the peripheral rays tends to zerowhile the focal intensity drops quadratically with decreasingwidth In the latter case theminimal focal spot size is attainedat a fixed annular pupil width with the focal intensity beingproportional to the ring width to the (34)th power

The numerical simulation has shown that notwithstand-ing the destructive interference by choosing a fairly wideannular pupil the focal point intensity can be essentiallyincreased without exceeding the limit corresponding to thenarrow annular pupil

2 Analysis of Generating a SharperFocal Spot Scalar Model

The focusing of a bounded plane wave with a lens of radius 119877and focal length119891 is described by a Fourier-Bessel transformwith the intensity in the focal plane given by

119868119908(120588) =

100381610038161003816100381610038161003816100381610038161003816

119896

119891

int

119877

0

1198690(

119896120588119903

119891

) 119903d119903100381610038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

119877

120588

1198691(

119896120588119903

119891

)

10038161003816100381610038161003816100381610038161003816

2

(1)

which describes an Airy pattern generated in the lens focalplane

Formally obtaining a Bessel beam in the focal planerequires taking a Fourier-Bessel transform of an infinitelynarrow annular pupil

119868120575(120588) =

10038161003816100381610038161003816100381610038161003816

119896

119891

int

infin

0

120575 (

119903

119877

minus 1) 1198690(

119896120588119903

119891

) 119903d11990310038161003816100381610038161003816100381610038161003816

2

=

100381610038161003816100381610038161003816100381610038161003816

119896119877

2

119891

1198690(

119896120588119877

119891

)

100381610038161003816100381610038161003816100381610038161003816

2

(2)

The numerical aperture (NA) of the lens of radius 119877 andfocal length 119891 is defined by the following relationship

NAlens = 119899 sin 120579 = 119899 sin[arctg119877

119891

] (3)

or in the paraxial case

NAlens asymp119899119877

119891

(4)

where 119899 is refractive index of mediaThe resulting focal spot size in the cases under study is

characterized by the following parametersWhen focusing on the full aperture of (1) the central focal

spot radius is determined by the first zero of the first-orderBessel function (119869

1(12057411) = 0 120574

11= 383)

120588lens =383120582119891

2120587119877

= 061

120582119891

119877

(5)

036120582

038120582

051120582

061120582

minus1 0 1

100

80

60

40

20

Figure 1 Illustration of the focal spot for the full-aperture illumina-tion and with narrow ring (119877 = 119891)

In the paraxial case

120588lens asymp061

NA120582 (6)

The estimate for the focal spot size at half level of maxi-mum intensity is

FWHMlens = 0515120582119891

119877

(7)

and for the spot area is

HMAlens = 120587 sdot (FWHMlens

2

)

2

asymp 021(

120582119891

119877

)

2

(8)

When focusing on the narrow annular diaphragm of (2)the focal spot radius is determined by the first zero of the zero-order Bessel function (119869

0(12057401) = 0 120574

01= 2405)

120588ring =24120582119891

2120587119877

= 038

120582119891

119877

(9)

In the paraxial case

120588ring asymp038

NA120582 (10)

The comparison of (5) and (9) suggests that introducingthe narrow ring enables generating a 16 times smaller focalspot even at high NA (Figure 1)

The estimate for the focal spot size at half level of maxi-mum intensity is

FWHMring = 0357120582119891

119877

(11)

which is 144 times lower than for the full-aperture illumi-nation whereas the spot area at half-maximum intensity isdefined as

HMAring = 120587 sdot (FWHMring

2

)

2

= 01(

120582119891

119877

)

2

(12)

which accordingly is 21 times smaller

Journal of Engineering 3

Note however that (2) disregards the amplitude decreaseassociatedwith increasing illuminated area as the value of the120575-function is assumed to be equal to infinity

Assume that the ring is not infinitely narrow as is the casein (2) but of a finite width that comprises a small fraction Δof the lens radius In this case the complex amplitude is givenby

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)int119877

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 (13)

In view of the small Δ the integral (13) can be approx-imately calculated by multiplying the integrand value at themiddle point

119903119904= 119877(1 minus

Δ

2

) (14)

by the line length

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)

times [119877

2

Δ(1 minus

Δ

2

) 1198690(

119896120588119877 (1 minus Δ2)

119891

)]

asymp

119896119877

2

Δ

119894119891

exp (119894119896119891) 1198690(

119896120588119877

119891

)

(15)

We can see that although this approximation gives thesame focal spot size as that for the infinitely narrow ring(2) this relation also takes into account that the amplitudedecreases at the center proportionally to the decreasingannular aperture width Δ If 119877(1 minus Δ2) is not replaced with119877 in the Bessel function argument the spot radius estimatebecomes a bit larger

Below we discuss in which way the focal spot size can bemade smaller by introducing a phase jump of 120587-radian alongthe central radius of the narrow annular aperture

120591 (119903) =

0 0 le 119903 lt 119877 (1 minus Δ)

1 119877 (1 minus Δ) le 119903 lt 119877(1 minus

Δ

2

)

exp(119894120587) 119877 (1 minus Δ2

) le 119903 le 119877

(16)

A radial phase jump which leads to the destructiveinterference between two light beams generated by each ofthe pupilrsquos rings was used for sharp focusing [15 16] Thedestructive action of the 120587-phase jump is well known in thecoding methods also [17 18]

The resulting distribution in the focal spot will corre-spond to the destructive interference of beams generated bythe individual rings

119864120591(120588)

=

119896

119894119891

exp (119894119896119891)int119877

0

120591 (119903) 1198690(

119896119903120588

119891

) 119903d119903

=

119896

119894119891

exp (119894119896119891)

times [int

119877(1minusΔ2)

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 minus int119877

119877(1minusΔ2)

1198690(

119896119903120588

119891

) 119903d119903]

(17)

In view of (15) we obtain

119864120591(120588) =

Δ

2

119896119877

2

119894119891

exp (119894119896119891)

times [1198690(

119896120588119877 (1 minus 3Δ4)

119891

)(1 minus

3Δ

4

)

minus 1198690(

119896120588119877 (1 minus Δ4)

119891

)(1 minus

Δ

4

)]

(18)

From (18) it is clear that in spite of destructive interferencethere is any nonzero intensity at the focal point because theinner and outer areas of the annular aperture are not equal

To estimate the resulting equation (18) for small 120588 let usrecall that the Bessel function can be approximately expressedthrough a Taylor series [24]

119869] (119909 997888rarr 0) sim1

Γ (] + 1)(

119909

2

)

]

times [1 minus

119909

2

4 (] + 1)+

119909

4

32 (] + 1) (] + 2)minus sdot sdot sdot]

(19)

Then (18) is approximately reduced to

119864120591(120588) asymp minus (

Δ

2

)

2

119896119877

2

119894119891

exp (119894119896119891)

times [1 minus (

119896120588119877

2119891

)

2

(3 minus

15

8

Δ +

13

16

Δ

2

)]

(20)

The first coefficient in the right-hand side of (20) showsthat the central spot amplitude falls proportionally to theannular aperture width squared This effect is due to the factthat the radius of 120587-phase jump is the middle of the ringThenegative value at the zero point means that there is a phaseshift by 120587 with respect to the incident beam

The focal spot radius is derived by setting the expressionin the square brackets equal to zero so that

120588jump =120582119891

120587119877radic3 minus (158) Δ + (1316) Δ

2

(21)

4 Journal of Engineering

Because Δ lt 1 the smallest radius in (21) is attained atΔ rarr 0 being equal to

120588

jumpmin 119886 =

120582119891

120587119877radic3

asymp 0184

120582119891

119877

(22)

Note that the estimate in (22) is understated since withthe quadratic approximation of the Bessel function in (19)the resulting value appears to be 20 lower than the truevalueTherefore we will consider the following value as moreaccurate

120588

jumpmin = 12120588

jumpmin 119886 = 022

120582119891

119877

(23)

This estimate predicts that the radius should be 28 timessmaller than for the full-aperture lens and 17 times smallerthan for the narrow annular aperture

The estimates for the focal spot diameter and area at half-maximum intensity are as follows

FWHMjumpmin = 026

120582119891

119877

(24)

HMAjumpmin = 0053(

120582119891

119877

)

2

(25)

Note that in this case the focal spot diameter FWHMin (24) appears to be larger than the radius measured asfar as the zero-intensity point (23) This situation differsfrom the one occurred for the first- and zero-order Besselfunctions for which the spot size FWHM was smaller thanthe zero-intensity radius showing that the central peak wallsare steeper

3 Analysis of the Focal Spot SizeDecrease When a Radially PolarizedBeam Is Tightly Focused

When doing the tight focusing of a radially polarized beamwith the use of a narrow annular pupil themajor contributionto the focal plane has been known to be from the longitudinal119864-field component [1ndash3] The significant predominance ofone component over the other vector field componentscreates a situation similar to the scalar regime Thus scalartheory instruments can be successfully utilized producingfairly accurate results The tightly focused focal spot usuallyappears to be larger in size than the scalar estimate as differentcomponents really contribute to the focal spot

Regarding the above considerations only the longitudinalcomponent will be taken into account when analyzing thetight focusing of radially polarized light

For a narrow annular pupil we obtain [9]

119864

119911

Δ(120588) = int

1

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

asymp

Δ(1 minus Δ2)

2

[1 minus (1 minus Δ2)

2

]

14

1198690(119896120588 (1 minus

Δ

2

))

(26)

Thus similar to the scalar case (see (15)) the focal spotsize is determined by the first zero of the zero-order Besselfunction Hence with decreasing annular pupilrsquos width Δ rarr0 we obtain in the limit

120588

119911ringmin =

038

NA120582 997888997888997888997888997888rarr

NArarr1120588

119911ringlim = 038120582 (27)

The focal spot size can be further decreased withoutan essential side-lobe increase based on the destructiveinterference if a 120587-radian phase jump is introduced along thecentral radius of the narrow annular aperture (16) Then thefield distribution in the focal plane is given by

119864

119911

120591(120588) = int

1minusΔ2

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

minus int

1

1minusΔ2

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

(28)

To calculate the integrals in (28) we write down

119864

119911

120591(120588) asymp 119869

0(119896120588 (1 minus

3Δ

4

))

1

(1 minus 3Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1minusΔ2

1minusΔ

minus 1198690(119896120588 (1 minus

Δ

4

))

1

(1 minus Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1

1minusΔ2

(29)

Then retaining the Δ terms up to the second order weobtain

119864

119911

120591(120588) asymp minusΔ

34

(0213 + 0386Δ minus 1139Δ

2

)

minus

119896

2

120588

2

4

(0213 + 0734Δ minus 2175Δ

2

)

(30)

From (30) an estimate for the focal spot radius then takesthe form

120588119911jump asymp

120582

120587

radic

0213 + 0386Δ minus 1139Δ

2

0213 + 0734Δ minus 2175Δ

2

(31)

with the minimum attained at Δ119898= 0168

Thus the minimal estimate of the central spot radius isgiven by

120588119911jump (Δ119898) asymp 0301120582 (32)

4 Numerical Results

Although the estimates discussed in Section 3 are approxi-mate they enable us to analyze in which way the focal spot

Journal of Engineering 5

05

045

04

035

0 02 04 06 08

Δ

FWH

M120582

Figure 2 The focal spot size (FWHM) as a function of the annularaperturersquos width (Δ) without (dashed line) and with (dashed-dottedline) a radial phase jump

size and intensity depend on the annular aperturersquos width Foranalytical calculations it was convenient to use the middleradius of the ring We understand that the radius of phasejump is additional parameter for optimization of the focaldistribution however experience of other researchers [4 6ndash8 19ndash21] shows that even a plenty of free parameters doesnot allow to reduce the focal spot simultaneously keepingenergy Therefore we have decided to vary only the width ofthe ring with the middle 120587-phase jump This limitation maybe convenient for the experimental realization

In this section the sharp focusing of a radially polarizedbeam is numerically simulated using the Richards-Wolfintegral relations [23] for an aplanatic focusing system

Figure 2 shows the focal spot size (FWHM) as a functionof the annular aperture width (Δ) for an element without andwith the radial phase jump of (16) The dotted horizontalline denotes the scalar limit for the lens (FWHM = 051120582)whereas the solid line denotes the scalar limit for the infinitelynarrow ring (FWHM = 036120582)

Figure 2 suggests that with a simple annular aperture togo beyond the lens scalar limit it is necessary to block a lenscentral portion of radius 043119877 (Δ = 057) which amountsto sim20 of the lens area Note that all 119864-field componentsare accounted for in the total intensity Meanwhile the resultscorresponding to the infinitely narrow aperture (FWHM =

036120582) remain unattained even at a very small width (at Δ =001 FWHM = 0367120582)

If there is a radial phase jump (16) there is no need toblock any part of the lens to go beyond the scalar limit At thesame time blocking the central rays on the radius 046119877 (Δ =054 with sim20 of the area got covered) enables obtaininga focal spot smaller than the scalar limit for the infinitelynarrow ring

Note that theminimal size of FWHM = 0329120582 is attainedwhen the annular aperturersquos width is Δ

119898= 02 (Figure 2)

This value is approximately 12 times larger than the spot sizederived in the previous section and employed in the estimateof (32)

Figure 3 shows the plot of the normalized focal pointintensity against the annular aperturersquos width (Δ) withoutand with a radial phase jump The vertical dotted lines markthe annular aperturersquos width at which the focal spot of sizeFWHM = 038120582 is formed for a conventional annular

0 02 04 06 08

Δ

1

05

1198680

Figure 3 Normalized focal point intensity as a function of theannular aperturersquos width (Δ) without (solid bold line) and with aradial phase jump (dotted line)

aperture the width is Δ = 006 and for aperture with theintroduced phase jump the width is Δ = 065 Ring sizeof Δ = 006 is rather narrow Thus for practical reasonswe decided to start from this value rather than the limit ofFWHM = 036120582 It is seen that when the phase jump wasintroduced the focal point intensity generated with the widerannular aperture is several times larger than that generatedwith the conventional narrow ring

Table 1 gives the numerical simulation results when aradially polarized beam is tightly focusing (119877 = 100120582 119891 =101120582) and a narrow annular aperture with the phase jumpof (16) is used Δ is the annular slit width 119868

0is the total

intensity in the focus 119878 is the ratio of the total intensity ofthe first side lobe to that of the central focal spot FWHMis the central focal spot diameter in terms of full-width half-maximum intensity and FWHM

119911is the central spot diameter

at half-maximum intensity of the longitudinal componentTable 1 suggests that the smallest central focal spot

(FWHM = 0329120582 see 2nd row of Table 1) has been attainedwithout an unacceptable increase in the side lobes (119878 lt

03) However there is an essential loss of the central spotenergy with the intensity 119868

0being half of that provided by the

narrow annular aperture (1st row of Table 1)The longitudinaldistribution suggests that the focal spot becomes tighter dueto the destructive interference of the beams transmitted bydifferent aperturersquos rings

However by increasing the pupilrsquos width the central spotenergy can be significantly increased with spot size smallerthan that generated with the narrow ring In particular atΔ = 06 (3rd row of Table 1) the focal spot is slightly smaller(FWHM = 0371120582) whereas the intensity is sim6 times largerwhen compared with that generated by the narrow annularaperture A disadvantage consists in somewhat higher sidelobes (119878 = 035)

For the full-aperture case with 120587-phase jump (Δ = 1 4throw of Table 1) what we observe is not only a larger focalintensity but also lower side lobes In this case the focal spotsize (FWHM = 0454120582) does not exceed the Airy patternrsquosscalar limit

Results for the case of full-aperture illumination withoutany apodization are shown (5th row of Table 1) for compar-ison it is possible to see that the focal spot size (FWHM =

0544120582) exceeds the Airy patternrsquos scalar limit

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

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2 Journal of Engineering

focal intensity and higher side lobes that corresponds to theToraldo di Francia theory However due to increase of widthof the ring aperture we can increase energy in the central spotand keep a level of side lobes below 30 of the central peak

The analysis of the sharp focusing based on the Richards-Wolf integrals [23] has shown that the dependence of the focalspot size on the width of the narrow annular pupil is differentfor the scalar and vector models In the former case thediffraction-limited focal spot size is attained when the widthof the ring that transmits the peripheral rays tends to zerowhile the focal intensity drops quadratically with decreasingwidth In the latter case theminimal focal spot size is attainedat a fixed annular pupil width with the focal intensity beingproportional to the ring width to the (34)th power

The numerical simulation has shown that notwithstand-ing the destructive interference by choosing a fairly wideannular pupil the focal point intensity can be essentiallyincreased without exceeding the limit corresponding to thenarrow annular pupil

2 Analysis of Generating a SharperFocal Spot Scalar Model

The focusing of a bounded plane wave with a lens of radius 119877and focal length119891 is described by a Fourier-Bessel transformwith the intensity in the focal plane given by

119868119908(120588) =

100381610038161003816100381610038161003816100381610038161003816

119896

119891

int

119877

0

1198690(

119896120588119903

119891

) 119903d119903100381610038161003816100381610038161003816100381610038161003816

2

=

10038161003816100381610038161003816100381610038161003816

119877

120588

1198691(

119896120588119903

119891

)

10038161003816100381610038161003816100381610038161003816

2

(1)

which describes an Airy pattern generated in the lens focalplane

Formally obtaining a Bessel beam in the focal planerequires taking a Fourier-Bessel transform of an infinitelynarrow annular pupil

119868120575(120588) =

10038161003816100381610038161003816100381610038161003816

119896

119891

int

infin

0

120575 (

119903

119877

minus 1) 1198690(

119896120588119903

119891

) 119903d11990310038161003816100381610038161003816100381610038161003816

2

=

100381610038161003816100381610038161003816100381610038161003816

119896119877

2

119891

1198690(

119896120588119877

119891

)

100381610038161003816100381610038161003816100381610038161003816

2

(2)

The numerical aperture (NA) of the lens of radius 119877 andfocal length 119891 is defined by the following relationship

NAlens = 119899 sin 120579 = 119899 sin[arctg119877

119891

] (3)

or in the paraxial case

NAlens asymp119899119877

119891

(4)

where 119899 is refractive index of mediaThe resulting focal spot size in the cases under study is

characterized by the following parametersWhen focusing on the full aperture of (1) the central focal

spot radius is determined by the first zero of the first-orderBessel function (119869

1(12057411) = 0 120574

11= 383)

120588lens =383120582119891

2120587119877

= 061

120582119891

119877

(5)

036120582

038120582

051120582

061120582

minus1 0 1

100

80

60

40

20

Figure 1 Illustration of the focal spot for the full-aperture illumina-tion and with narrow ring (119877 = 119891)

In the paraxial case

120588lens asymp061

NA120582 (6)

The estimate for the focal spot size at half level of maxi-mum intensity is

FWHMlens = 0515120582119891

119877

(7)

and for the spot area is

HMAlens = 120587 sdot (FWHMlens

2

)

2

asymp 021(

120582119891

119877

)

2

(8)

When focusing on the narrow annular diaphragm of (2)the focal spot radius is determined by the first zero of the zero-order Bessel function (119869

0(12057401) = 0 120574

01= 2405)

120588ring =24120582119891

2120587119877

= 038

120582119891

119877

(9)

In the paraxial case

120588ring asymp038

NA120582 (10)

The comparison of (5) and (9) suggests that introducingthe narrow ring enables generating a 16 times smaller focalspot even at high NA (Figure 1)

The estimate for the focal spot size at half level of maxi-mum intensity is

FWHMring = 0357120582119891

119877

(11)

which is 144 times lower than for the full-aperture illumi-nation whereas the spot area at half-maximum intensity isdefined as

HMAring = 120587 sdot (FWHMring

2

)

2

= 01(

120582119891

119877

)

2

(12)

which accordingly is 21 times smaller

Journal of Engineering 3

Note however that (2) disregards the amplitude decreaseassociatedwith increasing illuminated area as the value of the120575-function is assumed to be equal to infinity

Assume that the ring is not infinitely narrow as is the casein (2) but of a finite width that comprises a small fraction Δof the lens radius In this case the complex amplitude is givenby

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)int119877

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 (13)

In view of the small Δ the integral (13) can be approx-imately calculated by multiplying the integrand value at themiddle point

119903119904= 119877(1 minus

Δ

2

) (14)

by the line length

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)

times [119877

2

Δ(1 minus

Δ

2

) 1198690(

119896120588119877 (1 minus Δ2)

119891

)]

asymp

119896119877

2

Δ

119894119891

exp (119894119896119891) 1198690(

119896120588119877

119891

)

(15)

We can see that although this approximation gives thesame focal spot size as that for the infinitely narrow ring(2) this relation also takes into account that the amplitudedecreases at the center proportionally to the decreasingannular aperture width Δ If 119877(1 minus Δ2) is not replaced with119877 in the Bessel function argument the spot radius estimatebecomes a bit larger

Below we discuss in which way the focal spot size can bemade smaller by introducing a phase jump of 120587-radian alongthe central radius of the narrow annular aperture

120591 (119903) =

0 0 le 119903 lt 119877 (1 minus Δ)

1 119877 (1 minus Δ) le 119903 lt 119877(1 minus

Δ

2

)

exp(119894120587) 119877 (1 minus Δ2

) le 119903 le 119877

(16)

A radial phase jump which leads to the destructiveinterference between two light beams generated by each ofthe pupilrsquos rings was used for sharp focusing [15 16] Thedestructive action of the 120587-phase jump is well known in thecoding methods also [17 18]

The resulting distribution in the focal spot will corre-spond to the destructive interference of beams generated bythe individual rings

119864120591(120588)

=

119896

119894119891

exp (119894119896119891)int119877

0

120591 (119903) 1198690(

119896119903120588

119891

) 119903d119903

=

119896

119894119891

exp (119894119896119891)

times [int

119877(1minusΔ2)

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 minus int119877

119877(1minusΔ2)

1198690(

119896119903120588

119891

) 119903d119903]

(17)

In view of (15) we obtain

119864120591(120588) =

Δ

2

119896119877

2

119894119891

exp (119894119896119891)

times [1198690(

119896120588119877 (1 minus 3Δ4)

119891

)(1 minus

3Δ

4

)

minus 1198690(

119896120588119877 (1 minus Δ4)

119891

)(1 minus

Δ

4

)]

(18)

From (18) it is clear that in spite of destructive interferencethere is any nonzero intensity at the focal point because theinner and outer areas of the annular aperture are not equal

To estimate the resulting equation (18) for small 120588 let usrecall that the Bessel function can be approximately expressedthrough a Taylor series [24]

119869] (119909 997888rarr 0) sim1

Γ (] + 1)(

119909

2

)

]

times [1 minus

119909

2

4 (] + 1)+

119909

4

32 (] + 1) (] + 2)minus sdot sdot sdot]

(19)

Then (18) is approximately reduced to

119864120591(120588) asymp minus (

Δ

2

)

2

119896119877

2

119894119891

exp (119894119896119891)

times [1 minus (

119896120588119877

2119891

)

2

(3 minus

15

8

Δ +

13

16

Δ

2

)]

(20)

The first coefficient in the right-hand side of (20) showsthat the central spot amplitude falls proportionally to theannular aperture width squared This effect is due to the factthat the radius of 120587-phase jump is the middle of the ringThenegative value at the zero point means that there is a phaseshift by 120587 with respect to the incident beam

The focal spot radius is derived by setting the expressionin the square brackets equal to zero so that

120588jump =120582119891

120587119877radic3 minus (158) Δ + (1316) Δ

2

(21)

4 Journal of Engineering

Because Δ lt 1 the smallest radius in (21) is attained atΔ rarr 0 being equal to

120588

jumpmin 119886 =

120582119891

120587119877radic3

asymp 0184

120582119891

119877

(22)

Note that the estimate in (22) is understated since withthe quadratic approximation of the Bessel function in (19)the resulting value appears to be 20 lower than the truevalueTherefore we will consider the following value as moreaccurate

120588

jumpmin = 12120588

jumpmin 119886 = 022

120582119891

119877

(23)

This estimate predicts that the radius should be 28 timessmaller than for the full-aperture lens and 17 times smallerthan for the narrow annular aperture

The estimates for the focal spot diameter and area at half-maximum intensity are as follows

FWHMjumpmin = 026

120582119891

119877

(24)

HMAjumpmin = 0053(

120582119891

119877

)

2

(25)

Note that in this case the focal spot diameter FWHMin (24) appears to be larger than the radius measured asfar as the zero-intensity point (23) This situation differsfrom the one occurred for the first- and zero-order Besselfunctions for which the spot size FWHM was smaller thanthe zero-intensity radius showing that the central peak wallsare steeper

3 Analysis of the Focal Spot SizeDecrease When a Radially PolarizedBeam Is Tightly Focused

When doing the tight focusing of a radially polarized beamwith the use of a narrow annular pupil themajor contributionto the focal plane has been known to be from the longitudinal119864-field component [1ndash3] The significant predominance ofone component over the other vector field componentscreates a situation similar to the scalar regime Thus scalartheory instruments can be successfully utilized producingfairly accurate results The tightly focused focal spot usuallyappears to be larger in size than the scalar estimate as differentcomponents really contribute to the focal spot

Regarding the above considerations only the longitudinalcomponent will be taken into account when analyzing thetight focusing of radially polarized light

For a narrow annular pupil we obtain [9]

119864

119911

Δ(120588) = int

1

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

asymp

Δ(1 minus Δ2)

2

[1 minus (1 minus Δ2)

2

]

14

1198690(119896120588 (1 minus

Δ

2

))

(26)

Thus similar to the scalar case (see (15)) the focal spotsize is determined by the first zero of the zero-order Besselfunction Hence with decreasing annular pupilrsquos width Δ rarr0 we obtain in the limit

120588

119911ringmin =

038

NA120582 997888997888997888997888997888rarr

NArarr1120588

119911ringlim = 038120582 (27)

The focal spot size can be further decreased withoutan essential side-lobe increase based on the destructiveinterference if a 120587-radian phase jump is introduced along thecentral radius of the narrow annular aperture (16) Then thefield distribution in the focal plane is given by

119864

119911

120591(120588) = int

1minusΔ2

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

minus int

1

1minusΔ2

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

(28)

To calculate the integrals in (28) we write down

119864

119911

120591(120588) asymp 119869

0(119896120588 (1 minus

3Δ

4

))

1

(1 minus 3Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1minusΔ2

1minusΔ

minus 1198690(119896120588 (1 minus

Δ

4

))

1

(1 minus Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1

1minusΔ2

(29)

Then retaining the Δ terms up to the second order weobtain

119864

119911

120591(120588) asymp minusΔ

34

(0213 + 0386Δ minus 1139Δ

2

)

minus

119896

2

120588

2

4

(0213 + 0734Δ minus 2175Δ

2

)

(30)

From (30) an estimate for the focal spot radius then takesthe form

120588119911jump asymp

120582

120587

radic

0213 + 0386Δ minus 1139Δ

2

0213 + 0734Δ minus 2175Δ

2

(31)

with the minimum attained at Δ119898= 0168

Thus the minimal estimate of the central spot radius isgiven by

120588119911jump (Δ119898) asymp 0301120582 (32)

4 Numerical Results

Although the estimates discussed in Section 3 are approxi-mate they enable us to analyze in which way the focal spot

Journal of Engineering 5

05

045

04

035

0 02 04 06 08

Δ

FWH

M120582

Figure 2 The focal spot size (FWHM) as a function of the annularaperturersquos width (Δ) without (dashed line) and with (dashed-dottedline) a radial phase jump

size and intensity depend on the annular aperturersquos width Foranalytical calculations it was convenient to use the middleradius of the ring We understand that the radius of phasejump is additional parameter for optimization of the focaldistribution however experience of other researchers [4 6ndash8 19ndash21] shows that even a plenty of free parameters doesnot allow to reduce the focal spot simultaneously keepingenergy Therefore we have decided to vary only the width ofthe ring with the middle 120587-phase jump This limitation maybe convenient for the experimental realization

In this section the sharp focusing of a radially polarizedbeam is numerically simulated using the Richards-Wolfintegral relations [23] for an aplanatic focusing system

Figure 2 shows the focal spot size (FWHM) as a functionof the annular aperture width (Δ) for an element without andwith the radial phase jump of (16) The dotted horizontalline denotes the scalar limit for the lens (FWHM = 051120582)whereas the solid line denotes the scalar limit for the infinitelynarrow ring (FWHM = 036120582)

Figure 2 suggests that with a simple annular aperture togo beyond the lens scalar limit it is necessary to block a lenscentral portion of radius 043119877 (Δ = 057) which amountsto sim20 of the lens area Note that all 119864-field componentsare accounted for in the total intensity Meanwhile the resultscorresponding to the infinitely narrow aperture (FWHM =

036120582) remain unattained even at a very small width (at Δ =001 FWHM = 0367120582)

If there is a radial phase jump (16) there is no need toblock any part of the lens to go beyond the scalar limit At thesame time blocking the central rays on the radius 046119877 (Δ =054 with sim20 of the area got covered) enables obtaininga focal spot smaller than the scalar limit for the infinitelynarrow ring

Note that theminimal size of FWHM = 0329120582 is attainedwhen the annular aperturersquos width is Δ

119898= 02 (Figure 2)

This value is approximately 12 times larger than the spot sizederived in the previous section and employed in the estimateof (32)

Figure 3 shows the plot of the normalized focal pointintensity against the annular aperturersquos width (Δ) withoutand with a radial phase jump The vertical dotted lines markthe annular aperturersquos width at which the focal spot of sizeFWHM = 038120582 is formed for a conventional annular

0 02 04 06 08

Δ

1

05

1198680

Figure 3 Normalized focal point intensity as a function of theannular aperturersquos width (Δ) without (solid bold line) and with aradial phase jump (dotted line)

aperture the width is Δ = 006 and for aperture with theintroduced phase jump the width is Δ = 065 Ring sizeof Δ = 006 is rather narrow Thus for practical reasonswe decided to start from this value rather than the limit ofFWHM = 036120582 It is seen that when the phase jump wasintroduced the focal point intensity generated with the widerannular aperture is several times larger than that generatedwith the conventional narrow ring

Table 1 gives the numerical simulation results when aradially polarized beam is tightly focusing (119877 = 100120582 119891 =101120582) and a narrow annular aperture with the phase jumpof (16) is used Δ is the annular slit width 119868

0is the total

intensity in the focus 119878 is the ratio of the total intensity ofthe first side lobe to that of the central focal spot FWHMis the central focal spot diameter in terms of full-width half-maximum intensity and FWHM

119911is the central spot diameter

at half-maximum intensity of the longitudinal componentTable 1 suggests that the smallest central focal spot

(FWHM = 0329120582 see 2nd row of Table 1) has been attainedwithout an unacceptable increase in the side lobes (119878 lt

03) However there is an essential loss of the central spotenergy with the intensity 119868

0being half of that provided by the

narrow annular aperture (1st row of Table 1)The longitudinaldistribution suggests that the focal spot becomes tighter dueto the destructive interference of the beams transmitted bydifferent aperturersquos rings

However by increasing the pupilrsquos width the central spotenergy can be significantly increased with spot size smallerthan that generated with the narrow ring In particular atΔ = 06 (3rd row of Table 1) the focal spot is slightly smaller(FWHM = 0371120582) whereas the intensity is sim6 times largerwhen compared with that generated by the narrow annularaperture A disadvantage consists in somewhat higher sidelobes (119878 = 035)

For the full-aperture case with 120587-phase jump (Δ = 1 4throw of Table 1) what we observe is not only a larger focalintensity but also lower side lobes In this case the focal spotsize (FWHM = 0454120582) does not exceed the Airy patternrsquosscalar limit

Results for the case of full-aperture illumination withoutany apodization are shown (5th row of Table 1) for compar-ison it is possible to see that the focal spot size (FWHM =

0544120582) exceeds the Airy patternrsquos scalar limit

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 3

Note however that (2) disregards the amplitude decreaseassociatedwith increasing illuminated area as the value of the120575-function is assumed to be equal to infinity

Assume that the ring is not infinitely narrow as is the casein (2) but of a finite width that comprises a small fraction Δof the lens radius In this case the complex amplitude is givenby

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)int119877

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 (13)

In view of the small Δ the integral (13) can be approx-imately calculated by multiplying the integrand value at themiddle point

119903119904= 119877(1 minus

Δ

2

) (14)

by the line length

119864Δ(120588) =

119896

119894119891

exp (119894119896119891)

times [119877

2

Δ(1 minus

Δ

2

) 1198690(

119896120588119877 (1 minus Δ2)

119891

)]

asymp

119896119877

2

Δ

119894119891

exp (119894119896119891) 1198690(

119896120588119877

119891

)

(15)

We can see that although this approximation gives thesame focal spot size as that for the infinitely narrow ring(2) this relation also takes into account that the amplitudedecreases at the center proportionally to the decreasingannular aperture width Δ If 119877(1 minus Δ2) is not replaced with119877 in the Bessel function argument the spot radius estimatebecomes a bit larger

Below we discuss in which way the focal spot size can bemade smaller by introducing a phase jump of 120587-radian alongthe central radius of the narrow annular aperture

120591 (119903) =

0 0 le 119903 lt 119877 (1 minus Δ)

1 119877 (1 minus Δ) le 119903 lt 119877(1 minus

Δ

2

)

exp(119894120587) 119877 (1 minus Δ2

) le 119903 le 119877

(16)

A radial phase jump which leads to the destructiveinterference between two light beams generated by each ofthe pupilrsquos rings was used for sharp focusing [15 16] Thedestructive action of the 120587-phase jump is well known in thecoding methods also [17 18]

The resulting distribution in the focal spot will corre-spond to the destructive interference of beams generated bythe individual rings

119864120591(120588)

=

119896

119894119891

exp (119894119896119891)int119877

0

120591 (119903) 1198690(

119896119903120588

119891

) 119903d119903

=

119896

119894119891

exp (119894119896119891)

times [int

119877(1minusΔ2)

119877(1minusΔ)

1198690(

119896119903120588

119891

) 119903d119903 minus int119877

119877(1minusΔ2)

1198690(

119896119903120588

119891

) 119903d119903]

(17)

In view of (15) we obtain

119864120591(120588) =

Δ

2

119896119877

2

119894119891

exp (119894119896119891)

times [1198690(

119896120588119877 (1 minus 3Δ4)

119891

)(1 minus

3Δ

4

)

minus 1198690(

119896120588119877 (1 minus Δ4)

119891

)(1 minus

Δ

4

)]

(18)

From (18) it is clear that in spite of destructive interferencethere is any nonzero intensity at the focal point because theinner and outer areas of the annular aperture are not equal

To estimate the resulting equation (18) for small 120588 let usrecall that the Bessel function can be approximately expressedthrough a Taylor series [24]

119869] (119909 997888rarr 0) sim1

Γ (] + 1)(

119909

2

)

]

times [1 minus

119909

2

4 (] + 1)+

119909

4

32 (] + 1) (] + 2)minus sdot sdot sdot]

(19)

Then (18) is approximately reduced to

119864120591(120588) asymp minus (

Δ

2

)

2

119896119877

2

119894119891

exp (119894119896119891)

times [1 minus (

119896120588119877

2119891

)

2

(3 minus

15

8

Δ +

13

16

Δ

2

)]

(20)

The first coefficient in the right-hand side of (20) showsthat the central spot amplitude falls proportionally to theannular aperture width squared This effect is due to the factthat the radius of 120587-phase jump is the middle of the ringThenegative value at the zero point means that there is a phaseshift by 120587 with respect to the incident beam

The focal spot radius is derived by setting the expressionin the square brackets equal to zero so that

120588jump =120582119891

120587119877radic3 minus (158) Δ + (1316) Δ

2

(21)

4 Journal of Engineering

Because Δ lt 1 the smallest radius in (21) is attained atΔ rarr 0 being equal to

120588

jumpmin 119886 =

120582119891

120587119877radic3

asymp 0184

120582119891

119877

(22)

Note that the estimate in (22) is understated since withthe quadratic approximation of the Bessel function in (19)the resulting value appears to be 20 lower than the truevalueTherefore we will consider the following value as moreaccurate

120588

jumpmin = 12120588

jumpmin 119886 = 022

120582119891

119877

(23)

This estimate predicts that the radius should be 28 timessmaller than for the full-aperture lens and 17 times smallerthan for the narrow annular aperture

The estimates for the focal spot diameter and area at half-maximum intensity are as follows

FWHMjumpmin = 026

120582119891

119877

(24)

HMAjumpmin = 0053(

120582119891

119877

)

2

(25)

Note that in this case the focal spot diameter FWHMin (24) appears to be larger than the radius measured asfar as the zero-intensity point (23) This situation differsfrom the one occurred for the first- and zero-order Besselfunctions for which the spot size FWHM was smaller thanthe zero-intensity radius showing that the central peak wallsare steeper

3 Analysis of the Focal Spot SizeDecrease When a Radially PolarizedBeam Is Tightly Focused

When doing the tight focusing of a radially polarized beamwith the use of a narrow annular pupil themajor contributionto the focal plane has been known to be from the longitudinal119864-field component [1ndash3] The significant predominance ofone component over the other vector field componentscreates a situation similar to the scalar regime Thus scalartheory instruments can be successfully utilized producingfairly accurate results The tightly focused focal spot usuallyappears to be larger in size than the scalar estimate as differentcomponents really contribute to the focal spot

Regarding the above considerations only the longitudinalcomponent will be taken into account when analyzing thetight focusing of radially polarized light

For a narrow annular pupil we obtain [9]

119864

119911

Δ(120588) = int

1

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

asymp

Δ(1 minus Δ2)

2

[1 minus (1 minus Δ2)

2

]

14

1198690(119896120588 (1 minus

Δ

2

))

(26)

Thus similar to the scalar case (see (15)) the focal spotsize is determined by the first zero of the zero-order Besselfunction Hence with decreasing annular pupilrsquos width Δ rarr0 we obtain in the limit

120588

119911ringmin =

038

NA120582 997888997888997888997888997888rarr

NArarr1120588

119911ringlim = 038120582 (27)

The focal spot size can be further decreased withoutan essential side-lobe increase based on the destructiveinterference if a 120587-radian phase jump is introduced along thecentral radius of the narrow annular aperture (16) Then thefield distribution in the focal plane is given by

119864

119911

120591(120588) = int

1minusΔ2

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

minus int

1

1minusΔ2

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

(28)

To calculate the integrals in (28) we write down

119864

119911

120591(120588) asymp 119869

0(119896120588 (1 minus

3Δ

4

))

1

(1 minus 3Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1minusΔ2

1minusΔ

minus 1198690(119896120588 (1 minus

Δ

4

))

1

(1 minus Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1

1minusΔ2

(29)

Then retaining the Δ terms up to the second order weobtain

119864

119911

120591(120588) asymp minusΔ

34

(0213 + 0386Δ minus 1139Δ

2

)

minus

119896

2

120588

2

4

(0213 + 0734Δ minus 2175Δ

2

)

(30)

From (30) an estimate for the focal spot radius then takesthe form

120588119911jump asymp

120582

120587

radic

0213 + 0386Δ minus 1139Δ

2

0213 + 0734Δ minus 2175Δ

2

(31)

with the minimum attained at Δ119898= 0168

Thus the minimal estimate of the central spot radius isgiven by

120588119911jump (Δ119898) asymp 0301120582 (32)

4 Numerical Results

Although the estimates discussed in Section 3 are approxi-mate they enable us to analyze in which way the focal spot

Journal of Engineering 5

05

045

04

035

0 02 04 06 08

Δ

FWH

M120582

Figure 2 The focal spot size (FWHM) as a function of the annularaperturersquos width (Δ) without (dashed line) and with (dashed-dottedline) a radial phase jump

size and intensity depend on the annular aperturersquos width Foranalytical calculations it was convenient to use the middleradius of the ring We understand that the radius of phasejump is additional parameter for optimization of the focaldistribution however experience of other researchers [4 6ndash8 19ndash21] shows that even a plenty of free parameters doesnot allow to reduce the focal spot simultaneously keepingenergy Therefore we have decided to vary only the width ofthe ring with the middle 120587-phase jump This limitation maybe convenient for the experimental realization

In this section the sharp focusing of a radially polarizedbeam is numerically simulated using the Richards-Wolfintegral relations [23] for an aplanatic focusing system

Figure 2 shows the focal spot size (FWHM) as a functionof the annular aperture width (Δ) for an element without andwith the radial phase jump of (16) The dotted horizontalline denotes the scalar limit for the lens (FWHM = 051120582)whereas the solid line denotes the scalar limit for the infinitelynarrow ring (FWHM = 036120582)

Figure 2 suggests that with a simple annular aperture togo beyond the lens scalar limit it is necessary to block a lenscentral portion of radius 043119877 (Δ = 057) which amountsto sim20 of the lens area Note that all 119864-field componentsare accounted for in the total intensity Meanwhile the resultscorresponding to the infinitely narrow aperture (FWHM =

036120582) remain unattained even at a very small width (at Δ =001 FWHM = 0367120582)

If there is a radial phase jump (16) there is no need toblock any part of the lens to go beyond the scalar limit At thesame time blocking the central rays on the radius 046119877 (Δ =054 with sim20 of the area got covered) enables obtaininga focal spot smaller than the scalar limit for the infinitelynarrow ring

Note that theminimal size of FWHM = 0329120582 is attainedwhen the annular aperturersquos width is Δ

119898= 02 (Figure 2)

This value is approximately 12 times larger than the spot sizederived in the previous section and employed in the estimateof (32)

Figure 3 shows the plot of the normalized focal pointintensity against the annular aperturersquos width (Δ) withoutand with a radial phase jump The vertical dotted lines markthe annular aperturersquos width at which the focal spot of sizeFWHM = 038120582 is formed for a conventional annular

0 02 04 06 08

Δ

1

05

1198680

Figure 3 Normalized focal point intensity as a function of theannular aperturersquos width (Δ) without (solid bold line) and with aradial phase jump (dotted line)

aperture the width is Δ = 006 and for aperture with theintroduced phase jump the width is Δ = 065 Ring sizeof Δ = 006 is rather narrow Thus for practical reasonswe decided to start from this value rather than the limit ofFWHM = 036120582 It is seen that when the phase jump wasintroduced the focal point intensity generated with the widerannular aperture is several times larger than that generatedwith the conventional narrow ring

Table 1 gives the numerical simulation results when aradially polarized beam is tightly focusing (119877 = 100120582 119891 =101120582) and a narrow annular aperture with the phase jumpof (16) is used Δ is the annular slit width 119868

0is the total

intensity in the focus 119878 is the ratio of the total intensity ofthe first side lobe to that of the central focal spot FWHMis the central focal spot diameter in terms of full-width half-maximum intensity and FWHM

119911is the central spot diameter

at half-maximum intensity of the longitudinal componentTable 1 suggests that the smallest central focal spot

(FWHM = 0329120582 see 2nd row of Table 1) has been attainedwithout an unacceptable increase in the side lobes (119878 lt

03) However there is an essential loss of the central spotenergy with the intensity 119868

0being half of that provided by the

narrow annular aperture (1st row of Table 1)The longitudinaldistribution suggests that the focal spot becomes tighter dueto the destructive interference of the beams transmitted bydifferent aperturersquos rings

However by increasing the pupilrsquos width the central spotenergy can be significantly increased with spot size smallerthan that generated with the narrow ring In particular atΔ = 06 (3rd row of Table 1) the focal spot is slightly smaller(FWHM = 0371120582) whereas the intensity is sim6 times largerwhen compared with that generated by the narrow annularaperture A disadvantage consists in somewhat higher sidelobes (119878 = 035)

For the full-aperture case with 120587-phase jump (Δ = 1 4throw of Table 1) what we observe is not only a larger focalintensity but also lower side lobes In this case the focal spotsize (FWHM = 0454120582) does not exceed the Airy patternrsquosscalar limit

Results for the case of full-aperture illumination withoutany apodization are shown (5th row of Table 1) for compar-ison it is possible to see that the focal spot size (FWHM =

0544120582) exceeds the Airy patternrsquos scalar limit

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Engineering

Because Δ lt 1 the smallest radius in (21) is attained atΔ rarr 0 being equal to

120588

jumpmin 119886 =

120582119891

120587119877radic3

asymp 0184

120582119891

119877

(22)

Note that the estimate in (22) is understated since withthe quadratic approximation of the Bessel function in (19)the resulting value appears to be 20 lower than the truevalueTherefore we will consider the following value as moreaccurate

120588

jumpmin = 12120588

jumpmin 119886 = 022

120582119891

119877

(23)

This estimate predicts that the radius should be 28 timessmaller than for the full-aperture lens and 17 times smallerthan for the narrow annular aperture

The estimates for the focal spot diameter and area at half-maximum intensity are as follows

FWHMjumpmin = 026

120582119891

119877

(24)

HMAjumpmin = 0053(

120582119891

119877

)

2

(25)

Note that in this case the focal spot diameter FWHMin (24) appears to be larger than the radius measured asfar as the zero-intensity point (23) This situation differsfrom the one occurred for the first- and zero-order Besselfunctions for which the spot size FWHM was smaller thanthe zero-intensity radius showing that the central peak wallsare steeper

3 Analysis of the Focal Spot SizeDecrease When a Radially PolarizedBeam Is Tightly Focused

When doing the tight focusing of a radially polarized beamwith the use of a narrow annular pupil themajor contributionto the focal plane has been known to be from the longitudinal119864-field component [1ndash3] The significant predominance ofone component over the other vector field componentscreates a situation similar to the scalar regime Thus scalartheory instruments can be successfully utilized producingfairly accurate results The tightly focused focal spot usuallyappears to be larger in size than the scalar estimate as differentcomponents really contribute to the focal spot

Regarding the above considerations only the longitudinalcomponent will be taken into account when analyzing thetight focusing of radially polarized light

For a narrow annular pupil we obtain [9]

119864

119911

Δ(120588) = int

1

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

asymp

Δ(1 minus Δ2)

2

[1 minus (1 minus Δ2)

2

]

14

1198690(119896120588 (1 minus

Δ

2

))

(26)

Thus similar to the scalar case (see (15)) the focal spotsize is determined by the first zero of the zero-order Besselfunction Hence with decreasing annular pupilrsquos width Δ rarr0 we obtain in the limit

120588

119911ringmin =

038

NA120582 997888997888997888997888997888rarr

NArarr1120588

119911ringlim = 038120582 (27)

The focal spot size can be further decreased withoutan essential side-lobe increase based on the destructiveinterference if a 120587-radian phase jump is introduced along thecentral radius of the narrow annular aperture (16) Then thefield distribution in the focal plane is given by

119864

119911

120591(120588) = int

1minusΔ2

1minusΔ

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

minus int

1

1minusΔ2

119909

2

(1 minus 119909

2

)

minus14

1198690(119896120588119909) d119909

(28)

To calculate the integrals in (28) we write down

119864

119911

120591(120588) asymp 119869

0(119896120588 (1 minus

3Δ

4

))

1

(1 minus 3Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1minusΔ2

1minusΔ

minus 1198690(119896120588 (1 minus

Δ

4

))

1

(1 minus Δ4)

times [minus

2

3

119909

2

(1 minus 119909

2

)

34

minus

8

21

(1 minus 119909

2

)

74

]

1003816100381610038161003816100381610038161003816

1

1minusΔ2

(29)

Then retaining the Δ terms up to the second order weobtain

119864

119911

120591(120588) asymp minusΔ

34

(0213 + 0386Δ minus 1139Δ

2

)

minus

119896

2

120588

2

4

(0213 + 0734Δ minus 2175Δ

2

)

(30)

From (30) an estimate for the focal spot radius then takesthe form

120588119911jump asymp

120582

120587

radic

0213 + 0386Δ minus 1139Δ

2

0213 + 0734Δ minus 2175Δ

2

(31)

with the minimum attained at Δ119898= 0168

Thus the minimal estimate of the central spot radius isgiven by

120588119911jump (Δ119898) asymp 0301120582 (32)

4 Numerical Results

Although the estimates discussed in Section 3 are approxi-mate they enable us to analyze in which way the focal spot

Journal of Engineering 5

05

045

04

035

0 02 04 06 08

Δ

FWH

M120582

Figure 2 The focal spot size (FWHM) as a function of the annularaperturersquos width (Δ) without (dashed line) and with (dashed-dottedline) a radial phase jump

size and intensity depend on the annular aperturersquos width Foranalytical calculations it was convenient to use the middleradius of the ring We understand that the radius of phasejump is additional parameter for optimization of the focaldistribution however experience of other researchers [4 6ndash8 19ndash21] shows that even a plenty of free parameters doesnot allow to reduce the focal spot simultaneously keepingenergy Therefore we have decided to vary only the width ofthe ring with the middle 120587-phase jump This limitation maybe convenient for the experimental realization

In this section the sharp focusing of a radially polarizedbeam is numerically simulated using the Richards-Wolfintegral relations [23] for an aplanatic focusing system

Figure 2 shows the focal spot size (FWHM) as a functionof the annular aperture width (Δ) for an element without andwith the radial phase jump of (16) The dotted horizontalline denotes the scalar limit for the lens (FWHM = 051120582)whereas the solid line denotes the scalar limit for the infinitelynarrow ring (FWHM = 036120582)

Figure 2 suggests that with a simple annular aperture togo beyond the lens scalar limit it is necessary to block a lenscentral portion of radius 043119877 (Δ = 057) which amountsto sim20 of the lens area Note that all 119864-field componentsare accounted for in the total intensity Meanwhile the resultscorresponding to the infinitely narrow aperture (FWHM =

036120582) remain unattained even at a very small width (at Δ =001 FWHM = 0367120582)

If there is a radial phase jump (16) there is no need toblock any part of the lens to go beyond the scalar limit At thesame time blocking the central rays on the radius 046119877 (Δ =054 with sim20 of the area got covered) enables obtaininga focal spot smaller than the scalar limit for the infinitelynarrow ring

Note that theminimal size of FWHM = 0329120582 is attainedwhen the annular aperturersquos width is Δ

119898= 02 (Figure 2)

This value is approximately 12 times larger than the spot sizederived in the previous section and employed in the estimateof (32)

Figure 3 shows the plot of the normalized focal pointintensity against the annular aperturersquos width (Δ) withoutand with a radial phase jump The vertical dotted lines markthe annular aperturersquos width at which the focal spot of sizeFWHM = 038120582 is formed for a conventional annular

0 02 04 06 08

Δ

1

05

1198680

Figure 3 Normalized focal point intensity as a function of theannular aperturersquos width (Δ) without (solid bold line) and with aradial phase jump (dotted line)

aperture the width is Δ = 006 and for aperture with theintroduced phase jump the width is Δ = 065 Ring sizeof Δ = 006 is rather narrow Thus for practical reasonswe decided to start from this value rather than the limit ofFWHM = 036120582 It is seen that when the phase jump wasintroduced the focal point intensity generated with the widerannular aperture is several times larger than that generatedwith the conventional narrow ring

Table 1 gives the numerical simulation results when aradially polarized beam is tightly focusing (119877 = 100120582 119891 =101120582) and a narrow annular aperture with the phase jumpof (16) is used Δ is the annular slit width 119868

0is the total

intensity in the focus 119878 is the ratio of the total intensity ofthe first side lobe to that of the central focal spot FWHMis the central focal spot diameter in terms of full-width half-maximum intensity and FWHM

119911is the central spot diameter

at half-maximum intensity of the longitudinal componentTable 1 suggests that the smallest central focal spot

(FWHM = 0329120582 see 2nd row of Table 1) has been attainedwithout an unacceptable increase in the side lobes (119878 lt

03) However there is an essential loss of the central spotenergy with the intensity 119868

0being half of that provided by the

narrow annular aperture (1st row of Table 1)The longitudinaldistribution suggests that the focal spot becomes tighter dueto the destructive interference of the beams transmitted bydifferent aperturersquos rings

However by increasing the pupilrsquos width the central spotenergy can be significantly increased with spot size smallerthan that generated with the narrow ring In particular atΔ = 06 (3rd row of Table 1) the focal spot is slightly smaller(FWHM = 0371120582) whereas the intensity is sim6 times largerwhen compared with that generated by the narrow annularaperture A disadvantage consists in somewhat higher sidelobes (119878 = 035)

For the full-aperture case with 120587-phase jump (Δ = 1 4throw of Table 1) what we observe is not only a larger focalintensity but also lower side lobes In this case the focal spotsize (FWHM = 0454120582) does not exceed the Airy patternrsquosscalar limit

Results for the case of full-aperture illumination withoutany apodization are shown (5th row of Table 1) for compar-ison it is possible to see that the focal spot size (FWHM =

0544120582) exceeds the Airy patternrsquos scalar limit

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 5

05

045

04

035

0 02 04 06 08

Δ

FWH

M120582

Figure 2 The focal spot size (FWHM) as a function of the annularaperturersquos width (Δ) without (dashed line) and with (dashed-dottedline) a radial phase jump

size and intensity depend on the annular aperturersquos width Foranalytical calculations it was convenient to use the middleradius of the ring We understand that the radius of phasejump is additional parameter for optimization of the focaldistribution however experience of other researchers [4 6ndash8 19ndash21] shows that even a plenty of free parameters doesnot allow to reduce the focal spot simultaneously keepingenergy Therefore we have decided to vary only the width ofthe ring with the middle 120587-phase jump This limitation maybe convenient for the experimental realization

In this section the sharp focusing of a radially polarizedbeam is numerically simulated using the Richards-Wolfintegral relations [23] for an aplanatic focusing system

Figure 2 shows the focal spot size (FWHM) as a functionof the annular aperture width (Δ) for an element without andwith the radial phase jump of (16) The dotted horizontalline denotes the scalar limit for the lens (FWHM = 051120582)whereas the solid line denotes the scalar limit for the infinitelynarrow ring (FWHM = 036120582)

Figure 2 suggests that with a simple annular aperture togo beyond the lens scalar limit it is necessary to block a lenscentral portion of radius 043119877 (Δ = 057) which amountsto sim20 of the lens area Note that all 119864-field componentsare accounted for in the total intensity Meanwhile the resultscorresponding to the infinitely narrow aperture (FWHM =

036120582) remain unattained even at a very small width (at Δ =001 FWHM = 0367120582)

If there is a radial phase jump (16) there is no need toblock any part of the lens to go beyond the scalar limit At thesame time blocking the central rays on the radius 046119877 (Δ =054 with sim20 of the area got covered) enables obtaininga focal spot smaller than the scalar limit for the infinitelynarrow ring

Note that theminimal size of FWHM = 0329120582 is attainedwhen the annular aperturersquos width is Δ

119898= 02 (Figure 2)

This value is approximately 12 times larger than the spot sizederived in the previous section and employed in the estimateof (32)

Figure 3 shows the plot of the normalized focal pointintensity against the annular aperturersquos width (Δ) withoutand with a radial phase jump The vertical dotted lines markthe annular aperturersquos width at which the focal spot of sizeFWHM = 038120582 is formed for a conventional annular

0 02 04 06 08

Δ

1

05

1198680

Figure 3 Normalized focal point intensity as a function of theannular aperturersquos width (Δ) without (solid bold line) and with aradial phase jump (dotted line)

aperture the width is Δ = 006 and for aperture with theintroduced phase jump the width is Δ = 065 Ring sizeof Δ = 006 is rather narrow Thus for practical reasonswe decided to start from this value rather than the limit ofFWHM = 036120582 It is seen that when the phase jump wasintroduced the focal point intensity generated with the widerannular aperture is several times larger than that generatedwith the conventional narrow ring

Table 1 gives the numerical simulation results when aradially polarized beam is tightly focusing (119877 = 100120582 119891 =101120582) and a narrow annular aperture with the phase jumpof (16) is used Δ is the annular slit width 119868

0is the total

intensity in the focus 119878 is the ratio of the total intensity ofthe first side lobe to that of the central focal spot FWHMis the central focal spot diameter in terms of full-width half-maximum intensity and FWHM

119911is the central spot diameter

at half-maximum intensity of the longitudinal componentTable 1 suggests that the smallest central focal spot

(FWHM = 0329120582 see 2nd row of Table 1) has been attainedwithout an unacceptable increase in the side lobes (119878 lt

03) However there is an essential loss of the central spotenergy with the intensity 119868

0being half of that provided by the

narrow annular aperture (1st row of Table 1)The longitudinaldistribution suggests that the focal spot becomes tighter dueto the destructive interference of the beams transmitted bydifferent aperturersquos rings

However by increasing the pupilrsquos width the central spotenergy can be significantly increased with spot size smallerthan that generated with the narrow ring In particular atΔ = 06 (3rd row of Table 1) the focal spot is slightly smaller(FWHM = 0371120582) whereas the intensity is sim6 times largerwhen compared with that generated by the narrow annularaperture A disadvantage consists in somewhat higher sidelobes (119878 = 035)

For the full-aperture case with 120587-phase jump (Δ = 1 4throw of Table 1) what we observe is not only a larger focalintensity but also lower side lobes In this case the focal spotsize (FWHM = 0454120582) does not exceed the Airy patternrsquosscalar limit

Results for the case of full-aperture illumination withoutany apodization are shown (5th row of Table 1) for compar-ison it is possible to see that the focal spot size (FWHM =

0544120582) exceeds the Airy patternrsquos scalar limit

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Engineering

Table 1 Numerical simulation results

Transmission functionLongitudinal intensity

distribution (negative image)3120582 times 6120582

Transverseintensity distributionin the focus (negative

image) 3120582 times 3120582

Intensity profiles for the field componentsin the focus (dashed line transverse solid

line longitudinal dotted line total)

00 20 40 60 80

119877

1

Ring Δ = 0051198680 = 0044 119878 = 0162

004

002

0minus1 minus05 0 05

119909 120582FWHM = 0378120582 FWHM119911 = 0372120582

0 20 40 60 80

1

0

minus1

Jump Δ = 02119877 1198680 = 002 119878 = 0279

minus1 minus05 00

001

002

05119909 120582

FWHM = 0329120582 FWHM119911 = 0329120582

0 20 40 60 80

1

0

minus1

Jump Δ = 06119877 1198680 = 025 119878 = 0351

minus1 minus05 00

01

02

05119909 120582

FWHM = 0371120582 FWHM119911 = 0363120582

0 20 40 60 80119877

1

0

minus1

Jump Δ = 11198680 = 0656 119878 = 0171

minus1 minus05 00

03

06

05119909 120582

FWHM = 0454120582 FWHM119911 = 0412120582

00 20 40

Full-aperture illumination

60 80119877

1

1198680 = 1 119878 = 016

minus1 minus05 00

05

1

05119909 120582

FWHM = 0544120582 045120582 FWHM119911 =

5 Conclusions

The analysis of the sharp focusing has shown that thedependence of the focal spot size on the narrow annularpupilrsquos width is different for the scalar and vector models

In the scalar model when describing the focusing basedon the Fourier transform the smallest focal spot is attainedwhen the peripheral ring width tends to zero while thefocus intensity drops quadratically with decreasing ringwidth

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 7

In the vector model of the sharp focusing of the radiallypolarized beam (Debyersquos approximation) the smallest focalspot is obtained at a fixed annular aperturersquos width equal to20 of the full-aperture radius The amplitude at the focus isproportional to the ringwidth to the power 34 It is one of themost important analytical results which allows us to estimatethe energy that is stored in a central spot

It has been analytically and numerically shown that thedestructive interference between the beams passing different-phase rings enables going beyond a scalar limit correspond-ing to the zero-order Bessel function (FWHM = 036120582) Atighter focal spot (FWHM = 033120582) is attained at a sacrificeof a lower focal intensity and higher side lobes with the latternot becoming larger than 30 of the central peak

We have numerically shown that by introducing a radial120587-phase jump along the aperturersquos middle radius the scalarlimit can be broken we have got FWHM = 0454120582 (for theAiry pattern FWHM = 051120582) In themeantime blocking outthe central ringswith a 046119877 stop (coveringsim20of the area)enables obtaining a smaller focal spot (FWHM = 0371120582)

with intensity nearly 6 times higher than for simple narrowring The lack is higher side-lobes reaching up to 35 of thecentral peak intensity

Certainly reduction of the focal spot size just by factor165 (from FWHM = 0544120582 for open aperture to FWHM =

0329120582 for ring with jump) is accompanied with huge lossesof energy in this spot (by factor 50) It is a significantdisadvantage for the applications based on high energy suchas a nonlinear scanning microscopy Nevertheless opticalmicroscopy applications can tolerate very high losses scan-ning microscopes can work with only a few photons persecond providing sensitivity to tens orders below of laserpower [11]

Acknowledgments

The work was financially supported by the Ministry ofEducation and Science of the Russian Federation (Project8231) and by the Russian Foundation for Basic Research(Grants 13-07-00266 and 13-07-97004-r povolzhe a)

References

[1] S Quabis R Dorn M Eberler O Glockl and G LeuchsldquoFocusing light to a tighter spotrdquo Optics Communications vol179 no 1 pp 1ndash7 2000

[2] R Dorn S Quabis and G Leuchs ldquoSharper focus for a radiallypolarized light beamrdquo Physical Review Letters vol 91 no 23 pp2339011ndash2339014 2003

[3] C J R Sheppard and A Choudhury ldquoAnnular pupils radialpolarization and superresolutionrdquo Applied Optics vol 43 no22 pp 4322ndash4327 2004

[4] L E Helseth ldquoMesoscopic orbitals in strongly focused lightrdquoOptics Communications vol 224 no 4ndash6 pp 255ndash261 2003

[5] Y Kozawa and S Sato ldquoSharper focal spot formed by higher-order radially polarized laser beamsrdquo Journal of the OpticalSociety of America A vol 24 no 6 pp 1793ndash1798 2007

[6] C-C Sun and C-K Liu ldquoUltrasmall focusing spot with a longdepth of focus based on polarization and phase modulationrdquoOptics Letters vol 28 no 2 pp 99ndash101 2003

[7] H Wang L Shi B Lukyanchuk C Sheppard and C TChong ldquoCreation of a needle of longitudinally polarized lightin vacuum using binary opticsrdquo Nature Photonics vol 2 no 8pp 501ndash505 2008

[8] S N Khonina and S G Volotovsky ldquoControlling the contri-bution of the electric field components to the focus of a high-aperture lens using binary phase structuresrdquo Journal of theOptical Society of America A vol 27 no 10 pp 2188ndash2197 2010

[9] S N Khonina A V Ustinov and E A Pelevina ldquoAnalysis ofwave aberration influence on reducing focal spot size in a high-aperture focusing systemrdquo Journal ofOptics vol 13 no 9 ArticleID 095702 2011

[10] G Toraldo di Francia ldquoDegrees of freedomof an imagerdquo Journalof the Optical Society of America A vol 59 no 7 pp 799ndash8041969

[11] F M Huang and N I Zheludev ldquoSuper-resolution withoutevanescent wavesrdquo Nano Letters vol 9 no 3 pp 1249ndash12542009

[12] T R M Sales and G M Morris ldquoDiffractive superresolutionelementsrdquo Journal of the Optical Society of America A vol 14no 7 pp 1637ndash1646 1997

[13] S N Khonina and I Golub ldquoEnlightening darkness to diffraction limit and beyond comparison and optimization ofdifferent polarizations for dark spot generationrdquo Journal of theOptical Society of America A vol 29 no 7 pp 1470ndash1474 2012

[14] J Bewersdorf A Egner and S W Hell ldquo4pi-confocalmicroscopy is coming of agerdquo Imaging ampMicroscopy vol 4 pp24ndash25 2004

[15] L E Helseth ldquoBreaking the diffraction limit in nonlinearmaterialsrdquo Optics Communications vol 256 no 4ndash6 pp 435ndash438 2005

[16] N Bokor and N Davidson ldquoTight parabolic dark spot withhigh numerical aperture focusing with a circular 120587 phase platerdquoOptics Communications vol 270 no 2 pp 145ndash150 2007

[17] V V Kotlyar S N Khonina A S Melekhin and V A SoiferldquoFractional encoding method for spatial filters computationrdquoAsian Journal of Physics vol 8 no 3 pp 273ndash286 1999

[18] S N Khonina S A Balalayev R V Skidanov V V KotlyarB Paivanranta and J Turunen ldquoEncoded binary diffractiveelement to form hyper-geometric laser beamsrdquo Journal of OpticsA vol 11 no 6 Article ID 065702 7 pages 2009

[19] W Chen and Q Zhan ldquoThree-dimensional focus shaping withcylindrical vector beamsrdquo Optics Communications vol 265 no2 pp 411ndash417 2006

[20] X Gao J Wang H Gu and W Xu ldquoFocusing properties ofconcentric piecewise cylindrical vector beamrdquo Optik vol 118no 6 pp 257ndash265 2007

[21] K Huang P Shi X-L Kang X Zhang and Y P Li ldquoDesign ofDOE for generating a needle of a strong longitudinally polarizedfieldrdquo Optics Letters vol 35 no 7 pp 965ndash967 2010

[22] B Tian and J Pu ldquoTight focusing of a double-ring-shapedazimuthally polarized beamrdquo Optics Letters vol 36 no 11 pp2014ndash2016 2011

[23] B Richards and EWolf ldquoElectromagnetic diffraction in opticalsystems II Structure of the image field in an aplanatic systemrdquoProceedings of the Royal Society A vol 253 no 1274 pp 358ndash379 1959

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Engineering

[24] M Abramowitz and I A Stegun Handbook of MathematicalFunctions Courier Dover Publications North ChelmsfordMass USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of