Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials

0.87 MB·

by  Lei Wei

Shaping the focal field of radially/azimuthally polarized
phase vortex with Zernike polynomials
LEI WEI* AND H. PAUL URBACH1
1DepartmentofImagingPhysics,DelftUniversityofTechnology,Lorentzweg1,2628CJ,Delft,TheNetherlands
*Correspondingauthor:[email protected]
CompiledJanuary22,2016
6
1
0
2 1. INTRODUCTION

n
Opticalfocalfieldshapingbyengineeringthepolarization,amplitudeandphaseontheexitpupilof
a
J anopticalsystem[1–3],especiallywiththehelpofspatiallightmodulators[4–6],hasattractedlotsof

1 attentionsinrecentyears. Andthiscanfindapplicationsinmanyareas. Forinstance,abeamwith
2
phasevorticeorazimuthal polarization ontheexitpupilcangeneratehollow focus[7,8],withnull

] intensityinthecenter,whichcanbeusedtotrapabsorbingparticles,coldatoms[9]andalsoinhigh
s
c resolutionSTEDmicroscopy[10]. Theabilitytocontrolthelocalelectric/magneticfielddistribution
i
t in the focal region makes it possible to determine the orientation of a single optical emitter[11] or
p
o excitecertainresonancesofaquantumemitter[12]whichisnotpossibleotherwise. Therearevarious
.
s methods to shape the focal field of a radially/azimuthally polarized pupil, including designing
c
binary phase masks[13] or reversing the electrical dipole array radiation[14–16]. Tight focusing
i
s
y behaviorofpolarizedphasevortexbeams[17,18]hasbeeninvestigatedtoachievesharpresolution
h [19, 20], andtodemonstratespin-to-orbitalangular momentumconversion[21], etal.. In aprevious
p
publication[22] , we have demonstrated a method to generate elongated focal spot using Zernike
[

polynomials. Zernike polynomials forms a complete and orthogonal set of polynomials on a unit
1
v circle,whichformanidealsettostudythepupilengineeringofnormallycircularlyaperturedoptical
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imagingsystem. ByprecalculatingthefocalfieldsofZernikepolynomials,wecanoptimisethefocal
1
6 fieldbyonlyoptimisingtheZernikecoefficients. However,inpaper[22],scalardiffractionintegralis
5 considered,thusneitherpolarizationnorphasevorticesareinvestigated. Inthispaper,wefirststudy
0
the focal field properties of radially/azimuthally polarized Zernike polynomials. Based on this, we
.
1
appliedthepupilengineeringmethodusingcomplexZernikepolynomialstoshapethefocalfield
0
6 ofaphasevortexwithradialandazimuthalpolarizationstogenerateanlongitudinallypolarized
1
axially uniform hollow focus, a transversally polarized elongated spot, and circularly polarized
:
v
multiplefocalspot.
i
X
r
a
2. FOCAL FIELD OF RADIAL AND AZIMUTHAL POLARIZED ZERNIKE POLYNOMI-
ALS
ThecomplexZernikepolynomials Zm(ρ,φ) formacompletesetoforthonormalpolynomialsdefined
n
onaunitdisc:
Zm(ρ,φ) = R|m|(ρ)eimφ, (1)
n n
Rm(ρ) = ∑p (−1)s(n−s)! ρn−2s, (2)
n s!(q−s)!(p−s)!
s=0
1
Figure1: TheDebyediffractionmodel. Lightisfocusedfromtheexitpupilplane (x ,y ) tothefocal
s s
region. Thefocalcoordinatesis (x ,y ,z ) withthegeometricalfocalpointasitsorigin.
f f f
where p = 1(n − |m|), q = 1(n + |m|), n − |m| ≥ 0 and even, (ρ,φ) are the normalised polar
2 2
coordinatesontheexitpupilplane. Aswecanseefromitsdefinition, Zm isaphasevortexwiththe
n
topologicalcharge m.
ThepolarizedpupilfunctioncanbedecomposedintoZernikepolynomials:
E(cid:126) (ρ,φ) = ∑ eˆ(ρ,φ)βmZm(ρ,φ), (3)
s s n n
n,m
forradialpolarizationthereholdsthat eˆ(ρ,φ) = eˆ (ρ,φ) = (cosφ,sinφ),whileforazimuthalpolar-
s ρ
ization eˆ(ρ,φ) = eˆ (ρ,φ) = (−sinφ,cosφ).
s φ
Zernikepolynomialshavebeenveryusefultoanalyzetheaberrationsofanopticalsystemandhave
beenused asa basic setin theExtendedNijboer Zerniketheory,in whichasemi-analytical solution
ofthefocalfieldofeachZernikepolynomialontheexitpupilisderived[23,24].
Inthispaperwecomputethethrough-focusfieldsofeachradiallyorazimuthallypolarizedZernike
polynomialbynumericallycomputingthevectorialDebyediffractionintegral[25]withFastFourier
Transformmethod[26]:
√
iR (cid:90)(cid:90) cosθeˆ Zm(ρ,φ)eikzz
(cid:126)Enm(x ,y ,z ) = − f n (4)
f f f f 2π Ω kz
×exp[−i(k x +k y )]dk dk
x f y f x y
where R is the focal length, the geometrical cone Ω is defined by {(k ,k ) : k2 + k2 ≤ k2NA2},
x y x y 0
(cid:113)
k = k2n2−k2 −k2, k = 2π/λ,and(x ,y ,z ) isapointinthefocalregion,withgeometricalfocal
z 0 x y 0 f f f
pointasorigin. Forradialpolarization, eˆ = (cosφcosθ,sinφcosθ,sinθ),forazimuthalpolarization,
f
eˆ = (−sinφ,cosφ,0).
f
(cid:126)
SincethevectorialDebyediffractionintegralisalinearoperatorofpupilfield E ,thefocalfields
s
canbeexpressedbysummationofthefocalfieldsofeachcomplexZernikemode(cid:126)Enm withthesame
f
setofZernikecoefficients βm whichasoccurintheexpressionofthepupilfield:
n
(cid:126)E (x ,y ,z ) = ∑ βm(cid:126)Enm(x ,y ,z ), (5)
f f f f n f f f f
n,m
2
Onceasetofcoefficients βm for(cid:126)Enm isfoundtogiveadesiredfocaldistribution,thepupilfieldthat
n f
givesthisfocaldistributioncanbeobtainedfromEq.(3).
In order to gain more insight into the focal fields of radially and azimuthally polarized Zernike
polynomial,werewriteEq.(4)usingpolarcoordinates:
√
iRs2 (cid:90) 1
(cid:126)Efnm(ρf,φf,zf) = − λ0 (1−ρ2s20)−1/4e−ik0zf 1−ρ2s20 (6)
0
(cid:90) 2π
×R|m|(ρ)ρdρ eˆ eimφei2πρρfcos(φf−φ)dφ,
n f
0
where, (ρ ,φ ,z ) arethecylindricalcoordinatesofapointinthefocalregion,and s isthenumerical
f f f 0
aperture.
Integralsoverazimuthalangle φ canbecomputesanalytically. Thecartesiancomponentsofthefocal
electricfieldoftheradiallypolarizedpupilfieldaregivenby:
iπRs2 (cid:90) 1
Enm(ρ ,φ ,z ) = − 0(−i)m+1eimφf ρdρ (7)
f,x f f f λ
√ 0
×(1−ρ2s20)1/4e−ik0zf 1−ρ2s20R|nm|(ρ)
(cid:104) (cid:105)
× eiφfJm+1(2πρρf)−e−iφfJm−1(2πρρf) ,
iπRs2 (cid:90) 1
Enm(ρ ,φ ,z ) = − 0(−i)m+2eimφf ρdρ (8)
f,y f f f λ
√ 0
×(1−ρ2s20)1/4e−ik0zf 1−ρ2s20R|nm|(ρ)
(cid:104) (cid:105)
× eiφfJm+1(2πρρf)+e−iφfJm−1(2πρρf) ,
i2πRs2 (cid:90) 1 s ρ
Enm(ρ ,φ ,z ) = − 0(−i)meimφf 0 (9)
f,z f f f λ 0 (1−ρ2s2)1/4
√ 0
×e−ik0zf 1−ρ2s20Rn|m|(ρ)Jm(2πρρf)ρdρ.
Similarly,theCartesiancoordinatesoftheazimuthalcomponentsofthefocalfieldofanazimuthal
polarizedpupilfieldis:
iπRs2 (cid:90) 1
Enm(ρ ,φ ,z ) = − 0(−i)meimφf ρdρ (10)
f,x f f f λ
√0
×(1−ρ2s20)−1/4e−ik0zf 1−ρ2s20Rn|m|(ρ)
(cid:104) (cid:105)
× eiφfJm+1(2πρρf)+e−iφfJm−1(2πρρf) ,
iπRs2 (cid:90) 1
Enm(ρ ,φ ,z ) = − 0(−i)m+1eimφf ρdρ (11)
f,y f f f λ
√ 0
×(1−ρ2s20)−1/4e−ik0zf 1−ρ2s20Rn|m|(ρ)
(cid:104) (cid:105)
× eiφfJm+1(2πρρf)−e−iφfJm−1(2πρρf) ,
From these expressions, we can see that the focal field of a radially polarized phase vortex of
topologicalcharge m hasthefollowingproperties:
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1. Its transverse focal field component is proportional to (1− ρ2s2)1/4, while its longitudinal
0
component is proportional to s ρ/(1−ρ2s2)1/4. This means that for pupil fields with lower
0 0
NAof whichlowerspatialfrequencypartdominates, thetransversallypolarizedcomponent
is stronger; while for a pupil function with higher NA of which the higher spatial frequency
part dominates, the longitudinal component is stronger. This gives a way to modulate the
polarizationinthefocalregion.
2. Thetransversefocalfieldcomponentofradiallypolarized Z±1 isnon-zeroontheopticalaxis,
n
duetotheexistenceof J intheintegral.
0
|m|
3. Theradialpolynomial R modifiesthefocalfielddistributioninsuchawaythatforafixed m
n
andn > |m|,twofocalpointsappearsymmetricallytothefocalplanez = 0,andasnincreases,
f
theseparationbetweenthesetwofocialsoincreases,asshowninFig. 2a.
4. Thewidthofthehollowspotof I = |E |2 generatedbyradiallypolarized Zm increaseswith
z z n
increasing |m| (cid:54)= 0,asshowninFig. 2b.
a.
1
Z1
1
0.8 Z91
Z1
17
0.6 Z217
0.4
0.2
0
−15 −10 −5 0 5 10 15
b. zf/l
1
0.9 Z11
0.8 Z22
0.7
0.6
0.5
0.4
0.3
0.2
0.1
−05 0 5
x/l
f
Figure2: (a). Normalizedz-componentofthe focalintensityforvariousradiallypolarized Z1 along
n
z axis,lateralpositionisatitsintensitymaximainthetransversalplane,thenumericalapertureis
f
0.99. (b). Normalizedz-componentofthefocalintensityforvariousradiallypolarizedpupilfield Zm
m
along x axis, z position is at its intensity maxima in the longitudinal plane, the numerical aperture is
f
0.99.
While the focal field of an azimuthally polarized phase vortex of topological charge m has the
followingproperties:
1. Theazimuthalpolarizationstateiskeptinthefocalregionforazimuthallypolarizedpupilfield
Z0 ,whichcanbeeasilyderivedfromEq. 10andEq. 11;
n
2. Thetransversefocalfieldcomponentof Z±1 isnon-zeroontheopticalaxis,duetotheexistence
n
of J in the integral, this can be confirmed by previous work by L.E. Helseth[17]. And from
0
Eq. 10 and Eq. 11, the focal fields on the optical axis, i.e. ρ = 0, is circularly polarized. For
f
4
azimuthallypolarizedpupilfield Z±1,thedoublefocusphenomenaalongtheopticalaxisalso
n
appearsfor n > 1andas n increases,theseparationbetweenthefociincreases.
3. FOCAL FIELD SHAPING
Inthispaper,weonlyuse Zm withthesamephasevortexcharge m forshapingthefocus,because
n
thismakestheintensitycircularlysymmetricaroundtheopticalaxis. Thebasisoffocalfieldsona
givengrid inthe (x ,z )-planecorresponding totheradially andazimuthally polarized pupilfields
f f
(cid:126)
E arepre-calculatedusingtheFFTandstoredinthedatabase. Thefocalfieldofageneralpolarized
f
electricpupilfieldcanbewrittenasasalinearcombinationofthesebasicfunctions:
(cid:126)E = ∑βm(cid:126)Enm, (12)
f n f
p
where m isfixedand p = (n−|m|)/2.
Firstlyweusethetransversecomponentofthefocalintensity I|m|m(x ,0) orlongintudinalcompo-
t f
nentoffocalintensity I|m|m(x ,0) todeterminethelateralposition x where I|m|m(x ,0) or I|m|m(x ,0)
z f 0 t f z f
is at its maximum. (cid:126)Efnm(zf)|xf=x0 is then used to get a desired through-focus intensity distribution,
wheretheZernikecoefficients βm aretheoptimisationvariables.
n
Due to the fact that (cid:126)Enm has been precalculated, the focal field can be calculated rather fast. A
f
random search of βm is applied in order to get a good starting point of the optimisation, then a
n
Nelder-Mead simplex direct search method is used to optimize βm in order to get a minimum of
n
(cid:12) (cid:12)
(cid:12)|∑ βm(cid:126)Enm|2− I (cid:12). Once the Zernike coefficients are found, the desired pupil function can be
(cid:12) p n f target(cid:12)
calculated easily by E(cid:126) = eˆ(ρ,φ)∑ βmZm, where the polarisation of the pupil is either radially
s s n,m n n
polarizedthat eˆ(ρ,φ) = eˆ (ρ,φ) = (cosφ,sinφ),orazimuthallypolarizedthat eˆ(ρ,φ) = eˆ (ρ,φ) =
s ρ s φ
(−sinφ,cosφ). Afewexamplesaregiveninthefollowingsection:
A. Longitudinally polarized elongated subwavelength hollow channel
Anelongatedsubwavelengthhollowchannelcanbeusedtotrapabsorbingnano-particles[8]and
coldatoms[9]. Itcanalsobeusedasanexcitationbeamforhighresolutionfluorescencemicroscope
likeSTED[10],toexcitefluorescenceemissionoveranextendeddepthoffocus,whilekeepingahigh
resolution.
Thez-component Enm ofthefocusedfieldduetoaradiallypolarized Zm pupilfieldforgivenfixed
f,z n
m (cid:54)= 0,vanishesontheopticalaxis,whilethetopologicalchargeoftheincidentbeammiskeptinEnm.
f,z
Howeveritstransversecomponentsarenon-zeroontheopticalaxis. Intheoptimisationprocess,we
thereforetrytomaximizethez-componentbyminimizing|I /I − I |,where I = |∑ β1En1|2,
z max target z p n fz
I isthemaximumintensityinthexzplane,andthetargetfunction I isarectanglefunction
max target
along z axiswithvalue1between |z | < z ,where2×z isthedesireddepthoffocus.
f f max max
Weuse a combination of 14radially polarizedZernike polynomials eˆ [Z1]| on theexit
ρ n n=1,3,5,...,27
pupil of an optical system with NA=0.99. As shown in Fig.3, with a set of Zernike coefficients
[β1]| = [-0.1437, -0.3, -0.398, -0.589, -0.568, -0.824, -0.674, -0.892, -0.785, -0.743, -0.754, -0.73,
n n=1,3,5,...,27
-0.182, -0.829], we get a hollow spot with lateral resolution(FWHM) of 0.28λ, and its FWHM of
focal depth is about 16λ, with a good uniformity over a range of 12λ around the focal plane. For
5
ahighNAsystemwithNA=0.99,thisisequivalentto12.24Rayleighunit λ/NA2. Thisfocalfield
islongitudinallypolarizedalongtheopticalaxis. Thecorrespondingoptimizedradiallypolarized
pupilfieldwithm=1togeneratethiselongatedhollowspotisgivenby E(cid:126) (ρ,φ) = ∑ eˆ β1Z1(ρ,φ),
s n ρ n n
asshowninFig.4.
a. b.
l
lx/f y/f
x/l
z/l f
c. f
1
0.5
0
−10 −5 0 5 10
z/l
d. f
1
I
t
I
z
0.5
0
−5 0 5
x/l
f
Figure3: Elongatedsubwavelengthhollowspotbyanshapedradiallypolarizedphasevortexwith
m = 1ontheexitpupil,NA=0.99. (a).Nomalizedfocalfielddistributioninthe x −z plane;(b)Focal
f f
field in the transversal plane; (c). Focal intensity distribution along the z axis;, while x is at the
f f
maxmimumintensityonthetransversalplane(d). Focalintensitydistributionalong x axis.
f
Figure 4: The pupil field of an shaped radially polarized phase vortex with m = 1 to generate
elongatedsubwavelengthhollowspotinFig. 3.
The ratio of the transverse andlongitudinal components can be adjusted by pupil engineering of
thelowspatialfrequencyandhighspatialfrequencycomponents. Onecommonlyusedmethodis
touseannularaperture. Forinstance,aradiallypolarizedannularpupilwithphasevortex m = 1:
A(0.99 < ρ < 1) = eˆ exp(iφ) and 0 elsewhere in the exit pupil can also generate an elongated
ρ
subwavelength hollow channel, as shown in Fig.5, but the uniformity of the spot along z dimension
6
ismuchworsethantheresultsobtainedwithourmethod.
a.
l
/
xf
z/l
f
b.
1
0.8
0.6
0.4
0.2
0
−10 −8 −6 −4 −2 0 2 4 6 8 10
z/l
f
Figure5: Elongatedsubwavelengthhollowspotbyaradiallypolarizedannularaperturewithphase
vortex m = 1ontheexitpupil,NA=0.99. (a).Nomalizedfocalfielddistributioninthe x −z plane;
f f
(b). Focal intensity distribution along the z axis;, while x is at the maxmimum intensity on the
f f
transversalplane
B. On-axis transversl polarized focal field with radially polarized pupil Z1
n
The ability of modulating the local polarization of the focal field can be used to determine the
orientationofasingleopticalemitter[11].Thiscanbeachievedbyengineeringthepupilfield.
In the first example, we achieved an elongated hollow spot with radially polarized pupil field
Z1, for which at the focal points where the intensity is maximum, the focal electric field is po-
n
larized primarily along the z-direction. With the same set of Zernike polynomials, we can also
maximize the tranverse compoents by optimizing the β1 to minimize |I /I − I |, where
n t max target
I = |∑ β1En1|2+|∑ β1En1|2, I isthe maximumtotalintensityin thexzplane, thetargetfunc-
t p n fx p n fy max
tion I isarectanglefunctionalong z axiswithvalue1between |z | < z ,where2×z isthe
target f f max max
desireddepthoffocus..
Weuse a combination of 14radially polarizedZernike polynomials eˆ [Z1]| on theexit
ρ n n=1,3,5,...,27
pupil of an optical system with NA=0.9. As shown in Fig.6, with the set of Zernike coefficients
[β1]| =[-0.2632,0.7,-0.677,-0.09, 0.982,-1.273,0.413,0.553,-1.167,0.208, -0.194,0.135,-2.244,
n n=1,3,5,...,27
0.586],wegetanelongatedfocalspotalongtheopticalaxiswithstrongpolarizationinthetransverse
direction,whileitsz-polarizedcomponentismuchweaker.
Thepupilfieldofanshapedradiallypolarizedphasevortexwith m = 1togeneratethiselongated
tranverselypolarizedfocalspotisthenobtainedfrom E(cid:126) (ρ,φ) = eˆ ∑ β1Z1(ρ,φ),asshowninFig.7.
s ρ n n n
7
a.
l
/
xf
z/l
f
b.
1
0.8
0.6
0.4
0.2
0
−10 −8 −6 −4 −2 0 2 4 6 8 10
z/l
c. f
1
I
0.8 t
I
0.6 z
0.4
0.2
0
−5 −4 −3 −2 −1 x0/l 1 2 3 4 5
f
Figure 6: Elongated focal spot with stronger transversal polarisation component by an shaped
radiallypolarizedphasevortexwith m = 1ontheexitpupil,NA=0.9. (a).Nomalizedfocalintensity
distributioninthe x −z plane;(b). Focalintensitydistributionalongtheopticalaxis;(c). Transversal
f f
andlongintudinalcomponentsofthefocalintensitydistributionalong x axis.
f
Figure 7: The pupil field of an shaped radially polarized phase vortex with m = 1 to generate
elongatedtransversallypolarizedfocalspotinFig. 6.
8
C. Multiple circularly polarized focal spots along the optical axis with azimuthally polarized Z1
n
Multiplesubwavelengthexcitationfocalspotalongtheopticalaxiscanbeveryusefulforfluorescence
microscopy, as it can simultaneously excite fluorescence at multiple planes in the specimen. P.P.
Mondal proposed a method using interference of two counter-propagating long depth of focus PSFs
to generate multiple excitation spot PSF[27]. However, this approach utilizes a 4pi configuration,
which requires more complex optical setup than just shaping the pupil field. From Section 2, we
know that for Zernike polynomials with a fixed m and n > |m|, dual focus appear symmetrically
to the focal plane z = 0, and as n increases, the separation between these two focus increases. By
f
assigningcertainZernike coefficients toZernikepolynomials Zm withdifferent n,we cangenerate
n
multipleaxialfocalspotsbyshapingtheexitpupil.
Weuseacombinationof14azimuthallypolarizedZernikepolynomials eˆ [Z1]| onthe
φ n n=1,3,5,...,27
exitpupilofanopticalsystemwithNA=0.99. AsshowninFig.8,withthesetofZernikecoefficients
[β1]| =[-0.222, 0.228,0.4726, -0.106,0.4315, 0.853,-0.169, -0.0438,0.9424, 0.482,-0.85, 0.824,
n n=1,3,5,...,27
0.557, 0.115], we can get 8 axial focal spots, each with a lateral resolution of 0.216λ and separated
about 1.7λ between adjacent spots along the optical axis; The maximum intensity of these focal
spots of the azimuthally polarized phase vortex of m = 1 lies on the optical axis and contains no
longitudinalcomponent.
The pupil field of an shaped azimuthally polarized phase vortex with m = 1 to generate this
multiplefocalspotsalongopticalaxisisobtainedfrom E(cid:126) (ρ,φ) = ∑ eˆ β1Z1(ρ,φ),asshowninFig.9.
s n φ n n
4. CONCLUSION
In thispaper,the focal fieldof radially/azimuthally polarizedcomplex Zernike polynomials is inves-
tigated. Basedontheunderstandingofthis,amethodtoshapethefocalfieldofradially/azimuthally
polarizedphasevorticesisproposed. Withpre-calculationofthefocalfieldofeachradially/azimuthally
polarized Zernikepolynomial, the optimization variables are reduced to thenumber of Zernike coef-
ficientsbeingused. Threeresultsfromthismethodaregiveninthispaper,includingalongitudinally
polarizedsubwavelengthhollowfocalspotwithadepthoffocusupto12λ andalateralresolution
of0.14λ forasytemwithNA=0.99. Byengineeringtheratioofhighandlowspatialfrequencies,a
tranversallypolarizationdominatedelongatedfocalspotisobtainedforradiallypolarizedZernike
polynomials with topological charge of 1. We also obtained multiple subwavelength focal spots
alongtheopticalaxis. Withtheseresults,pupilshapingofazimuthally/radiallypolarizedoptical
vorticescanachieveresolutionimprovementofopticalsystemandcanshapethepolarizationand
intensity sistribution, in the focal region, which may result in interesting applications in the area of
highresolutionfluorescencemicroscopy,opticaltrapping,etc.
9
a.
l
/
xf
z/l
f
b.
11
00..55
−−001100 −−55 00 55 1100
z/l
c. 11 f
00..55
−−0055 00 55
x/l
f
Figure 8: Multiple focal spot along optical axis by an shaped azimuthally polarized phase vortex
with m = 1ontheexitpupil,NA=0.99. (a).Nomalizedfocalfielddistributioninthe (x ,z ) plane;(b).
f f
Focalintensitydistributionalongtheopticalaxis;(c). Focalintensitydistributionalongthe x axis, z
f f
isatoneofthelocationswherethespothasmaximumonaxisintensity.
Figure 9: The pupilfieldofanshaped azimuthally polarizedphasevortexwith m = 1togenerate
multiplefocalspotalongopticalaxis.
10