984 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Passilly et al.
Simple interferometric technique for generation ofa radially polarized light beam
Nicolas Passilly, Renaud de Saint Denis, and Kamel Aït-Ameur
Centre Interdisciplinaire de Recherche Ions Lasers, Equipe Lasers, Instrumentation Optique et Applications, Unitémixte de Recherche 6637, Commissariat à l’Energie Atomique, Centre National de la Recherche Scientifique,
Ecole Nationale Supérieure d’Ingénieurs de Caen, Université de Caen, 6 Boulevard-Maréchal Juin,F-14050 Caen, France
François Treussart, Rolland Hierle, and Jean-François Roch
Laboratoire de Photonique Quantique et Moléculaire, Unité mixte de Recherche 8537, Centre National de laRecherche Scientifique, Ecole Normale Supérieure de Cachan, 61 avenue du Président Wilson,
F-94235 Cachan, France
Received August 27, 2004; revised manuscript received November 19, 2004; accepted December 2, 2004
. INTRODUCTIONhe polarization vector, i.e., the direction of the electriceld, of a laser beam is one of its important properties. Itan be homogeneous as linear or circular polarization ornhomogeneous as in radially or azimuthally polarizedeams. While homogeneously polarized light is wellnown and widely used, inhomogeneous polarization isess studied, although it has recently attracted increasednterest in a variety of applications. In the following, weill consider only radially polarized beams that havexial symmetry in the sense that at each point of the laseream’s cross section, the polarization vectors point radi-lly toward the optical axis of the beam. Such a polariza-ion state has a remarkable property in that it yields athe focus of a high-numerical-aperture lens a strong lon-itudinal component oriented along the lens axis.1–3 It ismportant to note that the longitudinal electrical fieldomponent is a nonpropagating component that oscillatest the incident beam frequency and exists only near theocus. Another interesting property of such polarizedeams is that the spot size associated with the longitudi-al component is much smaller than in the case of linearolarization.4,5
Radially polarized beams have many applications suchs (i) in laser cutting of metals, where they improve thefficiency in comparison with usual polarization6,7; (ii) inetermination of single fluorescent moleculerientation8,9 in optimizing photon collection efficiency ofolecular-based single-photon sources10,11; (iii) in
harged-particle acceleration12; (iv) in guiding or trappingf particles13,14; and finally, (v) in scanning opticalicroscopy.15
Radial polarization can be produced by two methods.he first is a single-pass, polarization-beam shapingased either on the interferometric combination of or-hogonally polarized beams15,16 or on transmitting a lin-arly polarized Gaussian beam through a twisted nematiciquid crystal17 or a subwavelength grating.18 The secondechnique consists of introducing into a laser cavity aomponent that favors a fundamental mode and is radi-lly polarized.19,20
In this paper we consider a technique that is both dif-ractive and interferometric for converting a linearly po-arized TEM00 laser beam into a radially polarized beamith uniform azimuthal intensity distribution. We study
he influence of different parameters on the quality of theolarization transformation in order to optimize it.
. THEORY OF INTERFEROMETRICENERATION OF A RADIALLY POLARIZEDEAMhe use of two orthogonally polarized Hermite–GaussianEM01 and TEM10 beams to synthesize a radially polar-
zed beam by linear combination is an old idea suggestedy Kogelnik and Li21 as far back as 1966. Let us brieflyecall the mathematics that describes the synthesis of aadially polarized beam. If we consider the electric fieldistribution E01 sE10d of a linearly polarizedEM01 sTEM10d beam:
E10sx,yd =E0
2
x
Wux expF− Sx2 + y2
W2 DG , s1d
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Passilly et al. Vol. 22, No. 5 /May 2005/J. Opt. Soc. Am. A 985
E01sx,yd =E0
2
y
Wuy expF− Sx2 + y2
W2 DG , s2d
here W is the 1/e intensity beam radius of the Gaussianerm and ux suyd nil is a unit vector along the xsyd axis,imple vector addition yields a total field given by
ETsx,yd = E10 + E01 =E0
WexpF− Sx2 + y2
W2 DGfxux + yuyg.
s3d
he last term in brackets shows that any point of coordi-ates sx ,yd of the cross section of the total beam has a po-
arization vector that points radially. Consequently, thentensity is a function only of radius r= sx2+y2d1/2: Theolarization vector is radial everywhere. The coherent ad-ition of TEM01 and TEM10 beams gives a resultingoughnut beam radially polarized as shown in Fig. 1(c).In the following experiment it is necessary to have a di-
gnostic of radial polarization. Let us note that after pass-ng a linear polarizer, the radially polarized beam yieldsn intensity distribution made of two lobes aligned alonghe polarizer axis on both sides of the beam center (seeig. 2), whatever the orientation of the polarizer may be.It is important to note that coherent superposition of
he two orthogonally polarized beams shown in Figs. 1(a)nd 1(b) leads to a linear polarization that points radially,rovided that their relative phase shift is null or equal to
ig. 1. Intensity distributions of the coherent superposition ofesulting beam is a doughnut beam radially polarized.
ig. 2. Intensity distributions of a radially polarized beam after5°, (c) horizontally.
multiple of 2p. Consequently, the optical path lengthsetween the two arms of the interferometer must be care-ully balanced as will be shown below. We will consider inection 3 the transformation of a TEM00 into a TEM10eam by using a binary diffractive optical element.We want to transform an incident Gaussian beam into
wo orthogonal beams having electrical fields as close asossible to that given by Eqs. (1) and (2) so that their co-erent superposition leads to a vector polarization point-
ng everywhere radially. In the case where the beamransformation TEM00 → TEM10 is not perfect, some re-ions of the beam cross section have a polarization thateparts from the radial one, ending in a smaller longitu-inal field strength at the lens focus.The beam transformation TEM00→TEM10 is achieved
y use of a simple binary diffractive optical element,hich is referred to hereafter as the phase step. Its com-lex transmittance is given by
tsx,yd = H1 for y . y0
exps− ifd for y ø y0. s4d
pplications of tailored laser beams such as the radiallyolarized one we are interested in usually require focus-ng at the focus of a converging lens. Consequently,nowledge of their far-field intensity profile is the rel-vant feature. However, to fully characterize the tailoredeam, we also provide its near-field pattern.
M10, (b) TEM01 beams that are orthogonally polarized. (c) The
ng a linear polarizer having its axis oriented (a) vertically, (b) at
(a) TE
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986 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Passilly et al.
When a circular Gaussian beam of radius W is incidentn the phase step, it undergoes a phase shift f over a frac-ion of its section. The key parameters governing thehape of the far-field intensity distribution are f and Dy0 /W.Let the incident field on the phase step be a collimated
ircular Gaussian beam of wavelength l whose unit-mplitude distribution is given by
Einsx,yd = expF− Sx2 + y2
W2 DG . s5d
he incident Gaussian beam on the phase step has a ra-ius W=0.4 mm. In the following we will consider thearticular case for which f=p. This phase-shift value hasiven interesting results in the transformation of aaussian beam diffracted either by a circular-phase aper-
ure in the free-propagation case22 or inside an aperturedavity.23,24 Such phase aperture transforms a Gaussianeam into a super-Gaussian profile of order 6 in the focallane of a converging lens.22 Note that other f values alsoield interesting properties when two phase steps areombined to make a phase slit. It has been shown that for=p /2 this kind of diffractive optical element can trans-
orm an elliptic Gaussian beam in the near field into a cir-ular Gaussian beam in the far field.25
The following analysis is done within the scalar diffrac-ion theory framework. The diffracted field distributionsxP ,yP ,zPd at point P of transverse coordinates sxP ,yPd in
he plane located at distance zP from the phase step isiven by the Fresnel–Kirchhoff two-dimensional integrals
EsxP,yP,zPd =i
lE
−`
+`E−`
+`
tsx,ydEinsx,ydexps− ikrd
rdxdy,
s6d
here r= fsxP−xd2+ syP−yd2+zP2g1/2.
If one assumes that point P is not very distant from theaxis, then the paraxial approximation can be applied.he variable r in the term exps−ikrd /r of Eq. (6) can bepproximated by:
r < zP in the denominator,
r < zP +sxP − xd2 + syP − yd2
2zPin the numerator.
n Eq. (6) the variables x and y can then be separated.he normalized intensity distribution is therefore ex-ressed as the product of two one-dimensional integralsnd is proportional to
IsxP,yP,zPd = UE−`
+`
FsxddxFE−`
+`
tsx,ydFsyddyGU2
, s7d
here the function F is given by
Fsud = expS−u2
W2DexpF− iksuP − ud2
2zPG , s8d
ith uP=xP syPd when u=x syd.The diffraction integral given by Eq. (7) is calculated
umerically by using a FORTRAN 77 routine in both theear-field szP=0.2 md and the far-field szP=5 md regions.ote that the notion of near- and far-field regions is usu-lly attached to the transverse size of the diffracting ob-ect as in the case of a circular aperture, for instance. Inur case the diffracting object has a lateral extent that isnfinite. Consequently, the space beyond the phase stepannot be divided into near- and far-field regions in thease where the incident wave was plane. However, it ishe Rayleigh range zR [see Eq. (13) below] of the incidentaussian beam that defines the boundary of the near field
zP,zRd; the far field corresponds to zP.zR.
. EXPERIMENTAL SETUPhe phase step shown in Fig. 3 is made of a transparentlass on which a thin polymer film (refractive index n) iseposited. The film thickness h is calculated to introducephase shift f=2phsn−1d /l=p. Note that the phase dis-
ontinuity is set parallel to the x axis and is located at po-ition y=y0. The wavelength used in the calculations andxperiments is l = 532 nm.
The phase step is fabricated by first spincoating a 540-m-thick polymethylmethacrylate polymer layer on a mi-roscope glass slide. This thickness produces the desiredphase shift at l = 532 nm. The polymer layer is then re-oved on half of the slide by an oxygen-reactive ion-
tching plasma attack, which stops when it reaches thelass substrate. This method yields a steep edge step,hose height grows to its nominal value within a distance
f about 10 µm.The setup used for the transformation of a Gaussian
eam linearly polarized into a beam radially polarized byombining the two orthogonally polarized TEM10 andEM01 beams is shown in Fig. 4. The linear combining
echnique incorporates a Mach–Zehnder interferometerased on two polarizing beam splitters PBS, one with po-arizing axis horizontal, the other with it vertical.15 A CWoubled Nd:YVO4 laser beam polarized at 45° relative tohe vertical axis is expanded in a telescope and then sento the first PBS, which splits the beam into two equal-mplitude, Gaussian beams orthogonally polarized (oneertically, the other horizontally). Each of these twoeams is incident on a phase step having its discontinuityine oriented normal to the beam polarization axis; i.e.,he two phase steps are set perpendicular. In one of the
ig. 3. Schematic of the phase step of height h etched in a thinolymer film having a refractive index n and deposited on a glasslate. This step produces a phase shift f=2phsn−1d /l betweenhe two parts of the laser beam illuminating the polymer edge.
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Passilly et al. Vol. 22, No. 5 /May 2005/J. Opt. Soc. Am. A 987
nterferometer arms a variable delay is introduced toompensate for any path difference between the tworms. This is made of two glass plates tilted in oppositeirections for transversal walk-off compensation. At thenterferometer output, a second PBS combines the beamsransformed by the two phase steps. Consequently, at thenterferometer output one gets two beams that look likeEM10 and TEM01 beams that overlap along their propa-ation axis.
Experimentally, the diffracted beam patterns are re-orded by a CCD array. To restrict the transversal extentf the far-field pattern at zP=5 m, a converging lens (focalength f = 0.75 m) is placed after the phase step and quite
ig. 4. Experimental setup to produce a radially polarized beamrom the superposition of the two beams coming out of orthogo-ally oriented f=p phase steps. This setup relies on a Mach–ehnder interferometer. A CW doubled Nd:YVO4 laser is firstpatially filtered by passing through a pinhole and then sent tohe interferometer after collimation. T, pinhole; PBS, polarizingeam splitter; PS, f=p phase step; VD, variable delay made ofwo tilted BK7 glass plates. Note that it is important to set thewo phase steps at the same distance from the interferometerutput port.
ig. 5. Two-dimensional beam intensity patterns of (a) incident Gan beam diffracted by the phase step positioned on the optical a
lose to it. The distance zp8 between the lens and the CCDs adjusted so that the relation
1
zp8−
1
zp=
1
fs9d
s fulfilled. As a consequence, at zp=5 m s0.2 md is associ-ted a distance zp8=0.65 m s0.15 md. Note that we have toake into account the presence of this additional lens inhe calculation of the diffracted field. The latter can be de-ermined by using Eq. (7) provided that tsx ,yd is multi-lied by the complex factor tLsx ,yd associated with theens, where
tLsx,yd = expF− iksx2 + x2d
2fG . s10d
. COMPARISON OF EXPERIMENTAL DATAITH THEORY
he purpose of the phase step is to create a sign change ofhe incident beam phase distribution between the twoides of the phase discontinuity line. It is clear fromimple Fourier optics arguments that a p phase step inhe near field yields a minimum of amplitude in the fareld. The experimental two-dimensional intensity distri-ution of the Gaussian incident beam is shown in Fig.(a). The resulting far-field diffracted-beam two-imensional intensity distribution shown in Fig. 5(b) sug-ests that the vertical centered phase step has trans-ormed the Gaussian incident beam into a TEM10ermite–Gauss beam.Figure 6 shows the experimental and theoretical inten-
ity distributions together with a fit (dotted curve) by theerfect TEM01 beam intensity distribution resulting fromhe diffraction of a Gaussian beam on a horizontal phase step. The diffracted intensity is plotted as a func-ion of yp, i.e., in a direction perpendicular to the phaseiscontinuity line. In fact, there is no diffraction along the
an beam, (b) far-field diffraction pattern of this collimated Gauss-=0d.
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988 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Passilly et al.
axis, so that Isxpd has the same Gaussian shape as thatf the incident intensity distribution.
Figure 6 also shows that the diffracted beam intensityransverse pattern has ripples in the near field and be-omes smoother in the far field. If one looks at the dif-racted intensity profile around the yp=0 axis, one canote that the propagation does not affect the intensity
ig. 6. Theoretical (left side) and experimental (right side) x=0eam (radius W=0.4 mm) diffracted on a centered sD=0d phaseurve is a fit by the intensity distribution I01= uE01u2 of a perfect
ig. 7. Theoretical (left side) and experimental (right side) diffrracted on a phase step f=p displaced from the incident beam oiffraction intensity patterns.
rofile. In other words for D=0, the diffracted beam be-aves as an eigenmode of the whole optical system in theense that it keeps its profile as it propagates; only itsidth is affected. Consequently, the use of a p phase step
or the transformation of a Gaussian beam into a TEM01eam is quite satisfactory, especially close to the yp=0xis. However, far from this axis (i.e., in the wings of the
section intensity diffraction patterns of a collimated Gaussian=p in (a) the near-field and (b) the far-field regions. The dottedbeam.
patterns of a collimated Gaussian beam (radius W=0.4 mm) dif-axis sD=0.7d. Two cases are displayed, (a) near-field, (b) far-field
, crossstep fTEM
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Passilly et al. Vol. 22, No. 5 /May 2005/J. Opt. Soc. Am. A 989
iffracted beams), the diffracted field distribution is lessroperly adjusted by a TEM01. This is responsible for dif-culties in the transformation from linear to radial polar-
zation as will be shown in Section 5. A possible solutionor reducing this drawback would be to introduce an ap-ropriate apodizing filter in the near field of the diffractedeam to reduce the ripples shown in Fig. 6(a). However,uch a method also has some drawbacks: (i) Its fabrica-ion is relatively complex, (ii) it is probably characterizedy a low damage threshold, which is not compatible withigh power pulses, and (iii) the apodizing filter is adaptedo one incident beam radius W only.
If we now consider a case for which the phase disconti-uity line is not transversally centered relative to theropagation axis, D=0.7 (see Fig. 7), one observes that theiffracted beam is no longer an “eigenmode”: Its inten-ity profile radically evolves from the near to the far field.ince we are interested in producing a focused beam withhigh longitudinal field component, we need to know
hether there is an optimal position relative to the inter-erometer output port at which to place the focusing lens.his position would correspond to a maximum of linearlyolarized incident beam power converted into radially po-arized beam power. Indeed, it has been shown in Fig. 6hat the transformation into a TEM10 or a TEM01 beamchieved by a phase step is not perfect, and this results inolarization that is not radial everywhere over the beamross section. Since the degree of beam transformationmperfection is different from the near to the far fieldFig. 6), one can expect that the degree of radial polariza-ion in a given transverse plane should depend on its dis-ance zp from the phase steps. The latter are set at theame distance from the interferometer output port.
We now define a criterion to evaluate theoretically theegree of radial polarization as a function of the distance.irst we calculate the fields E1sxp ,yp ,zpd and2sxp ,yp ,zpd, applying Eq. (6) to the vertical and horizon-
al polarization directions emerging from the two perpen-icular phase steps, and we deduce the total field as
ETsxp,yp,zpd = E1sxp,yp,zpd + E2sxp,yp,zpd. s11d
n the plane z=zp the field distribution ETsxp ,yp ,zpd isalculated over a 200 3 200 point grid. In each square ofoordinate sxp ,ypd we compare the achieved polarizationo the perfect radial polarization: The angle difference isoted as Du. We then determine the power PDu that flowshrough all the meshes having a Du less than a givenalue (1° or 5°). Finally, the degree of radial polarization hf the resulting field in plane z=zp is defined as
h = PDu/PT, s12d
here PT is the total beam power integrated over thehole grid square.Variation of h as a function of the distance from the
hase steps is shown in Fig. 8 in the two cases Du,1° andu,5° (incident Gaussian beam radius W=1 mm). Weote that there is an optimal distance zPmax from thehase steps where the degree of radial polarization heaches a maximum. If one wants to produce a longitudi-al field component as intense as possible, one has to sethe focusing lens at this position.
Note that the optimal distance zPmax is a sensitive func-ion of the incident beam radius W on the phase step. Inig. 9, zPmax variation has been plotted as a function of2. zPmax appears to be a linear function of W2, and the
otted curve shows a fit to the following equation:
zR = pW2/lM2, s13d
hich represents the Rayleigh range of the GaussianEM00 beam transformed by a phase step (TEM10 beam)nd M2 is its beam propagation factor. Let us recall thathe beam propagation factor26 of a Hermite–GaussEMm0 or TEM0m beam is M2=2m+1. The fit shown inig. 9 yields a value M2=2.98 very close to M2=3, which
s the TEM10 or TEM01 beam propagation factor. Figure 9lso shows that the optimal distance varies by more thanne order of magnitude, from 0.3 to 8 m, when W variesrom 0.4 to 2 mm. From a practical point of view, this wideange of values of the optimal distance can be reduced bysing an imaging lens as shown in Section 3.Another important practical concern is how to deter-ine precisely where the optimal plane zPmax is located.his can be done experimentally without applying any po-
arization analysis, just by considering the scalar diffrac-ion pattern observed with the CCD array. When the lat-er is set in the plane zPmax the intensity distribution is aniform doughnut beam as shown in Fig. 10(a); other-ise, the diffraction pattern is horned, as shown in Fig.0(b). At the optimal position, i.e. zP=zPmax, we observexperimentally a uniform doughnut as shown in Fig.
ig. 8. Variation of the degree of radial polarization h as a func-ion of the distance zP from the phase steps in the two cases Du1° and Du,5°. The incident Gaussian beam radius is W1 mm.
ig. 9. Variation of the optimal distance zPmax as a function of2 (squares), where W is the radius of the incident Gaussian
eam. The dotted line represents a fit with use of the linearariation of the Rayleigh range zR=pW2 / slM2d versus W2, yield-ng M2=3.
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990 J. Opt. Soc. Am. A/Vol. 22, No. 5 /May 2005 Passilly et al.
1(a). We verified that this doughnut beam is effectivelyadially polarized by recording the beam pattern afterassing through an analyzer oriented at 45° [see Fig.1(b)]. As expected, the latter pattern shows two lobeslong the analyzer axis. Note that such result is obtainednly if the two arms of the Mach–Zehnder interferometerre properly balanced by tilting the two glass plates (VD)hown in Fig. 4; otherwise, the TEM01 and TEM10 beamsnterfering at the interferometer output yield an ellipti-ally polarized beam characterized by a spread intensityattern without the distinctive lobes of Fig. 11(b).
. CONCLUSIONSn interferometric technique for converting a linearly po-
arized Gaussian beam into a radially polarized doughnuteam has been considered theoretically and experimen-ally. This technique involves the coherent summation ofwo orthogonally polarized TEM01 and TEM10 beams ob-ained from the transformation of a TEM beam by using
ig. 10. Two-dimensional beam intensity pattern of the beam emteps in a plane located at distance zP from the output port. The inorresponds to the optimum position zPmax=2 m for the best conv
ig. 11. Experimental beam pattern recorded at position zP=zPmriented at 45°.
00
simple binary diffractive optical element. The latter is ahase step, which consists of a thin film of a polymer de-osited on half of a glass plate. The phase step introducesp phase shift on half of the incident Gaussian beam,
nd the resulting diffracted beam has two lobes that lookike TEM10 beams. The characterization of this beam was
ade in the near-field region and in the far-field regionnd showed that the quality of the transformationEM00→TEM01 depends on the distance at which the ob-ervation is done. It results that the quality of the polar-zation conversion linear → radial is expected to dependn the distance. We have shown that the degree of radialolarization is maximum at a given distance from the in-erferometer output port that depends on the size of thencident beam at the interferometer input port. As a con-equence, it is at this particular position that one has toet the focusing lens if one is interested in producing a fo-us with a high longitudinal field component. We haveound that the optimal distance can be determined experi-entally without applying any polarization analysis but
from the Mach–Zehnder interferometer including the two phaseGaussian beam radius is W=1 mm. Two cases are displayed: (a)of linear into radial polarization; (b) corresponds to zP=5 m.
otal intensity, (b) intensity distribution after passing an analyzer
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Passilly et al. Vol. 22, No. 5 /May 2005/J. Opt. Soc. Am. A 991
erely by considering the scalar diffraction pattern ob-erved with a CCD array.
CKNOWLEDGMENTShis work was performed in the framework of the Contrate Plan Etat-Région Pôle Image, Technologie de’Information et de la Communication. The authors ac-nowledge the support of the Délégation Régionale à laecherche et à la Technologie Basse Normandie, the Ré-ion Basse Normandie, and the Fond Européen de Dével-ppement Régional.
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