Second-harmonic generation of polarization singularities

1640 OPTICS LETTERS / Vol. 27, No. 18 / September 15, 2002

Second-harmonic generation of polarization singularities

Isaac Freund

Jack and Pearl Resnick Advanced Technology Institute, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

Received June 7, 2002

Second-harmonic generation was studied theoretically for the vector singularities (daisy modes) of linearlypolarized light and for the elliptic singularities (C points) of elliptically polarized light. Topological chargedoubling for C points and daisy modes, similar to that found for vortices, was found. Unlike for vortices,however, it was found that for both C points and daisy modes the sign of the charge is reversed; for C pointsthe photon spin or handedness (right or left) is also reversed; and for daisy modes the number of intensitypetals is not doubled. These findings are all unexpected because neither charge nor spin nor orbital angularmomentum is conserved for C points, whereas a daisy mode in which the number of intensity petals does notequal twice the magnitude of the charge is anomalous. © 2002 Optical Society of America

OCIS codes: 190.0190, 190.2620, 190.7070, 260.0260, 260.2130, 260.5430.

Second-harmonic generation (SHG) of vortexfields1 (single-component, or scalar, fields) is wellknown.2 – 4 In this fundamental nonlinear process thevortex charge is doubled but the sign of the charge ispreserved.

Linearly or elliptically polarized two-componentoptical fields support singularities of polarization.Although it is of obvious interest, the SHG of thesesingularities does not appear to have been discussedpreviously.

Linearly polarized vector fields have singularitiesat which the direction of the electric (and also of themagnetic) vector becomes indeterminate. Importantexamples are vector solitons5,6 and the daisy modes ofvertical-cavity surface-emitting lasers.7,8 The two or-thogonal f ield components of a daisy mode (a typicalvector singularity) can be written as

ex � axrn cos�nu�exp�2br2� ,

ey � 6ayrn sin�nu 1 g�exp�2br2� , (1)

where amplitudes ax and ay are assumed positive andwhere b determines the width of the Gaussian hostbeam. Because the angle that the vectors make withthe x axis is Q � arctan�ey , ex�, setting ax � ay andg � 0 yields Q �6nu. Thus the vectors surroundingthe central singularity of a daisy mode rotate (wind)through DQ � 2qp, where q �6n is the topologicalcharge. This charge is conserved under smooth, con-tinuous deformation, and by continuity it remains un-changed when ax fi ay . For nonzero g, the magnitudeof the charge remains n, but the sign of the charge be-comes 6signum�cos�g��.

The intensity distribution of a daisy mode consists ofa ring of 2n maxima (bright petals) that alternate with2n saddles. When ax � ay and g � 0, the maxima andthe saddles have the same height, the gaps between thepetals f ill in, and the daisy mode degenerates into avector doughnut. Some important properties of daisymodes are illustrated in Figs. 1(a)–1(c).

The important point polarization singularities ofelliptically polarized light (ellipse f ields) are C points,9

which are isolated points of circular polarization em-

0146-9592/02/181640-03$15.00/0

bedded in a field of elliptical polarization. C pointsare singular because when an ellipse degenerates intoa circle its major axis, and therefore the orientation ofthis axis, becomes undefined. The ellipses surround-ing a C point rotate through 2ICp in one completecircuit about the point, where IC is the singularityindex (topological charge) of the C point.9 Inasmuchas an ellipse returns to itself after rotation by p,in contrast to vortices and daisy modes, which canhave only integer charges, C points can have bothfractional and integer charges: 61�2, 61, 63�2,etc.10 C points are also characterized by their photonspin or handedness h, with h �11 �h �21� for a right-(left-) handed C point.

Elliptically polarized fields can be decomposed intoorthogonal, linearly polarized components ex and eyor, often more conveniently, into right-handed, eR ,and left-handed, eL, circular components. BecauseC points are states of pure circular polarization, aright-handed (left-handed) C point corresponds to avortex, i.e., zero, of eL �eR �.9 The charge of a C pointand the charge qR,L of the underlying vortex of eR,Lare related by10,11

IC �2hR,LqR,L�2 , (2)

where hR �11 and hL �21. Figures 2(a)–2(c) illus-trate some important properties of C points.

SHG of vortex f ields is usually phasematched.2 – 4 For daisy modes and for ellipsefields, however, phase matching can be difficult,as both orthogonal components of the f ield must besimultaneously phase matched. Accordingly, herewe discuss for simplicity non-phase-matched SHGin which all waves are ordinary waves propagatingparallel to the optic axis of a uniaxial crystal.Assuming LiNbO3 for the nonlinear medium,12

a mode-locked Nd:YAG pump laser operated at1.064 mm with an average power of 1 W, and a256 3 256 pixel Stokes parameter13,14 imagingdetector with 2% absolute detection efficiency, weestimate a count rate of �4000 �counts�pixel��s atthe peak of a Giordmaine–Maker fringe.15,16 Thedesired input daisy mode or C point structure

© 2002 Optical Society of America

September 15, 2002 / Vol. 27, No. 18 / OPTICS LETTERS 1641

Fig. 1. Daisy modes, Eqs. (1) with ax � ay , g � p�8,b � 3. (a), (b), (d), (e) Vector f ields. (c), (f ) Intensi-ties. The vectors of a positive (negative) daisy modewith charge 1jqj (charge 2jqj) rotate by 2jqjp in thesame (opposite) direction as a path that encloses thesingularity. These rotations may be most easily fol-lowed along the outer boundary for each figure. Forpositive daisy modes the streamlines form 2�jqj 2 1�vector petals; for negative modes, 2�jqj 1 1� hyperbolicsectors. The intensity distribution of a daisy modenormally contains 2jqj maxima (intensity petals) alter-nating with 2jqj saddle points. (a)– (c) Daisy modes atinput frequency v. (a) q �12 mode with two vectorpetals. (b) q � 22 mode with six hyperbolic sectors.(c) Intensity of the modes in (a) and (b), showing theexpected four maxima and four saddle points surroundingzero intensity at the origin. (d)– (f ) Output daisy modesat 2v produced by SHG of the adjacent input daisy modesin (a)– (c). Under SHG, charge q is doubled and reversedin sign. (d) q �24 daisy mode with 10 hyperbolic sectors.(e) q �14 daisy mode with six vector petals. (f ) Instead ofeight maxima and eight saddle points, this anomalous 2vintensity distribution shows four maxima and four saddlepoints at exactly the same locations as the v intensitydistribution in (c). This behavior is characteristic of SHGof real vector f ields, for which the intensity at 2v is thesquare of the intensity at w.

can be imposed on a low-power laser beam byholographic2,17 or other means. This low-power beamcan then be used to seed the high-power input pumplaser.

For all waves propagating parallel to the optic axisin LiNbO3, the components Ex and Ey of the outputsecond-harmonic f ield can be expressed in terms of theinput field components ex and ey as12

Ex � 22Cd22exey , Ey � Cd22�ey2 2 ex2� . (3)

In what follows, we set Cd22 � 1, where C contains thecoherence length and other system parameters.

Inserting a daisy mode with charge q �6n [Eq. (1)]into Eqs. (3) and taking initially for simplicity ax �ay � 1 and g � 0, we have

Ex � 7r2n sin�2nu�exp�22br2� ,

Ey � 2r2n cos�2nu�exp�22br2� . (4)

From Eqs. (4) it follows that the winding angleof the vectors of the output second-harmonicfield is DQ � 74np, so the charge of the outputdaisy mode is Q �22q. By continuity, this result

Fig. 2. C points. (a), (b), (d), (e) Ellipse f ields. (c), (f )Intensity distributions. Charge IC of a C point (centralfilled circles) equals the number of 2p rotations of thesurrounding ellipses along a closed path that encircles thepoint. The surrounding ellipses rotate in the same (oppo-site) direction as the path for a positive (negative) C point.Right-handed, h �11 (left handed, h �21) ellipses areshown by solid (dotted) curves. (a)– (c) C points of in-put ellipse f ields at frequency v created with (a) eR � G2,eL � V1G2, which produces a C point with IC �11�2, h �11[Eq. (2)], and (b) eR � V1G2, eL � G2, which produces aC point with IC �21�2, h �21, where eR and eL are theright and left circular components of the field, respectively,G2 � exp�2br2�, with b � 2, is the Gaussian envelopeof the host beam, and V1 � x 1 iy. (c) Intensity of (a)and of (b). (d)–(f ) C points of output ellipse fields at 2vproduced by SHG of adjacent input C points in (a)– (c).Under SHG the charge IC of a C point is doubled, andits sign is reversed, as is the sign of its handedness h.(d) IC �21, h �21. (e) IC �11, h �11. (f ) Intensity of(d) and (e). Because the input f ields are complex, the out-put intensity in (f) at 2v is not the square of the inputintensity at v in (c), unlike the case for Figs. 1(c) and 1(f).

1642 OPTICS LETTERS / Vol. 27, No. 18 / September 15, 2002

for Q holds also when ax fi ay and g fi 0. Thus, underSHG, the magnitude of the input charge of a vectorsingularity is doubled but, unexpectedly, the sign ofthe charge is reversed.

The input daisy mode with charge q has 2jqj in-tensity petals [maxima; Figs. 1(a)–1(c)]. This is thenormal structure of such a mode. One would there-fore expect the second-harmonic output daisy mode tohave 2jQj � 4jqj intensity petals. This expectation isnot realized, however, and we find instead that theoutput daisy mode has jQj � 2jqj intensity petals atthe same positions as the intensity petals of the inputmode. This unexpected result follows from Eqs. (3),which yield I �2v� � I2�v� for real (but not for com-plex) ex and ey, where I �V� is the intensity at frequencyV. Figures 1(d)–1(f ) show SHG of the daisy modes ofFigs. 1(a)–1(c).

The linearly polarized components cx and cy , andthe right- and left-circularly polarized components cRand cL, of an ellipse field are related by

cx � �cR 1 cL��21�2, cy � i�cR 2 cL��21�2,

(5a)

cR � �cx 2 icy ��21�2, cL � �cx 1 icy ��21�2.

(5b)

First using Eqs. (5a) to express ex and ey in terms of eRand eL, and then using Eqs. (3) together with Eqs. (5b),we obtain for the circular components ER and EL of thesecond-harmonic output field

ER � i�2�1�2eL2, EL �2i�2�1�2eR2. (6)

Excluding an improbable coincidence in which avortex of eR exactly overlaps a vortex of eL, we havefrom Eqs. (2) and (6) the following facts: The inputellipse field has right-handed C points with chargeIC �1qL�2 at the positions of the vortices of eL andleft handed C points with charge IC �2qR�2 at thevortices of eR . In contrast, the output ellipse f ieldhas left-handed C points with charge IC � 2qL atthe positions of the vortices of eL and right-handedC points with charge IC �1qR at the positions of thevortices of eR . Thus, under SHG, the magnitudesof the charges of all C points are doubled, and theirsigns as well as their handedness (photon spin) are re-versed. Figures 2(d)–2(f ) show SHG of the C pointsin Figs. 2(a)–2(c).

During SHG, two input photons with energy h̄vare converted into a single photon with energy 2h̄v,thereby conserving energy. The doubling of vortex

charge and orbital angular momentum18,19 under SHGis often also attributed to conservation laws. Butour results show that, for SHG of daisy modes andC points, neither charge nor photon spin (handedness)nor orbital angular momentum �eR , eL� is conserved.This, of course, does not imply a violation of fun-damental physical laws: The nonlinear mediumabsorbs all the missing angular momentum, whereasconservation of topological charge for vortices, vectorsingularities such as daisy modes, and C pointsneed hold only for smooth, continuous changes suchas free-space propagation and not for discontinuouschanges, such as the generation of free and boundwaves at 2v, that occur at the surfaces of a nonlinearcrystal during SHG.20

I am pleased to acknowledge a useful e-mail corre-spondence with Marat S. Soskin. My e-mail addressis freund@mail.biu.ac.il.

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