Alonso et al. Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. A 691
Nonparaxial fields with maximum jointspatial–directional localization. I. Scalar case
Miguel A. Alonso
The Institute of Optics, University of Rochester, Rochester, New York 14627
Riccardo Borghi
Istituto Nazionale per la Fisica della Materia, Dipartimento di Elettronica Applicata, Università Roma Tre,Via della Vasca Navale, 84 I-00146 Rome, Italy
Massimo Santarsiero
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Università Roma Tre,Via della Vasca Navale, 84 I-00146 Rome, Italy
. INTRODUCTIONncertainty relations play a fundamental role in severalranches of optics. They fix resolution limits for imagingystems in terms of their aperture size or, equivalently,ive the minimum possible angular spread of a (coherentr partially coherent) optical beam once the size of itsource has been set. In particular, as far as the character-zation of beams is concerned, there are several standard
easures of a beam’s transverse spatial and angularpread. Among them, second-moment-based measuresave the important property of being related, within theontext of the scalar paraxial approximation, through thetandard uncertainty relation: The product of the mea-ures of spatial and angular spread has a fundamentalower bound, reached only for paraxial Gaussian beams.n the basis of this fact, Siegman1 defined a measure ofeam quality, proportional to the product of these secondoments. This measure, known as the beam propagation
actor or M2, corresponds to the square of the ratio be-ween the width of the beam and that of a Gaussian beamhat spreads at the same rate under propagation.
Since their introduction, attempts to generalize suchdeas to the case of nonparaxial fields have beenresented.2 These generalizations are important becausepplications like photolithography require systems witharge numerical apertures, where the paraxial approxi-
ation is no longer valid. Further, in applications where amall target must be efficiently irradiated (e.g., in laserusion reactors), it is necessary to use focused fields span-ing solid angles beyond a hemisphere. In these cases, thesual measures of spread are no longer appropriate. Re-
ently, Alonso and Forbes3 defined new measures of spa-ial and angular spread valid for nonparaxial scalarelds, and that satisfy an uncertainty relation analogouso the standard one. These new measures were shown toeduce in the paraxial limit to the standard ones. A keyole in defining such measures is played by the represen-ation of a typical nonparaxial scalar field as a coherentuperposition of plane waves propagating in all direc-ions.
For nonparaxial cylindrical [i.e., two-dimensional (2D)]eams, the relation between spatial and angular spreadsescribed earlier is just a particular physical consequencef an uncertainty relation for periodic functions and theirourier coefficients,4,5 which also has important physical
mplications in quantum optics.6–9 This uncertainty rela-ion has a simple algebraic form that reduces to the oneor Fourier transformation in the limit of high localizationf the periodic function. It is, however, unattainable as anquality except in the limiting cases when one of the twoeasures goes to zero. The true lower bounds for the
pread measures and the periodic functions that attainhem were studied later by Opatrný9 and Forbes et al.10
y means of a variational approach.6 These functionsere found to be solutions of the Mathieu equation.The goal of the present paper is the generalization of
hese ideas to the case of three-dimensional (3D) scalarelds. We shall find that the uncertainty relation for 3Delds found in Ref. 3 can be strengthened. Even thistronger inequality, however, is unattainable as an equal-ty, so we go on to find the fields that achieve the maxi-
um possible joint spatial–directional localization. The
006 Optical Society of America
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692 J. Opt. Soc. Am. A/Vol. 23, No. 3 /March 2006 Alonso et al.
esults we present can be viewed as the first step towardhe study of uncertainty relations for nonparaxial beamsithin an electromagnetic vectorial framework. Such a
tudy, which could have interesting applications in theheory of imaging by optical systems with high numericalperture, is the subject of part II (this issue) of this seriesf papers.
. NONPARAXIAL SCALAR FIELDS,PERATORS, AND AVERAGE VALUESgeneral monochromatic scalar field in three dimensionsith no evanescent components can be written as a sumf propagating plane waves of the form
U�r� =�4�
A�u�exp�iku · r�d�, �1�
here k is the wavenumber, A is the angular spectrum ofhe wave field, and the integral is over the sphere of di-ections of the unit vector u. The angular spectrum,hich depends on only two angles, gives the most com-act representation of a free wave field. For the sake ofimplicity, from now on we set k=1 in the subsequent for-ulas, i.e., all distances are in units of reduced wave-
engths.
. Operatorsome operations acting on a field of the form in Eq. (1)an be translated into operations on its angular spectrum.or example, let us define the direction operator as −i�r,here �r denotes a vector derivative with respect to the
patial coordinates. It is then easy to see that the effect ofhis operator is equivalent to multiplying the angularpectrum by its argument u, i.e.,
− i�rU�r� =�4�
uA�u�exp�iu · r�d�. �2�
It must be noted, however, that not all operations on Uan be translated into operations on A. Consider, for ex-mple, the position coordinate r. It turns out that there iso operation on A that causes an effect equivalent to mul-iplying U by its argument. (This is due to the differentimensionalities of the r and u spaces.) Nevertheless,here are some operations involving r that can be trans-ated into operations on the angular spectrum. One suchxception is the angular-momentum operator, whichhen applied to a field in the position representation isefined in the form
L = − ir � �r. �3�
hen applied to the angular spectrum, this operator cane found to be given by
L = − iu � �u, �4�
here �u is the 2D gradient within the sphere of direc-ions, which corresponds to the angular part of the regu-ar gradient. (For simplicity, we used the same symbol toenote the angular-momentum operator in the twopaces.)
In particular, on introducing spherical coordinates� ,�� in the angular spectrum domain, the angular-omentum operator becomes
L = ie�
sin �
�
��− ie�
�
��, �5�
here e� and e� are unit vectors in the � and � directions,espectively, while the unit vector in the radial directionu coincides with u.
. Inner Products and Average Valueset us define the inner product between two fields as the
ntegral over all directions of the product of their angularpectra, one of them complex conjugated, i.e.,
�U1,U2� =�4�
A1*�u�A2�u�d�. �6�
his inner product was shown to be invariant to changesn the coordinate axes11 and to be suitable for the math-matical description of processes of measurement.12 It ishown in Appendix A that this inner product can also beritten in terms of the fields in the position representa-
ion as
�U1,U2� = limR→�
1
8�2R�SR
U1*�r�U2�r�d3r, �7�
here SR is a solid sphere of radius R centered at the ori-in. The norm of U is given by
�U� = �U,U�1/2. �8�
The average value of a given operator O (of the kindhat can be written as an operation on the angular spec-rum) is defined in the usual way:
�O�U =�U,OU�
�U�2 =1
�U�2�4�
A*�u�OA�u�d�. �9�
n this paper we consider only Hermitian operators,hich satisfy
�U,OU� = �OU,U�. �10�
t is easy to show that both the direction and the angular-omentum operators mentioned earlier are Hermitian.
. Variations of Average Valuesn what follows, it will be important to know how averagealues like the one in Eq. (9) change under infinitesimalariations of both real and imaginary parts of A=AR
iAI. We do this by following a variational approachhere the value of, say, AR for each value of its argumentis considered as an independent variable in an infinite-
imensional space. Then the relation
�
�AR�u��4�
AR�u��B�u��d�� = B�u� �11�
s analogous to the more familiar expression for the ele-ents of the gradient of an inner product in a discrete-
imensional space:
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Alonso et al. Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. A 693
�
�xj�j�
xj�bj� = bj. �12�
The variations of �U ,OU� due to variations in botharts of the angular spectrum are then easily found to be
�
�AR�u��U,OU� = OA�u� + �OA�u�*, �13�
�
�AI�u��U,OU� = iOA�u� − i�OA�u�*. �14�
lternatively, one can consider variations of two differentinear combinations of the real and imaginary parts. Oneatural option is to use A and A*, regarding their varia-ions for this purpose as independent (even though theyre connected by complex conjugation):
�
�A�u��U,OU� = �OA�u�*, �15�
�
�A*�u��U,OU� = OA�u�. �16�
otice that the right-hand sides of Eqs. (15) and (16) areust linear combinations of those of Eqs. (13) and (14) andhat Eq. (15) is the complex conjugate of Eq. (16). In whatollows, it is therefore sufficient to consider variations dueo changes in A*. The variation with respect to A* of thexpected value of an operator is then easily found to be
��O�U
�A* =OA
�U�2 −�O�U
�U�2
��U�2
�A* =O − �O�U
�U�2 A�u�. �17�
. MEASURES OF ANGULAR AND SPATIALOCALIZATION
n this section we review the measures for nonparaxialelds defined in Ref. 3. The main properties of these mea-ures is that they are suitable for fields composed of planeaves traveling in any set of directions, and are indepen-ent of reference frame.
. Measure of Angular Spreadhe measure of angular localization is defined in terms ofhe centroid of the square modulus of the angular spec-rum distributed over the unit sphere, i.e., �u�U. While theirection of this average corresponds to the main direc-ion of propagation of the field, its magnitude qualifies thengular localization. For a paraxial field, for example, A2s concentrated within a small patch around the directionf propagation, so the centroid is near the unit sphere’surface, that is, �u�U is nearly unity. For a nonparaxialeld, on the other hand, �u�U is smaller because A2 isore evenly distributed over the unit sphere of directions.
n Ref. 3 the angular spread, say �u, was defined through
� 2
�u = 1 − �u�U . �18�
owever, following the notation used in Refs. 4 and 10 forhe case of one-parameter periodic functions, we nowhoose to use the closely related measure ��, given by
�� = arccos�u�U = arcsin �u. �19�
his angular spread measure takes values between 0 (for�u�U=1) and � /2 (for �u�U=0). The geometric meaningf �� can be immediately seen from Fig. 1. It is propor-ional to the effective half-angle of propagation directionsor the plane waves composing the field, measured fromhe average direction of propagation. For example, forearly propagation-invariant beams,13,14 �� is approxi-ately the half-angle of the cone around which the direc-
ions of the plane waves composing the field are concen-rated. For a full 4�-focused field with A�u�=1 for all u,n the other hand, ��=� /2, since the field is completelyelocalized in direction.
. Measure of Spatial Spreadn Ref. 3 a spatial localization measure was proposed inerms of the square of the angular-momentum operator,ay L2. For simplicity, let us introduce a spherical refer-nce frame with the polar axis coincident with the beam’sverage propagation direction �u�U, and the coordinaterigin at the centroid of the field, i.e., rC found in Eq.10.24) of Ref. 3. Accordingly, the spatial spread measure,ay �r, is defined as
�r = ��L2�U, �20�
here L2 is the square of the angular momentum opera-or defined earlier. (The point rC mentioned earlier is de-ned precisely as the position of the origin that minimizesL2�U.)
To grasp the reason why this quantity can be regardeds a measure of spatial localization, one can show that �r
2
s related to the variance of the angular momentum of theux lines of U (see Appendix B). Stated in terms of rayptics, such a quantity could be interpreted as the vari-nce of the impact parameter of the rays building up theeld. In particular, this measure is expected to vanishhen all rays pass through the centroid of the beam. In
uch a limit, however, since evanescent waves are not con-
ig. 1. Geometric interpretation of the angular spread measuren terms of �u�U, the centroid of the distribution A�u�2 over thenit sphere of directions. �� corresponds to the half-angle of theone subtended from the origin by the intersection of the unitphere of directions and a plane containing this centroid and per-endicular to the line joining the origin and the centroid.
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694 J. Opt. Soc. Am. A/Vol. 23, No. 3 /March 2006 Alonso et al.
idered in the present model, the actual size of the focalpot is expected to be of the order of the wavelength, ase shall see in Section 5. Furthermore, in the paraxial
imit, this measure was shown in Ref. 3 to reduce thetandard rms width over a transverse plane.
. Uncertainty Relationhe uncertainty relation between the angular and thepatial spreads follows naturally once the correspondingperators acting on the angular spectrum have been de-ned. In particular, starting from Eqs. (18) and (20) andaking their commutation properties into account, it isasy to arrive at the following uncertainty relation (seeppendix C):
��r2 + 1��u
2 1. �21�
he corresponding inequality for cylindrical (i.e., 2D)elds is
��r2 + 1
4��u2
14 . �22�
nequality (22) was found in Ref. 3, where it was shown topply for both 2D and 3D fields. That is, the inequalityound in Ref. 3 for 3D fields was not as strong as the onen inequality (21). The reason for this is explained in Ap-endix C. On taking Eq. (19) into account, inequalities21) and (22) lead, after some algebra, to
�r tan �� 1 �23�
or 3D fields, and
�r tan �� 1
2�24�
or 2D fields.For graphical purposes, it is convenient to express in-
quality (23) in a slightly different form. In particular, onntroducing the quantity �r=arctan �r, inequality (23) be-omes
�� + �r �
2, �25�
hich, in the plane �r versus ��, represents a straight lineith slope −1. As discussed earlier, it turns out that even
he new inequality for 3D fields given in any of the formsn inequalities (21), (23), or (25) cannot be satisfied as anquality except in trivial limits.
In Section 4 we look for the true bounds of thesepreads and for the fields that have minimum spatialpread for a given angular one. These fields will be re-erred to as minimum uncertainty fields (MUFs).
. MINIMUM-UNCERTAINTY FIELDS. Derivation of the Differential Equationo find the scalar MUFs we use a variational approachased on Lagrange multipliers. For a MUF, an infinitesi-al variation of the angular spectrum that causes a
hange in, say, ��, necessarily causes a change also in �r,ince otherwise it would be possible to reduce the firstpread without changing the second, implying that the
tate is not a MUF. The variations of both spreads due tovariation in the angular spectrum of a MUF must then
e proportional, i.e.,
�
�A* �C1�� + C2�r� = 0, �26�
here C1 and C2 are the so-called Lagrange multipliers,hich are assumed to be nonnegative. Alternatively, Eq.
26) could be derived by noting that a MUF is defined by aeld for which, once the angular spread �� has been fixedo a value, say �̄�, it minimizes �r. In the Lagrange mul-ipliers language this easily leads to the equation
�
�A* ��r + ��� − �̄�� = 0, �27�
hich coincides to Eq. (26) when we let =C1 /C2.According to Eqs. (17) and (19), the variation in the an-
ular spread is
���
�A* = −�u�U
�u�U�1 − �u�U2·
��u�U
�A*
= −�u�U
�u�U·
u − �u�U
�U�2 sin ��
A�u�. �28�
ecall that we chose the polar axis to coincide with theain direction of propagation, so �u�U ·u= �u�Ucos �,here � is the angle from the polar axis. Equation (28)
an then be written as
���
�A* = −cos � − �u�U
�U�2 sin ��
A�u� = −cos � − cos ��
�U�2 sin ��
A�u�.
�29�
he variation of the spatial spread is, on the other hand,iven by
��r
�A* =1
2�r
��r2
�A* =L2 − �r
2
2�U�2�rA�u�. �30�
he substitution of Eqs. (29) and (30) into (26) gives
− C1
cos � − cos ��
sin ��
A�u� + C2
L2 − �r2
2�rA�u� = 0. �31�
y defining
w4 =C2
C1
sin ��
�r, �32�
� = 2w−2�1 − cos ��� + w2�r2, �33�
q. (31) can be written as
�2w−2�1 − cos �� + w2L2A�u� = �A�u�. �34�
The strategy now is to solve Eq. (34) for different valuesf w. This equation corresponds to a Stürm–Liouvilleroblem where � is the eigenvalue. Because the operatorsre Hermitian, the eigenvalues are real and the solutionsor the angular spectrum form a complete basis. Of theseolutions, only the one with the smallest eigenvalue, re-
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Alonso et al. Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. A 695
erred to as the ground state, corresponds to the MUF. Asvaries, so does the ground state and its eigenvalue, and
ence the spread measures. Therefore the curve corre-ponding to the lower boundary of the allowed region inhe �� versus �r space is traced by finding these spreadsor the ground states for all w�0. Notice that, in the 2Dase, Eq. (34) reduces to the Mathieu equation,15 since2=−�2 /��2. The corresponding MUFs then have angularpectra given by Mathieu functions, as the functions inef. 10:
A�u� = A0MC�A�0,4w−4�,4w−4,�/2, �35�
here A0 is a constant, MC is the even Mathieu function,nd A is the corresponding Mathieu characteristicalue.15 In the 3D case, however, the solutions to Eq. (34)re not, to our knowledge, standard functions. To findhem, we must rewrite this differential equation in a formore amenable to numerical treatment.
. Multipolar Expansionhe continuous eigenvalue problem in Eq. (34) can beurned into a discrete one by expanding the angular spec-rum into spherical harmonics,16 i.e.,
A�u� = ��=0
�
�m=−�
+�
a�,mY�,m�u�, �36�
here Y�,m�u� denotes the scalar spherical harmonicunction of order � and index m, given by17
Y�,±m�u� = �±1�m�2� + 1
4�
�� − m�!
�� + m�!exp�±im��P�
�m��cos ��,
�37�
ith P��m��·� being the associated Legendre function with
nteger indices � and m.16 By inserting Eq. (36) into Eq.34), taking into account the orthogonality relationships
�4�
Y�,m* Y��,m�d� = �m,m���,�� �38�
�4�
Y�,m* cos �Y��,m�d� = �m,m�� �+1,m����+1 + �,m���,�−1�,
�39�
here
�,m =��2 − m2
��2� + 1��2� − 1�, �40�
e obtain the following homogeneous linear system forhe a�,m’s:
here �̄=�w2 /2 and use has been made of the fact that2Y�,m=���+1�Y�,m.Equation (41) can be interpreted as an ensemble of un-
oupled homogeneous linear systems, each of them char-
cterized by a value of m, and for �= m , m+1 , . . .. Sincehe eigenvalues �̄ depend on m, the solutions of Eq. (34)ill necessarily present a modal expansion involving onlyultipoles with the same value of m, except in case of
egeneracy. In particular, one expects that the MUFshould have rotational symmetry around the polar axis.he expansion in Eq. (36) must then be restricted to m0, so that only spherical harmonics that are indepen-ent of the azimuthal angle will be involved in the expan-ion. This is confirmed by the results shown in Fig. 2,hich is obtained by numerically evaluating the mini-um eigenvalues, say �min, of the system in Eq. (41) for
everal values of m. It can be seen that the smallest ei-envalue is obtained with m=0 for any choice of the pa-ameter w.
. Asymptotic Analysishile the MUFs must be found numerically in the gen-
ral case, the asymptotic limits of Eq. (34) correspondingo small or large values of the parameter w lead to ana-ytic expressions for its solutions and the correspondingigenvalues.
Let us consider first the case w→�. In this limit, it isrivial to verify that A�u�=A0Y0,0���=A0 is a solution ofq. (34). Accordingly, the spatial spread turns out to ber=0, while the field is completely delocalized in the an-ular domain, so that the average value of u vanishes,nd the angular spread reaches its maximum value, i.e.,�=� /2. Now assume that there is a small correction to
his constant spectrum, proportional to Y1,0=�3/4� cos �.he substitution of this form into Eq. (34) tells us that for
arge (but not infinite) w we have
A�u� � A0�1 +cos �
w4 � , �42�
� �2
w2 . �43�
he spreads can then be calculated to give
ig. 2. Behavior of the minimum eigenvalue �min, as a functionf w, for m=0 (thick solid curve), m=1 (thin solid curve), m=2dashed curve), and m=3 (dotted curve).
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696 J. Opt. Soc. Am. A/Vol. 23, No. 3 /March 2006 Alonso et al.
�� � arccos� 2w4
1 + 3w8� � arccos� 2
3w4� , �44�
�r �� 2
1 + 3w8 ��2
�3w4. �45�
his implies that, for large w, i.e., for directionally unlo-alized MUFs,
cos �� ��
2− �� ��2
3�r ��2
3�r. �46�
his means that the curve for the lower bound for the un-ertainties in the �� versus �r space must have a slope of�2/3 at the top left corner.Let us now consider the opposite limit, namely, w→0.
n such a case, the second term on the left-hand side ofq. (34) is significant with respect to the first one only if
he angular spectrum presents fast variations. Let us as-ume the ansatz
A�u� = A0 exp�����
w2 , �47�
ith ��·� being a suitable function of �. By recalling that,or A independent of �, L2=−�2 /��2−cot �� /��, the differ-ntial equation can be written as
otice that ���0� must vanish to keep the expression fi-ite. For sufficiently small w, we find that �±4 cos�� /2�, where the positive sign must be chosen to
nsure high directional localization. The angular spec-rum then has the asymptotic form
A�u� � A0 exp� 4
w2 cos�
2� � A0 exp�4w−2�exp�−�2
2w2� ,
�49�
o that it represents approximately a narrow Gaussian ofidth w, localized around the pole. In this limiting case
he spreads can also be evaluated in closed form, yielding
�� � w, �50�
�r ��
2− w �
�
2− ��, �51�
hich implies that the curve for the lower bound in the ��
ersus �r space must have a slope of −1 at the bottomight corner. Furthermore, in this limiting case it is easyo show that, on using Eqs. (49) and (1), the resulting field�r� will turn out to coincide with a paraxial Gaussian
eam having its waist across the plane z=0 and a spotize, say s0, given by
s0 ��2
w� �2�r. �52�
. NUMERICAL RESULTSn Fig. 3 we show the lower bound for the uncertainties ofD scalar fields, i.e., the curve �r versus �� pertinent toD and 2D scalar MUFs (solid curves), together withhose corresponding to the equality sign in inequalities23) and (24) (dotted curves). Notice that, in both the 2Dnd the 3D cases, the algebraic lower bound is a good ap-roximation to the true lower bound only for fields withmall directional spread. In Fig. 4 the behavior of �� andr is plotted as a function of w. Notice that �� is nearlyqual to w [as predicted by relation (50)] for w�1. On thether hand, [as predicted in relation (45)] �r tends to 0hen w→�.As we already pointed out at the end of Subsection 3.B,
rbitrarily small values of �r do not imply that the field isoncentrated at a single point. In this limit, instead wexpect the actual size of the focal spot to have a linear sizef the order of the wavelength. This fact becomes evident,or instance, by studying the half-width at half-maximumHWHM) of the intensity distribution of a MUF, evalu-ted across the transverse plane z=0. This is shown inig. 5, where the HWHM of scalar MUFs is plotted as a
unction of the parameter �r. In particular, it is easy toee that for �r→0, the HWHM tends to a limiting value of
ig. 3. Behavior of the lower bound for the uncertainties perti-ent to 2D and 3D scalar MUFs (solid curves), together withhose corresponding to the equality sign in inequalities (23) and24) (dotted curves).
ig. 4. Behavior, as functions of w, of �� (solid curve) and �rdotted curve).
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F
Alonso et al. Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. A 697
he order of � /4 (dotted line). In the opposite limit, i.e., forr→�, the field asymptotically tends to a Gaussian beam,
or which the HWHM at its waist equals �log 2�r (dashedine).
Figure 6 shows polar plots of the angular spectra foreveral values of w. For completeness, an alternative rep-esentation of the angular spectra A��� is given in Fig. 7.rom Figs. 6 and 7 we can appreciate that, indeed, theUFs are highly directional for small w, with an angular
pectrum that is nearly Gaussian. For w larger thannity, the angular spectrum behaves as predicted by rela-ion (42), i.e., as a constant minus a cosine perturbation.
To conclude this section, we present some plots of thepatial distribution of intensity of the MUFs. To this aim,e recall the expression of the field, say ��,m, pertinent to
he scalar multipole of order � and degree m, i.e.,17
��,m�r� = j��r�Y�,m�s�, �53�
here j��·� is the �th-order spherical Bessel function, r ishe position vector, and s is the unit vector along the r di-ection. Furthermore, on recalling that spherical harmon-cs Y�,m constitute the angular spectrum of the multipoles
�,m (see, for instance, Ref. 18 and references therein),.e.,
4�i���,m�r� =�4�
Y�,m�u�exp�iu · r�d�, �54�
nd on substituting Eq. (36) for m=0 into Eq. (1), afterome algebra we find
U�r� = 4���=0
�
i�a�,0��,0�r�. �55�
n Fig. 8 we present intensity maps, evaluated across thez ,r�� plane, with r� being the transverse radial distance,or MUFs with different values of w.
. CONCLUSIONShe characterization of optical scalar beams beyond thearaxial approximation is presently a subject of consider-ble interest in optics. In this paper, a class of beams thatresent a maximum joint spatial–directional localizationas been obtained by means of a variational approach. Inoing so, use has been made of recently introduced defi-itions for characterizing the spatial and angular spreadsf nonparaxial fields that reduce, in the paraxial limit, tohose based on the standard second-order moments of in-ensity.
The MUFs found in this work are nonparaxial exten-ions of scalar Gaussian beams. Of course, other alterna-ive extensions exist (see, for example, Ref. 19 and refer-nces therein). It would be interesting to compare thosether generalizations with the ones found here, and to seeow close they can get to the lower bounds obtained inhis paper. This comparison, nevertheless, is beyond thecope of this paper, and will be the topic of future work.20
urthermore, while the definition of angular spread usedere is conceptually pleasing, the measure of spatialpread is less direct. They are, however, naturally tied byhe uncertainty relation. If, for practical or other reasons,
ig. 5. Behavior of the HWHM of scalar MUFs as a function of
ig. 7. Behavior of the angular spectra for several values of w.
oiamt
lsetct
APCtsE
�
w
Icebt
w
To
e �z ,r
698 J. Opt. Soc. Am. A/Vol. 23, No. 3 /March 2006 Alonso et al.
ne decided to adopt different measures, the correspond-ng MUFs could be calculated by a varational methodnalogous to the one presented here, as long as these neweasures are expressible in terms of average values like
hose used in this work.Finally, the analysis presented here is based on the sca-
ar approximation. Of course, the natural context for thetudy of highly nonparaxial optical beams is the vectoriallectromagnetic theory. In this perspective, the results ob-ained in the present work must be generalized to thease of vectorial nonparaxial beams. Such a generaliza-ion will be explored in part II of this series of papers.21
PPENDIX A: INNER PRODUCT INOSITION REPRESENTATIONonsider the integral of the product of two fields, one of
hem complex conjugated, over a volume SR given by aphere centered at the origin and with radius R. By usingq. (1), this integral is seen to give
SR
U1*�r�U2�r�d3r
=�4�
�4�
A1*�u1�A2�u2�I�u1,u2,R�d�1d�2, �A1�
here
Fig. 8. Two-dimensional intensity maps evaluated across th
I�u1,u2,R� =�SR
exp�i�u2 − u1� · rd3r
=4�R
u2 − u12� sin�Ru2 − u1�
Ru2 − u1− cos�Ru2 − u1� .
�A2�
t is easy to see that, as R→�, the function I becomes in-reasingly localized around small values of u2−u1. How-ver, the normalization of this function must be verifiedefore we assume that it reduces, in the limit of large R,o the distribution
��u1,u2� =��1 − u1 · u2�
�, �A3�
hich satisfies
�4�
A�u1���u1,u2�d�1 = A�u2�. �A4�
o check this normalization, we consider the integral of Iver all u , which gives
�� plane for MUFs having w= �a�100, (b) 0.8, (c) 0.5, (d) 0.2.
1
IgaifE
AETibt
wh
w
wt
wjtgtosc
og
AITtfci
wc
Lg
wHTTca
Ttca=
Ia
i
o
B=p
Alonso et al. Vol. 23, No. 3 /March 2006/J. Opt. Soc. Am. A 699
�4�
I�u1,u2,R�d�1 =�SR
exp�iu2 · r��4�
exp�− iu1 · r�
�d�1d3r
= 4��SR
exp�iu2 · r�sin�r�
rd3r
= �4��2�0
R
r2� sin�r�
r 2
dr
= 8�2R�1 −sin�2R�
2R . �A5�
n the limit of large R, the right-hand side of Eq. (A5)ives 8�2R, so it is necessary to multiply I by 1/ �8�2R�nd take the limit of large R to achieve the delta functionn Eq. (A3). By multiplying both sides of Eq. (A1) by thisactor and taking this limit, we get to the expression inq. (7).
PPENDIX B: ON THE MEANING OFQUATION (20)o grasp the reason why the definition in Eq. (20) can benterpreted as a measure of spatial localization, we starty writing it explicitly using Eqs. (7) and (9), as well ashe Hermiticity of L:
�r2 =
1
�U�2 limR→�
1
8�2R�SR
�LU�* · �LU�d3r
=1
�U�2 limR→�
1
8�2R�SR
�r � U* � U�* · �r � U* � U�
Id3r,
�B1�
here I�r�= U�r�2 is the intensity of the field. Notice,owever, that
U*�r� � U�r� =�I�r�
2+ iJ�r�, �B2�
here J is the flux density, defined as
J�r� = Im�U*�r� � U�r�, �B3�
ith Im�·� denoting the imaginary part operator. Equa-ion (B1) can then be rewritten as
�r2 =
1
�U�2 limR→�
1
8�2R�1
4�SR
Ir � g2d3r +�SR
Ir � j2d3r� ,
�B4�
here g=�I /I is the relative gradient of the intensity, and=J /I is the intensity-normalized flux density. The firsterm in Eq. (B4) is proportional to the variance of the an-ular momentum of the relative intensity gradient, whilehe second gives the variance of the angular momentumf the flux lines of the field, both weighted by the inten-ity. For fields with no significant counterpropagatingomponents, the second term dominates, and the measure
f spatial spread is the radius of a region around the ori-in through which a significant part of the flux lines go.
PPENDIX C: DERIVATION OFNEQUALITY (21)his appendix gives a derivation of the uncertainty rela-
ion in inequality (21), which is stronger than the oneound in Ref. 3. To this aim, we recall that, given two non-ommuting vector operators, say B and C, the followingnequality holds:
�B2 �C
2 1
4��Bj,Cj�U2, �C1�
here the convention of implicit sum over repeated indi-es is used, and
�B2 = �B · B�U − �B�U · �B�U,
�C2 = �C · C�U − �C�U · �C�U. �C2�
Here we are interested in the vectorial operators u and. The commutation of their Cartesian components isiven by
�Li,uj = i�ijmum, �C3�
here �ijm is the Levi–Civita tensor. Let us choose B=u.owever, instead of choosing C=L, let Cj=TjkLk, wherejk are the elements of a unitary matrix T, such thatjkTjm
* =�jm. This means that �C=�L, regardless of thehoice of the unitary matrix. By using these substitutionsnd Eq. (C3) into inequality (C1), it follows that
�u2�L
2 1
4Tij�ijm�um�U2. �C4�
he goal now is to choose the unitary matrix to maximizehe right-hand side of this expression. (Notice that thehoice Tij=�ij actually minimizes it.) Let us choose the zxis to coincide with the direction of �u�U, so that �um�U�mz�uz�U=�mz�u�U. Then, inequality (C4) becomes
�u2�L
2 1
4Tij�ijz2�u�U2. �C5�
t is easy to see that, when we choose the unitary matrixs
T = �0 − i 0
i 0 0
0 0 1� , �C6�
nequality (C5) becomes
�u2�L
2 �u�U2, �C7�
r, by using the fact that �u2 =1− �u�U2,
�u2 �1 + �L
2 � 1. �C8�
ecause the measure of spatial spread is defined as �r2
�L2�U, and because therefore �L2 ��r
2, inequality (C8) im-lies that
Ndps
AMfAHp
@
R
1
1
1
1
1
1
1
11
1
2
2
700 J. Opt. Soc. Am. A/Vol. 23, No. 3 /March 2006 Alonso et al.
��r2 + 1��u
2 1. �C9�
otice that different unitary matrices were used in theerivation in Ref. 3. This different choice of matrices ex-lains why the uncertainty relation found there is not astrong as inequality (C9).
CKNOWLEDGMENTS. A. Alonso acknowledges support from the start-up
unds by The Institute of Optics, as well as the Careerward PHY-0449708 by the National Science Foundation.e is also grateful for the hospitality and financial sup-ort during a visit to Università Roma Tre.M. A. Alonso’s e-mail address is alonso
optics.rochester.edu.
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