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2044 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 D. Deng and Q. Guo

Exact nonparaxial propagationof a hollow Gaussian beam

Dongmei Deng and Qi Guo*

Laboratory of Photonic Information Technology, South China Normal University, Guangzhou 510631, China*Corresponding author: guoq@scnu.edu.cn

Received July 20, 2009; accepted September 11, 2009;posted September 16, 2009 (Doc. ID 114242); published October 9, 2009

A group of virtual sources that generate a hollow Gaussian wave are determined on the basis of the superpo-sition of beams. A closed-form expression is derived for the hollow Gaussian wave that in the appropriate limityields the paraxial hollow Gaussian beam (HGB). From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a HGB is derived. Theinfinite series expression of a HGB can provide accuracy up to any order of diffraction angle. The radiationintensity of the hollow Gaussian wave is ascertained, and the radiation intensity pattern is characterized. Thetotal time-averaged power is evaluated. The characteristics of the quality of the paraxial beam approximationto the full hollow Gaussian wave are discussed. © 2009 Optical Society of America

OCIS codes: 350.5500, 260.2110, 260.1960, 140.3300.

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athIgirt

. INTRODUCTIONecently, optical beams with zero central intensity, calledark-hollow beams (DHBs), have attracted much atten-ion due to their increasing applications in atom optics.uch beams can be used to guide and trap cold atoms [1],onfine Bose–Einstein condensate [2], and create opticalraps [3] and tweezers [4,5] for the manipulation of micro-bjects, including biological species. Various methods,uch as transverse-mode selection [6], the geometric opti-al method [7], computer-generated hologram method [8],binary spatial light modulator [9], and spatial filtering

10], have been introduced to generate DHBs. A newathematical model called the hollow Gaussian beam

HGB) was presented to describe DHBs [11]. HGBs pro-ide a convenient and powerful way to describe and treathe propagation of DHBs with circular symmetry, andropagation properties of the HGBs have been studied inetail [12–20].The nonparaxial propagation of electromagnetic wave

eams has drawn much interest in optics [21–40] with thedvent of new optical structures, such as microcavitiesnd photonic bandgap crystals [41], characterizing linearimensions or spatial scales of variation comparable withr even smaller than wavelength �. Lax et al. [21] devel-ped a perturbative approach in terms of a small dimen-ionless parameter 1/ �kw0� in 1975, where k is the wave-umber. Couture and Belanger [22] obtained correctionso the paraxial solution by using the perturbative expan-ion procedure in 1981. Seshadri [23] reported the non-araxial corrections for the fundamental Gaussian beamn 2002. In 2003 Borghi and Santarsiero [24] showed thathe method proposed by Lax [21] could be profitably em-loyed even in the case of extremely nonparaxial beamsccording to a different resummation scheme introducedy Weniger [42]. Duan and Lü [25] derived a far-field ex-ression for nonparaxial Gaussian beams diffracted at aircular aperture on the basis of angular spectrum repre-

0740-3224/09/112044-6/$15.00 © 2

entation and the stationary-phase method in 2003. Se-hadri [26] explored the quality of paraxial electromag-etic beams in 2006. Deng D. M. [27,28] studied theonparaxial propagation of the radially polarized and theadially polarized elegant beams by using the vectorialayleigh–Sommerfeld formulas in 2006 and 2007. In007, Deng D. M. et al. [29] reported nonparaxial beamropagation in free space by the application of the multi-cale singular perturbation method. Mei and Zhao [30]tudied the nonparaxial propagation properties of vecto-ial Laguerre–Bessel–Gaussian beams on the basis of vec-orial Rayleigh–Sommerfeld formulas in 2007. Deng D. G.t al. [31] studied the nonparaxial propagation of vectorialollow Gaussian beams in 2008. The virtual sourceethod was introduced by Deschamps [32] and system-

tically extended by Felsen and his collaborators [33,34].y application of this method, the virtual sources thatenerate Bessel–Gaussian waves [35], cylindrical sym-etric elegant Laguerre–Gaussian waves [36], elegantermite–Gaussian waves [37], elegant Laguerre–aussian waves with radial mode number n and angularode number m [38], cosh-Gaussian beams [39], and el-

gant Hermite–Laguerre–Gaussian beams [40] have beenerived. However, to the best of our knowledge, the vir-ual sources of a HGB and the exact nonparaxial propa-ation of a HGB have not been reported; the radiation in-ensity pattern of the hollow Gauss wave has not beennvestigated.

In this paper, we have introduced the virtual sources ofHGB, the exact nonparaxial propagation of a HGB, and

he radiation intensity pattern of the hollow Gauss waveave been investigated. The paper is organized as follows.n Section 2, on the basis of the superposition of beams, aroup of virtual sources that generate the hollow Gauss-an wave is presented, and a closed-form expression is de-ived for the hollow Gaussian wave. In Section 3, usinghe Green’s function approach, we obtain the differential

009 Optical Society of America

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D. Deng and Q. Guo Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2045

epresentation of the hollow Gaussian waves and derivehe paraxial approximation and the infinite series non-araxial corrections. In Section 4, the radiation intensityattern of the hollow Gaussian wave is treated. A conclu-ion is presented in Section 5.

. VIRTUAL SOURCES OF A HOLLOWAUSSIAN BEAM

uppose E�� ,z� to be a monochromatic paraxial scalarave function that denotes a paraxial hollow Gaussianave propagating along the positive z axis. The hollowaussian field at the z=0 plane is characterized by [11]

En��,0� = C0� �2

w02�n

exp�−�2

w02� , �1�

here n=0,1,2, . . . is the order of the paraxial hollowaussian beam (HGB), C0 is a constant, and w0 is theaist width of the Gaussian amplitude distribution. Ap-lying the identity [43] x2n=n! /2n�m=0

n �−1�m� mn �Lm�2x2�,

here n! is the factorial of an integer n, � mn � denotes a bi-

omial coefficient, and Lm� · � is the mth-order Laguerreolynomial. Equation (1) can be written alternatively inhe form

En��,0� = C0n!�m=0

n

�− 1�m� mn �Lm� �2

w02�exp�−

�2

w02� . �2�

quation (2) shows that the field distribution of a HGB ofrder n across the z=0 plane is simply a superposition of+1 circularly symmetric elegant Laguerre–Gaussianeams. Equation (2) simplifies to the fundamental Gauss-an beam when n=0. The paraxial HGB is generated by aroup of sources of strength Scs�m� situated at �=0 and=zcs. Proper choice of Scs�m� and zcs yields the desiredeam. The exact wave function Un�� ,z�, which can reduceo the paraxial hollow Gaussian wave function En�� ,z�,atisfies the inhomogeneous Helmholtz equation

��t2 +

�2

�z2 + k2�Un��,z� = − C0n!�m=0

n

� mn �Scs�m���t

2�m

�����

���z − zcs�, �3�

here �t2=�2 /��2+1/�� /��, k is the wavenumber, and

�t2�m= ��2 /��2+1/�� /���m. Apply the Hankel transform

airs [36]

Un��,z� =�0

+�

Un��,z�J0�����d�, �4�

Un��,z� =�0

+�

Un��,z�J0�����d�, �5�

here J0� · � is the zeroth-order Bessel function, Un�� ,z� ishe radial spectrum of Un�� ,z�, and � is the radial compo-ent of wave vector k� . From Eq. (3), Un�� ,z� is given andubstituted into Eq. (4) [35,36]; we can obtain that

Un��,z� = C0n!�m=0

n

� mn �

i

2Scs�m��

0

+�

d���− �2�m

�exp�i��z − zcs��J0����/�, �6�

here Re�z−zcs��0, �= �k2−�2�1/2. Expanding � into a se-ies and keeping the first and second terms, we obtain �k�1−�2 / �2k2��. Under the paraxial approximation, i.e.,2�2, we replace � of the exponential part in Eq. (6) withhe approximation and the other terms with k; Eq. (6)implifies to

Unp��,z� = C0n!�m=0

n

� mn ��− 1�m

i

2kScs�m�exp�ik�z − zcs��

��0

+�

d��2m+1

�exp�− i�2�z − zcs�/�2k��J0����. �7�

he additional subscript p in Unp represents the paraxialpproximation. Applying the relation [36],

�0

+�

�2m+1 exp�− a2�2�J0���d�

=m!

2a−2�m+1� exp�−

2

4a2�Lm� 2

4a2� , �8�

he integral of Eq. (7) can be obtained:

Unp��,z� = C0n!�m=0

n

� mn �

i

4km!�− 1�mScs�m�exp�ik�z − zcs��

�� 2ik

z − zcs�m+1

exp ik�2

2�z − zcs�

�Lm−ik�2

2�z − zcs� . �9�

o determine the HGB for z�0, we assume that the inputistribution is given by Eq. (2) for the paraxial approxi-ation. By comparing Eq. (9) at z=0 and Eq. (2), the pa-

ameters Scs�m� and zcs can be given as

zcs = ikw02/2 = izR, �10�

Scs�m� = − 2izR/m!�w02/4�m exp�− kzR�.

�11�

rom Eqs. (6), (10), and (11), the exact integral expressionor Un can be obtained:

Un��,z� = C0n!�m=0

n

� mn ��− 1�m

zR

m!�w02

4 �m

exp�− kzR�

��0

+�

d��2m+1exp�i��z − zcs��J0����/�. �12�

he solution given by Eq. (12) is the exact solution to theomogeneous Helmholtz equation. The exact solution re-uces to the correct paraxial approximation if kw →�.

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2046 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 D. Deng and Q. Guo

. EXACT NONPARAXIAL PROPAGATIONF A HOLLOW GAUSSIAN BEAMsing the Green’s function approach, we can obtain theifferential representation of the hollow Gaussian wave.he solution of the differential equation

���2 +

�2

�z2 + k2�G��,z� = − Scs�m�����

2����z − zcs� �13�

s given by [35–37] G�� ,z�=Scs�m�exp�ikR� / �4�R�, where= ��2+ �z− izR�2�1/2. Comparing Eqs. (13) and (3) and ap-lying Eq. (11), the differential or multipole representa-ion of the hollow Gaussian wave is found to be

Un��,z� = iC0n!�m=0

n

� mn �

zR

m!�w02

4 �m

exp�− kzR�

���t2�mexp�ikR�

R . �14�

quation (14) shows that a hollow Gaussian mode can beenerated by applying first the operator ��t

2�m and thenum �m=0

n � mn �n! /m! to the complex-source-point spherical

aves exp�ikR� /R. The elegant Laguerre–Gaussianeams can be obtained by applying the operator ��t

2�m toxp�ikR� /R [36]. Hence, hollow Gaussian modes can beegarded as the superposition of n+1 elegant Laguerre–aussian beams.By using the perturbative series method and the solu-

ions obtained for the lowest order Gaussian beam [22] ac-urate to any order �2, where �=1/ �kw0�= d /2 is a di-ensionless perturbation parameter, d is the far field

iffraction angle, we can obtain the higher order HGB ac-urate to any order �2 immediately as follows:

Un��,z� = C0 exp�ikz�exp ik�2

2�z − izR���

s=0

�

�2sn!�m=0

n

� mn ��

t=0

s

�ts�− 1�m

��2s − t + m�!

m!�2s − t�! � − izR

z − izR�s+m+1

�L2s−t+m − ik�2

2�z − izR� , �15�

here �ts= �−1�t+s��s+1�� t

s�� s2s� and �� · � is the gamma

unction. Equation (15) is the infinite series correctionorm of a HGB that may be a proper one for achieving theesired accuracy. Equation (15) is consistent with the ex-ct solution obtained by adopting the power expansion/� and exp�i��z−zcs�� in Eq. (6) and keeping the productf both series terms up to order �2s for s=�.

When n=0, Eq. (15) reduces to the nonparaxial correc-ion solution of the fundamental Gaussian beam, U0�� ,� = C0 exp �ikz�exp �ik�2 / �2�z− izR��� �s=0

� �2s �t=0s �t

s�−izR /zizR�s+1L2s−t�−ik�2 / �2�z− izR���. If we take �=0 in Eq. (15),

he on-axis nonparaxial correction solution of HGB of anyrder n can be given as

Un�0,z� = C0 exp�ikz��s=0

�

�2sn!�m=0

n

� mn ��

t=0

s

�− 1�m

��ts�2s − t + m�!

m!�2s − t�! � − izR

z − izR�s+m+1

. �16�

For the hollow Gaussian wave, the time-averagedower flow per unit area in the z direction is expressed as44]

�z,n��,z� =1

2Re− i�En

*��,z��

�zEn��,z� , �17�

here Re denotes the real part and � stands for the com-lex conjugate part.Figure 1(a) shows the normalized time-averaged power

ow per unit area in the plane z=0.3zR and Fig. 1(b) z0.6zR of the HGB. Figure 2(a) presents the normalized

ime-averaged power flow of the HGB per unit area ver-us z /zR for �=0 and Fig. 2(b) for �=�. In Figs. 1 and 2,he dimensionless perturbation parameter is �=0.25. Itan be seen from Figs. 1 and 2 that the normalized time-veraged power flow per unit area of the sixth-order cor-ection solution is closer to the exact solutions than that

(a)

(b)

ig. 1. (Color online) Normalized time-averaged power flow pernit area in the plane (a) z=0.3zR and (b) z=0.6zR of the HGBith n=4 evaluated by Eqs. (15) and (17) with s=0 (solid curve),

.e., the paraxial solution; s=2 (dashed curve), i.e., up to theourth-order correction solution; s=3 (dotted curve), i.e., up to theixth-order correction solution; and by Eqs. (12) and (17)dashed–dotted curve), i.e., the exact solution.

otttpsnhfi

4HTst

St

sfsv(

w

�(i=nfthsootprklnvps

h=tmac

t

F→tam

Fu(Hstle

D. Deng and Q. Guo Vol. 26, No. 11 /November 2009 /J. Opt. Soc. Am. B 2047

f the fourth-order correction solution. The paraxial solu-ion, higher-order nonparaxial correction solution, andhe exact solution have little effect on the beam propaga-ion in the far-field region �z�5zR� because the beam ex-ands and the paraxial approximation is valid in freepace; however, the paraxial solution, the higher-orderonparaxial correction solution, and the exact solutionas a great effect on the beam propagation in the near-eld region, as shown in Fig. 2.

. RADIATION INTENSITY PATTERN OF AOLLOW GAUSSIAN BEAM

he time-averaged power Pn manifested in the physicalpace can be determined by integrating �z,n�� ,z� acrosshe entire transverse plane as

Pn =�0

2�

d��0

�

�z,n��,z��d�. �18�

ubstituting Eqs. (6) and (17) into Eq. (18), the integra-ion � is carried out first, then the integration with re-

(a)

(b)

ig. 2. (Color online) Normalized time-averaged power flow pernit area versus z /zR for (a) �=0, i.e., on-axis, evaluated by Eqs.

16) and (17) and (b) �=� evaluated by Eqs. (15) and (17) of theGB with n=4 and s=0 (solid curve), i.e., the paraxial solution;

=2 (dashed curve), i.e., up to the fourth-order correction solu-ion; s=3 (dotted curve), i.e., up to the sixth-order correction so-ution; and by Eqs. (12) and (17) (dashed–dotted curve), i.e., thexact solution.

pect to one of the transform dummy variables is per-ormed [18,19,26]. For the integration with respect to theecond transform dummy variable, the contribution to Pnanishes for k���� and, setting �=k sin , from Eq.18), Pn can be written in the form

Pn =�0

2��0

�/2

d d��n� ,��sin , �19�

here

�n� ,�� =1

2��2

22n

�2n�!exp�− k2w0

2�1 − cos ���n2� �,

�20�

n� �=n!�m=0n � m

n �1/ �m!4m��−k2w02sin2 �m .�n� ,�� in Eq.

20) indicates the radiation intensity of the hollow Gauss-an wave of radial mode number n. �n�0,���2��2�2n�!�−122nn! depends on the radial mode number. For n=0, the hollow Gaussian wave simplifies to the

undamental scalar Gaussian wave with the radiation in-ensity as �0� ,��= �2��2�−1 exp�−k2w0

2�1−cos ��, whichas been reported by Seshadri [18,19,26]. If n=1 andin =2/ �kw0��1, �1� ,��=0 for the case of 0� �� /2,r else the radiation intensity pattern from Eq. (20) hasne major lobe in the propagation direction � =0�. Similaro the Laguerre–Gaussian beams [18], for any n, being de-endent on the value of kw0, �n� ,�� has n zeros in theange 0� �90°. Hence, for any n and sufficiently largew0, the radiation intensity pattern has n circular lobes,eading to one major lobe in the propagation direction and

minor lobes. The peak intensity of the first minor lobe isery small compared with that of the major lobe, and theeak intensities of the succeeding minor lobes becomemaller and smaller.

Figure 3 shows the radiation intensity patterns of theollow Gaussian wave for the mode numbers n=0,1,2, n3,4,5, and kw0=2.980. It is easy to see from Fig. 3 that

he major lobe becomes sharper with increasing radialode number n, and the minor lobes also become sharper

nd move toward the propagation direction with the in-reasing radial mode number.

By integrating Eq. (19), the time-averaged power forhe first three radial mode numbers are found to be

P0 = 1 − exp�− k2w02�, �21�

P1 = 1 − �2 + 3�4 − �3/2 + 2�2 + 3�4 − k2w02

− 1/8k4w04�exp�− k2w0

2�, �22�

P2 = 1 − 2/3�2 + 5�4 − 45�6 + 105�8 + �7/24

+ 2/3�2 − 25/4�4 − 60�6 − 105�8

+ 13/6k2w02 − 13/16k4w0

4 + 1/12k6w06

− 1/384k8w08�exp�− k2w0

2�. �23�

rom Eqs. (21)–(23), we can find that Pn→1 when kw0�. The time-averaged power Pn of the full wave is less

han that Pnp of the corresponding paraxial beam. If Pnpproaches Pnp, then the paraxial beam is a good approxi-ation for the full wave [18]. Thus, 1/P can be used as a

n

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pnorrtptasqtpi

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R

1

1

1

1

FG=

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2048 J. Opt. Soc. Am. B/Vol. 26, No. 11 /November 2009 D. Deng and Q. Guo

arameter to estimate the quality of the paraxial beampproximation to the full wave. The smaller the value of/Pn �1/Pn�1� is, the higher the quality of the paraxialeam approximation. Figure 4 shows that the quality ofhe paraxial beam approximation becomes poor as w0 /�0.5. The bigger the radial mode number of the wave is,

he larger the required value of w0 /� for obtaining theame level of the quality of the paraxial beam approxima-ion. For a given w0 /� greater than about 0.5, the qualityf the paraxial beam approximation decreases with an in-reasing radial mode number.

. CONCLUSIONn conclusion, we have introduced a group of complexources that generate a hollow Gaussian wave. We haveerived the integral and the differential representationor the hollow Gaussian wave. Based on Couture and Be-anger’s [22] series expression for the complex-source-

(a)

(b)

ig. 3. (Color online) Radiation intensity pattern of the hollowaussian wave for the mode numbers (a) n=0,1,2 and (b) n3,4,5 and kw0=2.980.

ig. 4. (Color online) (1/Pn as functions of w0 /� for n=0,1,2,3nd for 0.05�w0 /��1. Pn is the power of the hollow Gaussianave of mode number n; � is the wavelength.

oint spherical wave, we have obtained the infinite seriesonparaxial correction expression for a HGB with radialrder n. The infinite series expression can provide accu-acy up to any correction order when we expect an accu-ate description of a HGB with large-order n. We have in-roduced the radiation intensity and the time-averagedower carried by the corresponding full wave with respecto the scalar hollow Gaussian wave. The ratio of the time-veraged power of the paraxial beam to that of the corre-ponding full wave is used as a parameter to assess theuality of the paraxial beam approximation. We find thathe lower-order hollow Gaussian waves have betteraraxial HGB quality than do higher-order hollow Gauss-an waves for a fixed kw0.

CKNOWLEDGMENTShis research was supported by the National Natural Sci-nce Foundation of China (grants 10674050 and0904041), the Specialized Research Fund for the Doc-oral Program of Higher Education (grant 20060574006),he Program for Innovative Research Team of the Higherducation in Guangdong (grant 06CXTD005), and thepecialized Research Fund for growing seedlings of theigher Education in Guangdong (grant C10087).

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