Optics Communications 285 (2012) 4856–4860

Contents lists available at SciVerse ScienceDirect

Optics Communications

0030-40

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optcom

Propagation of nonparaxial vector hollow Gaussian beamsthrough a circular aperture

Jiang Guo a,n, Zao Li b

a Department of Geophysics, Chengdu University of Technology, Chengdu 610059, Chinab Jincheng College of Sichuan University, Chengdu 611731, China

a r t i c l e i n f o

Article history:

Received 30 December 2011

Received in revised form

6 July 2012

Accepted 6 August 2012Available online 29 August 2012

Keywords:

Hollow Gaussian beams (HGBs)

Vectorial Rayleith–Sommerfeld formulae

Nonparaxial propagation

Vectorial diffraction theory

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.08.032

esponding author. Tel.: þ86 28 84079327.

ail addresses: guo-jiang-li@163.com, guojiang

a b s t r a c t

Based on the vectorial Rayleith–Sommerfeld formulae, the nonparaxial propagation properties of the

vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the

derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular

aperture. The derived expression is independent the approximation of paraxial and far field, which are

valid for either far and near field and for the systems in which aperture radius is comparable to or even

smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical

calculation results indicate that there is no difference between derived formulae and the Collins

formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the

propagation properties of HGBs. The calculated results indicate that the propagation field of vector

hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field.

These research results could well shed light on the further understanding of the vectorial property of

HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, there have been increasing interests in thestudy of optical beams with zero central intensity called dark-hollow beams (DHBs) because of many practical applications inscience and technology, including cold atom guiding and opticalatom trapping [1–2], laser writing and drilling [3], laser manipula-tion of microscopic dielectric and metal objects [4]. Many differentmethods have been used to obtain the hollow beams [5–8]. Inrecent theoretical investigations, several different mathematicalmodels for describing DHBs have been proposed, such as the TEM01

beam and the higher-order Bessel beam, high-order Laguerre–Gaussian beam, hollow Gaussian beams (HGBs) [9–11]. Followedby the advent of HGBs, a rich amount of investigations about HGBshave been increasingly presented. The paraxial propagation prop-erties of HGBs through aperture have been studied [12], and theanalytical vectorial structure of HGBs in the far field has beenstudied [13], and the free-space nonparaxial propagation proper-ties of the HGBs have been also studied [14]. The description ofoptical beam propagation properties in the nonparaxial regionbecomes more and more important with the development of newoptical structures (e.g., micro-cavities, photonic crystals, and so on)and near-field optics, characterizing linear dimensions or spatialscales of variation comparable to or even smaller than wavelength l.

ll rights reserved.

@cdut.cn (J. Guo).

In the near field, the nonparaxial propagation properties andintensity distribution of vector HGBs through an aperture opticalsystem have been rarely reported so far.

In this paper, the nonparaxial propagation properties of theHGBs through a circular aperture were studied through thevectorial Rayleith–Sommerfeld formulae in detail. We describethe derivation of the integral expressions of the propagation ofnonparaxial vector HGBs through a circular aperture. The derivedexpressions do not depend on the approximation of the paraxialand far field (i.e., ac l or Rc a, where a R and l are apertureradius, beam propagation distention and wavelength, respectively).And the derived expressions are valid either for the far and nearfield or for the systems in which the condition a�l or aol wasconsiderate. By using the deduced formula, the propagation prop-erties of HGBs through a circular aperture are numerically calcu-lated and graphically illustrated. Moreover, numerically-calculatedresults are analyzed and compared with those obtained by usingthe Collins formulae of scalar approximation. In addition, we alsoresearch the influence of the truncation parameter on the near fieldpropagation properties.

2. Nonparaxial vectorial propagation integral expressionsof HGBs

According to the boundary values E(x0,y0,0)¼Ex(x0,y0,0)iþEy(x0,y0,0)j on the plane z¼0, the solution of the Helmholtz

J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860 4857

Eq. (1) is expressed as Eq. (2a)–(2c) in the half-space z40 [15].

r2Eþk2E¼ 0 ð1Þ

Exðx,y,zÞ ¼�z

2p

Z þ1�1

Z þ1�1

Exðx0,y0,0ÞikR�1

R3exp ðikRÞdx0dy0 ð2aÞ

Eyðx,y,zÞ ¼�z

2p

Z þ1�1

Z þ1�1

Eyðx0,y0,0ÞikR�1

R3expðikRÞdx0dy0 ð2bÞ

Ezðx,y,zÞ ¼1

2p

Z þ1�1

Z þ1�1

½Exðx0,y0,0Þðx�x0ÞþEyðx0,y0,0Þðy�y0Þ�

�ikR�1

R3expðikRÞdx0dy0 ð2cÞ

where k is wave number, R¼ 9r�q09¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx�x0Þ

2þðy�y0Þ

2þz2

q,

q0¼x0iþy0j is vector on the plane z¼0, r¼xiþyjþzk is spatialvector. The Eq. (2) is called the vectorial Rayleigh–Sommerfeldintegrals. By using the formulae, the propagation properties ofvectorial beams could be studied.

Considering the hollow Gaussian beams,

Eðx,y,0Þ ¼ Exðx0,y0,0ÞiþEyðx0,y0,0Þj ð3Þ

Exðx0,y0,0Þ ¼ A0r2

0

w20

!n

exp �r2

0

w20

!ð3aÞ

Eyðx0,y0,0Þ ¼ 0 ð3bÞ

was incident on a circular aperture with radius of a at the planez¼0. Where n is the order of the HGBs, A0 is the amplitudeconstant, w0 is the beam waist width. For n¼1 Eq. (3a) becomes aGaussian beam with beam waist w0.

The field at the plane after aperture may be represented as

Exðx0,y0,0Þ ¼ tðx0,y0ÞA0r2

0

w20

!n

exp �r2

0

w20

!ð4aÞ

Eyðx0,y0,0Þ ¼ 0 ð4bÞ

0

0.3

0.6

0.9

1.2

-0.36 -0.24 -0.12 0 0.12 0.24 0.36x/mm

Nor

mal

ized

inte

nsit

y

0

0.2

0.4

0.6

0.8

1

1.2

-0.45 -0.3 -0.15 0 0.15 0.3 0.45

x/mm

Nor

mal

ized

inte

nsit

y

Fig. 1. (Color online) Normalized transversal intensity distributions along x direction fo

(d) z¼50 mm. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the da

where tðx0,y0Þ ¼1 x2

0þy20ra2

0 else

(is the transmission function of

aperture.Replacing R in the exponent part with [16,17],

R¼ 9r�q090�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þr2

0

q1�

xx0þyy0

r2þr20

!ð5Þ

and R everywhere else with R�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þr2

0

qin Eq. (2) (it requires

that ðxx0þyy0=r2þr20Þ{1, these are the only approximation

conditions required in this paper), and using Eq. (4) and cylind-rical coordinates, the Eq. (2a)–(2c) are transformed as:

Exðx,y,zÞ ¼ �i2pzA0

l

Z 1

0

a2r2

w20

!na2rg2

1þi

kg

� �

� J0kabr

g

� �exp ikg�

a2r2

w20

!dr ð6aÞ

Eyðx,y,zÞ ¼ 0 ð6bÞ

Ezðx,y,zÞ ¼2pA0

l

Z 1

0

a2r2

w20

!na2rg2

1þi

kg

� �

� ixJ0kabr

g

� ��

xra

bJ1

kabrg

� �� �exp ikg�

a2r2

w20

!dr

ð6cÞ

where b¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þy2

p, g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2þz2þðarÞ2

q, J0 (kabr/g) and J1 (kabr/

g) is Bessel function of the zero- and first-order, respectively,r0¼ar, r is a normalized variable (0rrr1). Eq. (6) is the mainresult in this paper, which is general propagation integral expres-sion for nonparaxial vectorial HGBs through a circular aperture.We note that the propagation field owns both the longitudinaland the transversal components. However, the longitudinal com-ponents is asymmetric along x and y directions. Thus thepropagation field is asymmetric. The intensity distribution is

Iðx,y,zÞ ¼ 9Eðx,y,zÞ92¼ 9Exðx,y,zÞ92

þ9Eyðx,y,zÞ92þ9Ezðx,y,zÞ92

ð7Þ

0

0.4

0.8

1.2

-0.45 -0.3 -0.15 0 0.15 0.3 0.45x/mm

Nor

mal

ized

inte

nsit

y

0

0.3

0.6

0.9

1.2

-0.45 -0.3 -0.15 0 0.15 0.3 0.45

x/mm

Nor

mal

ized

inte

nsit

y

r the different propagation distances z. (a) z¼2.4 mm; (b) z¼3 mm; (c) z¼15 mm;

ta from Collins formula. The calculated parameters are a¼0.35 mm, w0¼0.1 mm.

J. Guo, Z. Li / Optics Communications 285 (2012) 4856–48604858

In the course of deducing Eq. (6) without the use of the approxima-tions Rba, and abl, Eq. (6) is valid for far field, near field and forthe optical systems in which the size of the aperture is comparablewith or smaller than the wavelength (i.e., a�l or aol). Eq. (6)is general propagation integral expression for nonparaxial vectorialHGBs through a circular aperture. In addition, for the observat-

ion point on axis R¼ 9r�q09�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2þr2

0

q, Eq. (6) is a strict integral

formula.

0.E+00

1.E-03

2.E-03

3.E-03

4.E-03

0 2 4 6 8 10 12z /mm

Nor

mal

ized

inte

nsity

Fig. 3. (Color online) On-axis normalized intensity distribution. ‘‘dot line’’ repre-

sents the data from Eq. (6); ‘‘red dot line’’ represents the data from Collins

formula. The calculation parameters are a¼0.35 mm, w0¼0.1 mm.

3. Numerical calculation results and analysis

The normalized intensity distributions of nonparaxial vectorialHGBs through a circular aperture are calculated by using Eq. (6),and the calculated results are compared in detail with thoseobtained by the generalized Collins formula. Within the frame-work of the paraxial approximation, under rotationally symmetriccylindrical coordinate condition, Collins formula can be expressedas [18]

Eðx,y,zÞ ¼ik

Bexp �ikz�

ik

2BDb2

� � Z a

0E0ðr0,0Þexp �

ik

2BAr2

0

� �J0

kbr0

B

� �r0dr0

ð8Þ

where A, B, C and D are the transition matrix elements of the HGBsin paraxial ABCD optical system. The calculated results are shownin Figs. 1–5. During the calculation, we choose the parametersA¼1, B¼z, C¼0, D¼1, l¼1060 nm, n¼5.

If Rba, i.e., when the paraxial approximation condition issatisfied, the propagation field of HGBs is symmetric around thepropagating axis, and the transversal intensities distributions byusing Eq. (6) are in agreement with those by using the Collinsformula Eq. (8), as shown in Fig. 1a–d. And the maximal errors areless than 0.048, 0.025, 9.7�10�4, 1.29�10�5 when z is 2.4, 3, 15and 50 mm, respectively. It is evident that there is no difference

0

0.2

0.4

0.6

0.8

1

-0.45 -0.3 -0.15 0 0.15 0.3 0.45

Nor

mal

ized

inte

nsity

x/mm

0

0.4

0.8

1.2

-0.45 -0.3 -0.15 0 0.15 0.3 0.45x/mm

Nor

mal

ized

inte

nsity

Fig. 2. (Color online) Normalized transversal intensity distributions along x direction f

(d) z¼5a. ‘‘dot line’’ represents the data from Eq. (6); ‘‘red line’’ represents the data fr

between Eq. (6) and the Collins formulae in the situation ofparaxial approximation. Therefore, the accuracy of Eq. (6) is wellconfirmed.

When the paraxial approximation condition is not satisfied,i.e., z�a or zoa, it can be seen from Fig. 2a–d that the numericalerrors of different formulae gradually increase with the decreas-ing of z, such as the maximal difference of intensities transversalalong the x direction is 0.077, 0.195, 0.43, 0.95 when z is 5a, 3a,1.72a, and 0.857a, respectively. From Fig. 2a, the high-spatial-frequency oscillation appears in the calculated results usingCollins formula under the condition of z¼0.857a. In contrast,the calculated beam profile using the Eq. (6) are smooth. FromFig. 3, it can be seen that the on-axis intensity by using the strictintegral formula (i.e., Eq. (6)) is not in agreement with those byusing the Collins formula when zoa, and the on-axis intensitycalculated using Collins formula tends to infinity. Only when thepropagation distance is larger than a (i.e., far field), the resultsobtained from two different formulae are in agreement. It

0

0.4

0.8

1.2

-0.45 -0.3 -0.15 0 0.15 0.3 0.45x/mm

Nor

mal

ized

inte

nsity

0

0.4

0.8

1.2

-0.36 -0.24 -0.12 0 0.12 0.24 0.36x/mm

Nor

mal

ized

inte

nsity

or the different propagation distances z. (a) z¼0.872a, (b) z¼1.72a, (c) z¼3a and

om Collins formula. The calculated parameters are a¼0.35 mm, w0¼0.1 mm.

-0.05

-0.025

0

0.025

0.05

00.250.50.751

-0.05-0.025

00.025

0.05

-0.05-0.025

0

0.025

-0.025

0

0.025

0.05

00.250.50.751

0.05

-0.05

0

-0.05

-0.05

-0.025

0

0.025

0.05

00.250.50.751

-0.05-0.025

0.0250.05

0

-0.05

-0.025

0

0.025

0.05

00.250.50.751

-0.05-0.025

0.0250.05

-0.04 -0.02 0 0.02 0.04

0

0.02

-0.04

-0.02

0.04

-0.04 -0.02 0 0.02 0.04

0

0.02

0.04

-0.04

-0.02

-0.04 -0.02 0 0.02 0.04

-0.04

-0.02

0

0.02

0.04

-0.04 -0.02 0 0.02 0.04

-0.04

-0.02

0

0.02

0.04

x

y

y

x

y

y

x

x

a

b

c

d

Fig. 4. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different truncation parameter a/w0 at z¼0.2a plane. (a) a/

w0¼30, (b) a/w0¼10, (c) a/w0¼8 and (d) a/w0¼5. The calculated parameters are w0¼0.01 mm.

J. Guo, Z. Li / Optics Communications 285 (2012) 4856–4860 4859

indicates that Collins formula of scalar paraxial approximation isinvalid for near field, and a vectorial diffraction theory should beused in near field.

Fig. 4a–d show the numerically-calculated results of normal-ized 3-D intensity distribution and the intensity contour lines ofHGBs through a circular aperture for the different truncationparameter a/w0 at z¼0.2a plane by using Eq. (6). From Fig. 4a–d,we find that the propagation field of HGBs gradually becomesasymmetric around the propagating axis in the near field with thedecrease of truncation parameter. The maximal difference oftransversal intensities between along the x direction and thatof the y direction is 0.003, 0.016, 0.146, 0.237, 0.488 when

a/w0¼30, 20, 10, 8 and 5, respectively. Moreover, when truncationparameter a/w0¼constant, with the increase of propagation distancethe propagation field of HGBs gradually becomes symmetric.

For example, when a/w0¼5, the calculated results of normal-ized 3-D intensity distribution and the intensity contour lines ofHGBs by using Eq. (6) are shown in Fig. 4d and Fig. 5a–b at thedifferent propagation distance z. The maximal difference ofintensities transversal between along the x direction and that ofthe y direction is 0.488, 0.146, 0.016 when z¼0.2a, 0.4a, 0.8a,respectively. The asymmetric propagation distance is about 0.8a

for a/w0¼5. In addition, the asymmetric propagation distance isabout 1.5a, a, 0.5a, 0.4a, 0.2a when a/w0¼3, 4, 8, 10 and 20,

-0.05

-0.025

0

0.025

0.05

00.250.50.751

-0.05-0.025

00.025

0.05

-0.05

-0.025

0

0.025

0.05

00.250.50.751

-0.05-0.025

00.025

0.05 -0.04 -0.02 0 0.02 0.04

0

0.02

0.04

-0.04

-0.02

-0.04 -0.02 0 0.02 0.04

0

0.04

-0.02

0.02

-0.04

y

y

x

x

a

Fig. 5. (Color online) Normalized 3-D intensity distribution and the intensity contour lines of HGBs for the different propagation distances z when a/w0¼5. (a) z¼0.4a,

(b) z¼0.8a. The calculated parameters are w0¼0.01 mm.

J. Guo, Z. Li / Optics Communications 285 (2012) 4856–48604860

respectively. We can draw a conclusion that the asymmetricpropagation distance gradually decreases with the increase oftruncation parameter. Therefore, the asymmetric property ofHGBs through a circular aperture in near field should be con-sidered in practical applications, especially for those small atruncation parameter a/w0. For example, we often encounteroptical beams with very small spot size (spot size is comparablewith the wavelength of light) on the micro-cavities lasers, photo-nic crystals lasers. Then the propagation properties of thesebeams cannot be described by scalar diffraction theory, butdescribed by the vector diffraction theory. For scalar diffractiontheory and vector diffraction theory, the nonparaxial propagationproperties and the asymmetric propagation properties of thebeams exist some different in the near field by which would havepotential influence on designing the micro lasers.

4. Conclusion

From the vectorial Rayleigh–Sommerfild integrals, the generalpropagation integral expressions for nonparaxial vectorial hollowGaussian beams through a circular aperture are derived. And theexpressions do not depend on the far-field paraxial approximationRba, or scalar approximation a/lb1. Thus, it is valid for eitherfar-field or near-field, and for the systems in which the size of theaperture is comparable with or smaller than the wavelength. And itprovides the strict integral formula for the light field on the axis.The derived formulae consist with the Collins formulae of thescalar diffraction theory in the situation of paraxial approximationor scalar approximation. Therefore, it is the general propagationintegral expressions for nonparaxial vectorial hollow Gaussian

beams through a circular aperture. The calculated results fromEq. (6) show that the propagation field of HGBs through a circularaperture is asymmetric in near field, and the propagation field issymmetric in far field. These research can shed light on the furtherunderstanding of the vectorial property of HGBs through a circularaperture, and will play a guiding role in the future research andpractical application of HGBs.

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