Propagation of hollow Gaussian beams through

(here L␯n denotes the Laguerre polynomial), we obtain the following expression after some tedious but straightforward integration:
introduced a new mathematical model called the hollow
Gaussian beam (HGB) to describe DHBs of circular symmetry15 and introduced the hollow elliptical Gaussian
guiding of atoms by a HGB and a HEGB have also been studied.17,18 In the present paper, by expanding the hard-
aperture function into a finite sum of complex Gaussian functions,19,20 we derive approximate analytical formulas
form in a cylindrical coordinate system:
͵ ͵ ͭ i
2␲ a1
ik
E͑r,z͒ = ␭B exp͓− ikl0͑z͔͒ 0 0 E0͑r1,0͒exp − 2B ͓Ar12
ͮ − 2r1r cos͑␪ − ␪1͒ + Dr2͔ r1dr1d␪1,
͑2͒
where r1 , ␪1 and r , ␪ are the radial and the azimuth angle coordinates in the input and output planes, respectively, l0͑z͒ is the optical length along the propagation axis z (for simplicity we will not show its z dependence in some equations below), k = 2␲ / ␭ is the wavenumber; and ␭ is the wavelength. A , B , C , D are the elements of the trans-
Joint Research Center of Photonics of the Royal Institute of Technology and Zhejiang University, East Building No. 5, Zijingang Campus, Zhejiang University, Hangzhou, 310027, China, and Division of Electromagnetic Theory, Alfvén Laboratory, Royal Institute of Technology, SE-10044 Stockholm, Sweden
beam (HEGB) to describe DHBs of noncircular symmetry.16 Propagation and transformation of a HGB
and a HEGB through an unapertured paraxial general
optical system have been studied in detail. Trapping and
the following sum (finite terms) of complex Gaussian functions19,20:
͚ ͩ ͪ M
H͑r͒ = Am exp
m=1
−
Bm a12
r2
,
͑6͒
where Am and Bm are the expansion and Gaussian coefficients, which can be obtained by numerical optimization directly.19,20 The numerical result shows that the simulation accuracy improves as M increases.
1410 J. Opt. Soc. Am. A / Vol. 23, No. 6 / June 2006
Y. Cai and S. He
Propagation of hollow Gaussian beams through apertured paraxial optical systems
Yangjian Cai and Sailing He
Received August 26, 2005; revised November 29, 2005; accepted November 29, 2005; posted November 30, 2005 (Doc. ID 64402)
On the basis of the generalized Collins formula and the expansion of the hard-aperture function into a finite sum of complex Gaussian functions, an approximate analytical formula for a hollow Gaussian beam propagating through an apertured paraxial stigmatic (ST) ABCD optical system is derived. Some numerical examples are given. Furthermore, by using a tensor method, we derive approximate analytical formulas for a hollow elliptical Gaussian beam propagating through an apertured paraxial general astigmatic ABCD optical system and an apertured paraxial misaligned ST ABCD optical system. Our results provide a convenient way for studying the propagation and transformation of a hollow Gaussian beam and a hollow elliptical Gaussian beam through an apertured general optical system. © 2006 Optical Society of America
ቤተ መጻሕፍቲ ባይዱ
͑3͒
and applying the integral formula22
͵1 2␲
J0͑x͒ = 2␲ 0 exp͑ix cos ␪͒d␪,
͑4͒
we find that Eq. (2) becomes
1084-7529/06/061410-9/$15.00 © 2006 Optical Society of America
for a HGB and a HEGB propagating through an aper-
tured paraxial general optical system, and some numeri-
cal examples are given.
2. PROPAGATION OF A HOLLOW
Within the framework of the paraxial approximation,
the propagation of a HGB through an apertured paraxial
stigmatic (ST) ABCD optical system can be treated by the generalized Collins formula,21 which takes the following
GAUSSIAN BEAM THROUGH AN
APERTURED PARAXIAL STIGMATIC ABCD
OPTICAL SYSTEM
The electric field of a HGB at z = 0 is defined as follows15:
ͩ ͪ ͩ ͪ r2 n
r2
En͑r,0͒ = G0 w02 exp − w02 ,
fer matrix of the paraxial ST optical system, and a1 denotes the radius of the aperture.
By introducing the hard-aperture function
ͭ ͮ 1, ͉r1͉ ഛ a1
H͑r1͒ = 0, ͉r1͉ Ͼ a1
Y. Cai and S. He
ͩ ͪ͵ 2␲i
ikDr2
E͑r,z͒ = ␭B exp͑− ikl0͒exp − 2B
ϱ
En͑r1, 0͒H͑r1͒
0
ͩ ͪ ͩ ͪ ikAr12
krr1
ϫexp − 2B
J0
B
r1dr1.
͑5͒
The hard-edged-aperture function can be expanded as
Substituting Eqs. (1) and (6) into Eq. (5) and applying the integral formula22
͵ϱ exp͑− pt͒tv/2+nJv͑2␣1/2t1/2͒dt 0