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Topological charge inversion in polynomial astigmatic Gaussian beams
Optics Communications 281 (2008) 4205–4210
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Topological charge inversion in polynomial astigmatic Gaussian beams Filippus S. Roux Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0001, South Africa
a r t i c l e
i n f o
Article history: Received 4 January 2008 Received in revised form 28 April 2008 Accepted 14 May 2008
PACS: 42.25.Bs 42.15.Fr 42.25.Fx
a b s t r a c t The evolution of the global topological charge in a general polynomial astigmatic Gaussian beam is investigated. The leading order terms of the polynomial prefactor determines the global topological charge and can be expressed as a product of first order polynomials, each representing an optical vortex function. We show that the global topological charge is bounded by the order of the polynomial and change during propagation in steps of 2 every time one of the optical vortices undergo topological charge inversion. We investigate the locations of the flip planes where charge inversions occur and provide expressions for a number of special cases. Numerical results are provided for an example of such a polynomial astigmatic Gaussian beam. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Singular optics Optical vortices Astigmatic Gaussian beams Topological charge
1. Introduction The behavior of phase singularities, also called optical vortices [1,2], in optical beams have attracted much attention [3–11]. A beam that contains such vortices carries orbital angular momentum [12–18], which can be transferred to small particles [19]. In this way singular optics plays an essential role in nanotechnology. For the successful application of singular optics, it is essential that the vortices exist at the precise locations in the beam where they are needed and with the desired properties. The parameters of optical vortices are sensitive to perturbations of the beam. It is therefore necessary to understand the effects of such perturbations on the optical vortex parameters. Here we are in particular interested in the effects of astigmatism on the optical vortex parameters. It has been observed that, in an astigmatic beam, optical vortices can invert their topological charge [20]. Previous analyses either focussed on the properties of astigmatic Laguerre–Gaussian beams [21,22] or considered only a single vortex in the astigmatic beam [23]. In this paper we investigate general polynomial astigmatic Gaussian beams, consisting of an astigmatic Gaussian beam multiplied by a complex-valued bivariate polynomial prefactor of finite order. These polynomial prefactors include all sets of orthogonal functions that can be expressed as polynomials of finite order, such as the Laguerre polynomials and the Hermit polynomials. PolynoE-mail address: [email protected] 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.05.012
mial (stigmatic) Gaussian beams are ubiquitous. In addition to the well known Laguerre–Gaussian beams and Hermit–Gaussian beams, they also include the Indebetouw beams [3], spiral beams [24] and other stable beams [25,26], but not non-diffracting beams [27,28] such as Bessel beams because such prefactors tend to be transcendental functions and not of finite order. We define our notation for the general expression for an astigmatic Gaussian beam in Section 2. The complex zeros of the polynomial prefactor represent the optical vortices in the beam. The propagation of polynomial astigmatic Gaussian beams is discussed in Section 3. The cases of first order and second order polynomial prefactors are considered in detail. The evolution of the global topological charge in such a beam is analysed in Section 4. The flip plane locations are discussed in Section 5 and we consider some special cases to show that they are in agreement with previous work [23]. The results of a numerical simulation of the propagation of an example of a polynomial astigmatic Gaussian beam is reported in Section 6. We end with some conclusions in Section 7. 2. Polynomial astigmatic Gaussian beams Astigmatism is an aberration often found in optical systems [29]. It manifests as a longitudinal separation of the focal point – two different transverse directions are focused at different points along the optical axis. For instance, the beam can be focused along the x-direction to form a focal line along the y-direction at z ¼ z1 and then be focused along the y-direction to form a focal line along
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the x-direction some distance later at z ¼ z2 . In the most general astigmatic beam the two focal lines need not be orthogonal. For the purposes of the investigation reported here we are interested in astigmatic Gaussian beams. A general astigmatic Gaussian beam can be expressed by
gðu; v; tÞ ¼
2 pd0
1=2
Q u2 þ 2Q uv uv þ Q vv v2 ; exp uu d0
ð1Þ
ð2Þ
Q vv
ð3Þ
Q uv ¼ g sin b; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d0 ¼ ðt þ i 1 þ g2 Þ2 s2 þ g2 þ i2gs cos b:
ð4Þ ð5Þ
Here we specify the transverse coordinates, u and v, in units of some convenient scale parameter x, while the coordinate along the direction of propagation t is measured in units of q ¼ px2 =k, with k being the wavelength. In other words, u ¼ x=x, v ¼ y=x and t ¼ z=q, where ðx; y; zÞ represents the set of Cartesian coordinates. There are three parameters that quantify the properties of the astigmatic beam: s and g are non-negative real-valued parameters and b is an angular parameter within the range p=2 < b < p=2. The parameterization that we use here differs from the parameterization used by other authors [21,22], because it is more convenient for our purposes and provides us with a way to express astigmatic Gaussian beams in their most general form. The astigmatic Gaussian beam has two focal planes located at t ¼ s and t ¼ s. The shape of the wavefront of the beam in each of these focal planes is purely cylindrical.1 The orientation of these cylindrical wavefronts depend on b and g. For g ¼ 0 or b ¼ 0 the two cylindrical wavefronts are oriented at 90° with respect to each other. In the region between these two focal planes the wavefront of the Gaussian beam has a saddle shape and outside of this region the wavefront is a non-symmetrical paraboloid. In the derivation below we shall use helical coordinates pffiffiffi for the transverse p directions, defined by w ¼ ðu þ ivÞ= 2 and ffiffiffi w ¼ ðu ivÞ= 2. It will also be more convenient to use matrix– vector notation. For that purpose we express the astigmatic Gaussian beam more compactly as
gðw; w; tÞ ¼
2
T
exp
pd0
W UW ; d0
w ; W¼ w " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # g expðibÞ þ is it 1 þ g2 U¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g expðibÞ þ is it 1 þ g2
p¼0
lTn;p W W n;p ;
fn;p
;
ð10Þ
is a morphology vector that defines the morphology [13,10,18] of the complex zero (vortex) associated with the nth first order polynomial factor in the pth factored polynomial, and
# wn;p ; wn;p
ð11Þ
represents a shift in the coordinates for the ðn; pÞ-th first order polynomial factor. As such wn;p represents the location of the complex zero of the first order polynomial factor. The orders of the reducible polynomials in the summation in Eq. (9) differ by 2, because a reducible Nth order polynomial contains as many degrees of freedom as there are for all the Nth order and ðN 1Þth order terms in a general Nth order complex-valued bivariate polynomial. 3. Propagation of polynomial astigmatic Gaussian beams In general the parameters in the expression of the bivariate polynomial prefactor in Eq. (9) depends on the position along the propagation direction. To find out how the prefactor changes during propagation one can follow a procedure such as the one provided in [31]. Here we shall adapt this procedure to the case of an astigmatic Gaussian beam. The polynomial prefactor of the astigmatic Gaussian beam is specified on an input plane at t ¼ t 0 . To find the expression for the prefactor at an output plane at t > t0 , one can use the Fresnel integral [32]. In terms of the normalized helical coordinates the Fresnel integral is formally expressed as
f ðw; w; tÞ ¼
Z Z i2ww exp f ðw0 ; w’; t 0 Þ t t0 t t0 D i2w’w’ exp t t0 i2 ½ww0 þ ww’ dw0 dw’; exp t t0
W
ð12Þ
ð6Þ
ð7Þ
f ðw0 ; w’; t0 Þ ¼ Pðw; wÞgðw0 ; w’; t0 Þ;
ð8Þ
and d0 is defined in Eq. (5). A polynomial astigmatic Gaussian beam consists of the astigmatic Gaussian beam in Eq. (1) or Eq. (6), multiplied by a complex-valued bivariate polynomial prefactor. A bivariate polynomial that can be factored into a product of first order polynomials is called fully reducible. Using matrix–vector notation and helical coordinates, one can represent any bivariate polynomial as a sum of fully reducible polynomials [30] Ndiv2 X N2p Y
nn;p
where W represents a t-dependent complex-valued constant; D represents the infinite two-dimensional integration domain in the input plane; the primed helical coordinates ðw0 ; w’Þ are defined on the input plane and the unprimed helical coordinates ðw; wÞ are defined on the output plane. We now substitute
!
where
Pðw; wÞ ¼ A0
W n;p ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g2 þ iðt þ sÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g cos b 1 þ g2 þ iðt sÞ;
Q uu ¼ g cos b
1=2
ln;p ¼
"
where,
where A0 is an overall complex-valued factor, W is given in Eq. (7),
in Eq. (12), where Pðw; wÞ is the polynomial prefactor given in Eq. (9) and gðw0 ; w’; t 0 Þ is the astigmatic Gaussian beam given in Eq. (6) at t ¼ t 0 . The polynomial prefactor is now moved out of the Fresnel integral by replacing the transverse helical coordinates with partial derivatives
t t0 i2 t t0 w’ ! i2 w0 !
1 Here the term ‘cylindrical’ is meant in the paraxial sense. In actual fact the wavefront has a parabolic dependence along one transverse dimension and is constant along the orthogonal transverse dimension.
o ; ow o : ow
ð14Þ ð15Þ
The Fresnel integral now only contains the astigmatic Gaussian beam evaluated at t ¼ t 0 . This integral can be solved to obtain an expression for the field in the output plane given by
ð9Þ
n¼1
ð13Þ
f ðw; w; tÞ ¼
1=2 i2ww 2 exp t t0 t t0 pd0 ! t t0 o t t0 o W T QW ; exp P ; d0 i2 ow i2 ow
W
ð16Þ
F.S. Roux / Optics Communications 281 (2008) 4205–4210
where Pð Þ represents the differential operator that is obtained after the replacements of Eqs. (14) and (15) are made in the polynomial prefactor. It acts on the exponential function on its right-hand side, for which
Q¼
Q 11
Q 12
Q 21
Q 22
ð17Þ
;
with,
Q 11 ¼ is þ g expðibÞ;
ð18Þ
Q 22 ¼ is þ g expðibÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q 12 ¼ Q 21 ¼ i t þ i 1 þ g2 i
ð19Þ d0 ; ðt t 0 Þ
ð20Þ
and d0 is given in Eq. (5). The result after the differential operator acted on the exponential function is that the exponential function is multiplied by the polynomial prefactor at an arbitrary point along the propagation direction. The exponential function can be combined with the first exponential toward the left-hand side of the differential operator in Eq. (16) to reproduce the expression for the astigmatic Gaussian beam in Eq. (6). We are interested in the polynomial prefactor that is formed by the differential operator. The latter can, with the aid of Eq. (9), be expressed in matrix–vector notation as follows:
P ¼ A0
Ndiv2 X N2p Y p¼0
T n;p ðW
l
W n;p Þ;
ð21Þ
n¼1
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where ln and W n , respectively represent the morphology vectors and the shift vectors for the two first order polynomial factors. The prefactor that is generated with this differential operator is given by
P2 ¼ flT1 ½iðt t 0 ÞQW W 1 gflT2 ½iðt t 0 ÞQW W 2 g 1 þ ðt t 0 Þ2 lT1 Q l2 : 2
ð27Þ
Note that, in addition to the two first order polynomial factors, the propagation process also generates a coupling term, containing the morphology vectors of both first order polynomial factors. This is similar to the situation with polynomial stigmatic Gaussian beams [31]. The coupling term represents the coupling strength. In this case it is a function of t given by
1 2 ðt t0 Þ2 iðn2 n1 f2 f1 Þg sin b þ ðn2 n1 þ f2 f1 Þðg cos b þ isÞ ¼ 2d0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d0 : þ ðn2 f1 þ f2 n1 Þ it 1 þ g2 i t t0
j ¼ ðt t0 Þ2 lT1 Q l2
ð28Þ Note also that the coupling term is two orders lower than the order of the polynomial. For higher order polynomials more coupling terms are generated during propagation. However, the coupling terms are always of lower order than the order of the polynomial.
where,
" tt W¼
0 o i2 ow tt 0 o i2 ow
#
4. Global topological charge
:
ð22Þ
Note that when one applies the operators in Eq. (22) on the vector in Eq. (7) one obtains
WW T ¼
t t0 0 1 t t0 : ¼ i2 i2 1 0
ð23Þ
Next we consider a first order polynomial prefactor and a second order polynomial prefactor to see how the above process produce the desired polynomial prefactors. 3.1. First order polynomial prefactor For a first order polynomial prefactor the differential operator is given by
P1 ¼ lT0 ðW W 0 Þ
ð24Þ
where l0 is the morphology vector, defined as in Eq. (10) and W 0 is the shift vector, defined as in Eq. (11). The prefactor that is generated when this differential operator acts on the exponential function is given by
P1 ¼ lT0 ½iðt t 0 ÞQW W 0 ð25Þ
3.2. Second order polynomial prefactor Consider now a second order polynomial prefactor that is fully reducible in an input plane at t ¼ t0 . The differential operator that represents this prefactor is given by
P2 ¼ ½lT1 ðW W 1 Þ½lT2 ðW W 2 Þ;
m¼
1 2p
I
rT hðu; vÞ d^l;
ð26Þ
ð29Þ
C
where hðu; vÞ is the phase function of the beam on a cross-section perpendicular to the direction of propagation; C represents a circular contour with a radius that approaches infinity; rT is the twodimensional gradient operator on the cross-section; and d^l is the integration measure along the contour. One can compute the phase of any complex-valued function f ðu; vÞ with,
hðu; vÞ ¼ i ln½f ðu; vÞ þ i ln½jf ðu; vÞj:
¼ n0 ½iðt t0 ÞðQ 21 w þ Q 22 wÞ w0 þ f0 ½iðt t 0 ÞðQ 11 w Q 12 wÞ w0 :
Now we investigate the evolution of the global topological charge in the beam along the direction of propagation. It is known that for a polynomial stigmatic Gaussian beam (where the Gaussian envelope of the beam is rotationally symmetric) the global topological charge is conserved along the direction of propagation [31]. However, it is also known that this is not in general true for polynomial astigmatic Gaussian beams. In fact, when a Gaussian beam is sent through a cylindrical lens, the topological charge is inverted at same distance behind the lens [20]. Here we consider the most general polynomial astigmatic Gaussian beam to find the conditions for topological charge inversion. The global topological charge is given by
ð30Þ
However, since the second term in Eq. (30) will not make any contribution to the topological charge under the contour integral in Eq. (29), one only needs to use the first term in Eq. (30). Moreover, the astigmatic Gaussian envelope contains no topological defects in its phase and will therefore not contribute to the topological charge. So one only needs to substitute the prefactor into the first term in Eq. (30). Furthermore, since the contour integral in Eq. (29) is evaluated at points far from the origin, only the leading order terms in the prefactor need to be considered. Hence one can compute the global topological charge using,
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m¼
F.S. Roux / Optics Communications 281 (2008) 4205–4210
i 2p
Z
2p
0
o lnðPL Þd/; o/
ð31Þ
where PL is the prefactor in polar coordinates with only the leading order terms at an arbitrary point t along the propagation direction, which is given by, N
PL ðr; /Þ ¼ i ðt t0 Þ
N Y N
lTn QW
n¼1 N
¼ i ðt t0 ÞN
N Y
½nn ðQ 21 w þ Q 22 wÞ þ fn ðQ 11 w þ Q 12 wÞ;
ð32Þ
n¼1
with w ¼ r expði/Þ and w ¼ r expði/Þ. The resulting expression for the global topological charges after evaluating Eq. (31) is given by,
m ¼ N þ
N I 1 X 1 1 dz; þ i2p n¼1 C z zn z þ zn
ð33Þ
where z ¼ expði/Þ and
1=2 n Q þ fn Q 12 zn ¼ n 22 nn Q 21 þ fn Q 11
ð34Þ
and where we used contour integration with the contour of integration C going around the unit circle. Each term will give a value of i2p if j zn j< 1. So we see that the global topological charge can have values from N to N in steps of 2, depending on how many of the poles zn lie inside the unit circle. The leading order terms of the prefactor produce a fully reducible bivariate polynomial. An Nth order fully reducible bivariate polynomial has precisely N zeros, which represent N vortices, which in turn give rise to the global topological charge in the beam. Since each vortex has a topological charge of either 1 or 1, the global topological charge of an Nth order bivariate polynomial cannot be larger than N or smaller than N and it must change in steps of 2. If one includes all the lower order terms of the polynomial prefactor, it is in general not fully reducible. This implies that the total number of vortices can exceed N. However, those vortices in access of the global topological charge must appear in oppositely charged pairs, which are called vortex dipoles. These dipoles can appear or disappear during propagation without affecting the global topological charge of the beam. On the other hand, if one considers only the leading order terms of the prefactor, then one would not find all the vortex dipoles that are in the beam and one also looses the information about the lateral locations of the vortices in the beam, but one can obtain the correct value for the global topological charge. 5. Flip plane locations The global topological charge in a polynomial astigmatic Gaussian beam can change during propagation. The reason follows from the analysis done in Section 4. The global topological charge at any particular point along the propagation direction is given by the number of poles zn that lie inside the unit circle. However, from Eqs. (34), (18), (19) and (20) we see that the poles are functions of the propagation distance t. The poles can therefore move into, or out of, the unit circle during propagation. At any point along t where j zn j¼ 1 the topological charge of the associated vortex is inverted, which causes the global topological charge to change by a value of 2. The transverse plane at such a point is called a flip plane. One can find the locations for the flip planes along t by solving j zn j2 ¼ 1. The latter gives the following equation:
h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ 2sg sinðbÞ 2j2 s 1 þ g2 j3 ðs2 1Þ t 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffii þ 2g j1 ðs2 þ 1Þ sinðbÞ j2 ðs2 1Þ cosðbÞ 2j3 s cosðbÞ 1 þ g2 t j3 ½4g2 s2 cos2 ðbÞ þ ðs2 þ 1Þ2 ;
ð35Þ
where,
j1 ¼ nf þ n f ¼ sinðwÞ cosð/Þ; j2 ¼ iðnf n fÞ ¼ sinðwÞ sinð/Þ; j3 ¼ nn ff ¼ cosðwÞ:
ð36Þ ð37Þ ð38Þ
In Eq. (35) we picked the input plane at t ¼ t0 ¼ 0. The only real effect that this choice has is the particular values of the morphology parameters of the polynomial prefactor. The equation in Eq. (35) is second order in t, which means that there are potentially 2 flip planes for each vortex. The flip planes of the different vortices are in general located at different points. The locations depend on the beam parameters (s, g and b), which are the same for all vortices in the beam, but they also depend on the morphologies of the respective vortices, which are in general different for the different vortices. For general parameters the solutions of Eq. (35) are rather complicated expressions. Here we consider a few special cases, some of which serve as confirmation by being in agreement with previous results. 5.1. A symmetrical astigmatic beam When one sets g ¼ 0 one obtains a beam that is symmetric in the forward and backward directions apart from a 90° rotation around the propagation axis. The expression in Eq. (35) then simplifies to
0 ¼ ½2s sinðwÞ sinð/Þ cosðwÞðs2 1Þt 2 þ cosðwÞðs2 þ 1Þ2 :
ð39Þ
The resulting expression is now independent of b. The only remaining beam parameter is s. The flip plans are then located at
t¼
s2 þ 1 ½s2 2s tanðwÞ sinð/Þ 11=2
;
ð40Þ
for
s > sinð/Þ tanðwÞ þ ½sinð/Þ2 tanðwÞ2 þ 11=2 ;
ð41Þ
which is in agreement with [23]. Note that the two flip planes are located at equal distances on opposite sides of the central plane of the astigmatic Gaussian beam. 5.2. A non-astigmatic elliptical beam For s ¼ 0 the two focal planes are both located in the central plane. Although the resulting beam is not astigmatic it can still be elliptical as a result of a non-zero value of g. The expression in Eq. (35) then becomes
0 ¼ cosðwÞt 2 g sinð/ þ bÞ sinðwÞt þ cosðwÞ:
ð42Þ
It can be seen that with a rotation / ! / b one can remove the bdependence. Here the only remaining beam parameter is g. The flip planes are then located at
t¼
1=2 1 1 g sinð/Þ tanðwÞ g2 sin2 ð/Þ2 tan2 ðwÞ 1 ; 2 4
ð43Þ
2 ; sinð/Þ tanðwÞ
ð44Þ
for
g>
F.S. Roux / Optics Communications 281 (2008) 4205–4210
4209
which is also in agreement with [23]. Here both flip planes lie on the same side of the central plane. 5.3. Flip plane in the central plan One can force one of the flip planes to be located in the central plane by setting the morphology in the central plane equal to that of an edge dislocation ðw ¼ p=2Þ. The resulting expression for the equation for the flip planes, from Eq. (35), is
h i 0 ¼ s ð1 þ g2 Þ1=2 sinð/Þ g sinðbÞ t2 þ g½sinð/ bÞs2 sinð/ þ bÞt:
ð45Þ
The other flip plane is located at
t¼
g½sinð/ þ bÞ sinð/ bÞs2 : s½ð1 þ g2 Þ1=2 sinð/Þ g sinðbÞ
ð46Þ
6. Numerical example Here we consider an arbitrary case with the aid of a numerical simulation. We select the parameters of the astigmatic Gaussian beam to be s ¼ 1=2, g ¼ 7=24 and b ¼ p=3. We consider the propagation of a single non-canonical vortex2 located on the optical axis. The morphology of this vortex in the central plane ðt ¼ 0Þ is given by w ¼ 5p=8 and / ¼ 2p=3. The input beam in the central plane is represented by a 1024 1024 array of complex samples. The amplitude function and the phase function of the input beam are shown in Fig. 1a and b, respectively. The single non-canonical vortex located in the center can be clearly seen in both figures. The phase function in Fig. 1b has the typical saddle shape that one would expect from the phase function in the central plane of an astigmatic beam. With the aid of Eq. (35) one can determine that the flip plane for this case is located at t ¼ 1:435. We now use numerical beam propagation to investigate the behavior of the vortex. In Fig. 2 we show the amplitude functions and the phase functions obtained in three different planes: a small distance in front of the flip plane ðt ¼ 1:335Þ, at the location of the flip plane ðt ¼ 1:435Þ and a small distance behind the flip plane ðt ¼ 1:535Þ. The three amplitude functions, shown in Fig. 2a, c and e, show very little difference apart from a slight increase in size. They all have a thin dark line through them, which indicates the presence of a highly anisotropic vortex. In fact, in the flip plane the amplitude functions in Fig. 2c contains an edge dislocation. The phase function in front of the flip plane at t ¼ 1:335, shown in Fig. 2b, contains a vortex with the same topological charge as the vortex in the input plane, shown in Fig. 1b. At the flip plane ðt ¼ 1:435Þ the phase function contains an edge dislocation, as shown in Fig. 2d. Such an edge dislocation can be regarded as a pair of oppositely charged vortices going off to (or coming in from) infinity from opposite directions, thereby providing the mechanism for topological charge inversion. The phase function behind the flip plane at t ¼ 1:535, which is shown in Fig. 2f, contains a vortex with the opposite topological charge to that of the vortex in the input plane, shown in Fig. 1b. This confirms that topological charge inversion occurred at the flip plane and that the flip plane is located at its predicted location.
Fig. 1. The input beam in the central plane at t ¼ 0, showing (a) the amplitude function and (b) the phase function. One observes a single vortex on the optical axis.
7. Summary and conclusions The evolution of the global topological charge in a general polynomial astigmatic Gaussian beam is investigated. The global topo-
2
We could also have considered multiple vortices, but their behavior can become rather complicated in the region of the optical axis due to the subleading order terms, which are not considered in this analysis.
Fig. 2. The vortex bearing beam during propagation: (a) the amplitude function at t ¼ 1:335; (b) the phase function at t ¼ 1:335; (c) the amplitude function at t ¼ 1:435; (d) the phase function at t ¼ 1:435; (e) the amplitude function at t ¼ 1:535; and (f) the phase function at t ¼ 1:535.
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F.S. Roux / Optics Communications 281 (2008) 4205–4210
logical charge is determined by the optical vortices that exist in the beam as a result of a bivariate polynomial prefactor of finite order with which the astigmatic Gaussian beam is multiplied. Only the leading order terms of this polynomial is considered because they determine the global topological charge. The leading order terms can be factored into a product of first order polynomials, each of which represents an optical vortex with a specific morphology. The maximum value of the global topological charge is given by the order of the polynomial. During propagation the value of the global topological charge can change in steps of 2. Such a change would occur due to the charge inversion of an optical vortex at a specific point along the propagation direction, which is called a flip plane. The locations of these flip planes are determined by the parameters of the astigmatic Gaussian beam and the morphologies of the first order polynomial factors of the leading order terms of the polynomial prefactor. A numerical simulation is provided to confirm the predicted location of the flip plane, as well as that charge inversion does occur in the flip plane. References [1] [2] [3] [4] [5]
J.F. Nye, M.V. Berry, Proc. R. Soc. London, A 336 (1974) 165. P. Cullet, L. Gil, F. Rocca, Opt. Commun. 73 (1989) 403. G. Indebetouw, J. Mod. Opt. 40 (1993) 73. F.S. Roux, J. Opt. Soc. Am. B 12 (1995) 1215. D. Rozas, C.T. Law, G.G.A. Swartzlander Jr., J. Opt. Soc. Am. B 14 (1997) 3054.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
I. Freund, Opt. Commun. 181 (2000) 19. I. Freund, Opt. Commun. 187 (2001) 71. G. Molina-Terriza, E.M. Wright, L. Torner, Opt. Lett. 26 (2001) 163. F.S. Roux, J. Opt. Soc. Am. B 21 (2004) 655. F.S. Roux, J. Opt. Soc. Am. B 21 (2004) 664. F. Flossmann, U.T. Schwarz, M. Maier, Opt. Commun. 250 (2005) 218. L. Allen, M.W. Beijerbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 13 (1996) 967. M.S. Soskin, V.N. Gorshkov, M.V. Vasnetsov, J.T. Malos, N.R. Heckenberg, Phys. Rev. A 56 (1997) 4064. M.V. Berry, Proc. SPIE 3487 (1998) 6. M.J. Padgett, J. Courtial, Opt. Lett. 21 (1999) 430. A.T. O’Neil, I. MacVicar, L. Allen, M.J. Padgett, Phys. Rev. Lett. 88 (2002) 053601. F.S. Roux, Opt. Commun. 242 (2004) 45. J.E. Curtis, D.G. Grier, Phys. Rev. Lett. 90 (2003) 133901. G. Molina-Terriza, J. Recolons, J.P. Torres, L. Torner, E.M. Wright, Phys. Rev. Lett. 87 (2001) 023902. A.Y. Bekshaev, M.S. Soskin, M.V. Vasnetsov, J. Opt. Soc. Am. A 20 (2003) 1635. A. Wada, T. Ohtami, Y. Miyamoto, M. Takeda, J. Opt. Soc. Am. A 22 (2005) 2746. F.S. Roux, S. Afr. J. Sci. 102 (2006) 601. E. Abramochkin, V. Volostnikov, Opt. Commun. 102 (1993) 336. R. Piestun, Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 17 (2000) 294. F.S. Roux, Opt. Commun. 268 (2006) 196. W.D. Montgomery, J. Opt. Soc. Am. 57 (1967) 772. Z. Bouchal, R. Horak, J. Wagner, J. Mod. Opt. 43 (1996) 1905. M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, New York, 1980. F.S. Roux, Phys. Rev. A 77 (2008) 033837. F.S. Roux, Opt. Commun. 266 (2006) 433. J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw Hill, Boston, 1996.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Topological charge inversion in polynomial astigmatic Gaussian beams Filippus S. Roux Department of Electrical, Electronic and Computer Engineering, University of Pretoria, Pretoria 0001, South Africa
a r t i c l e
i n f o
Article history: Received 4 January 2008 Received in revised form 28 April 2008 Accepted 14 May 2008
PACS: 42.25.Bs 42.15.Fr 42.25.Fx
a b s t r a c t The evolution of the global topological charge in a general polynomial astigmatic Gaussian beam is investigated. The leading order terms of the polynomial prefactor determines the global topological charge and can be expressed as a product of first order polynomials, each representing an optical vortex function. We show that the global topological charge is bounded by the order of the polynomial and change during propagation in steps of 2 every time one of the optical vortices undergo topological charge inversion. We investigate the locations of the flip planes where charge inversions occur and provide expressions for a number of special cases. Numerical results are provided for an example of such a polynomial astigmatic Gaussian beam. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Singular optics Optical vortices Astigmatic Gaussian beams Topological charge
1. Introduction The behavior of phase singularities, also called optical vortices [1,2], in optical beams have attracted much attention [3–11]. A beam that contains such vortices carries orbital angular momentum [12–18], which can be transferred to small particles [19]. In this way singular optics plays an essential role in nanotechnology. For the successful application of singular optics, it is essential that the vortices exist at the precise locations in the beam where they are needed and with the desired properties. The parameters of optical vortices are sensitive to perturbations of the beam. It is therefore necessary to understand the effects of such perturbations on the optical vortex parameters. Here we are in particular interested in the effects of astigmatism on the optical vortex parameters. It has been observed that, in an astigmatic beam, optical vortices can invert their topological charge [20]. Previous analyses either focussed on the properties of astigmatic Laguerre–Gaussian beams [21,22] or considered only a single vortex in the astigmatic beam [23]. In this paper we investigate general polynomial astigmatic Gaussian beams, consisting of an astigmatic Gaussian beam multiplied by a complex-valued bivariate polynomial prefactor of finite order. These polynomial prefactors include all sets of orthogonal functions that can be expressed as polynomials of finite order, such as the Laguerre polynomials and the Hermit polynomials. PolynoE-mail address: [email protected] 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.05.012
mial (stigmatic) Gaussian beams are ubiquitous. In addition to the well known Laguerre–Gaussian beams and Hermit–Gaussian beams, they also include the Indebetouw beams [3], spiral beams [24] and other stable beams [25,26], but not non-diffracting beams [27,28] such as Bessel beams because such prefactors tend to be transcendental functions and not of finite order. We define our notation for the general expression for an astigmatic Gaussian beam in Section 2. The complex zeros of the polynomial prefactor represent the optical vortices in the beam. The propagation of polynomial astigmatic Gaussian beams is discussed in Section 3. The cases of first order and second order polynomial prefactors are considered in detail. The evolution of the global topological charge in such a beam is analysed in Section 4. The flip plane locations are discussed in Section 5 and we consider some special cases to show that they are in agreement with previous work [23]. The results of a numerical simulation of the propagation of an example of a polynomial astigmatic Gaussian beam is reported in Section 6. We end with some conclusions in Section 7. 2. Polynomial astigmatic Gaussian beams Astigmatism is an aberration often found in optical systems [29]. It manifests as a longitudinal separation of the focal point – two different transverse directions are focused at different points along the optical axis. For instance, the beam can be focused along the x-direction to form a focal line along the y-direction at z ¼ z1 and then be focused along the y-direction to form a focal line along
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the x-direction some distance later at z ¼ z2 . In the most general astigmatic beam the two focal lines need not be orthogonal. For the purposes of the investigation reported here we are interested in astigmatic Gaussian beams. A general astigmatic Gaussian beam can be expressed by
gðu; v; tÞ ¼
2 pd0
1=2
Q u2 þ 2Q uv uv þ Q vv v2 ; exp uu d0
ð1Þ
ð2Þ
Q vv
ð3Þ
Q uv ¼ g sin b; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d0 ¼ ðt þ i 1 þ g2 Þ2 s2 þ g2 þ i2gs cos b:
ð4Þ ð5Þ
Here we specify the transverse coordinates, u and v, in units of some convenient scale parameter x, while the coordinate along the direction of propagation t is measured in units of q ¼ px2 =k, with k being the wavelength. In other words, u ¼ x=x, v ¼ y=x and t ¼ z=q, where ðx; y; zÞ represents the set of Cartesian coordinates. There are three parameters that quantify the properties of the astigmatic beam: s and g are non-negative real-valued parameters and b is an angular parameter within the range p=2 < b < p=2. The parameterization that we use here differs from the parameterization used by other authors [21,22], because it is more convenient for our purposes and provides us with a way to express astigmatic Gaussian beams in their most general form. The astigmatic Gaussian beam has two focal planes located at t ¼ s and t ¼ s. The shape of the wavefront of the beam in each of these focal planes is purely cylindrical.1 The orientation of these cylindrical wavefronts depend on b and g. For g ¼ 0 or b ¼ 0 the two cylindrical wavefronts are oriented at 90° with respect to each other. In the region between these two focal planes the wavefront of the Gaussian beam has a saddle shape and outside of this region the wavefront is a non-symmetrical paraboloid. In the derivation below we shall use helical coordinates pffiffiffi for the transverse p directions, defined by w ¼ ðu þ ivÞ= 2 and ffiffiffi w ¼ ðu ivÞ= 2. It will also be more convenient to use matrix– vector notation. For that purpose we express the astigmatic Gaussian beam more compactly as
gðw; w; tÞ ¼
2
T
exp
pd0
W UW ; d0
w ; W¼ w " pffiffiffiffiffiffiffiffiffiffiffiffiffiffi # g expðibÞ þ is it 1 þ g2 U¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g expðibÞ þ is it 1 þ g2
p¼0
lTn;p W W n;p ;
fn;p
;
ð10Þ
is a morphology vector that defines the morphology [13,10,18] of the complex zero (vortex) associated with the nth first order polynomial factor in the pth factored polynomial, and
# wn;p ; wn;p
ð11Þ
represents a shift in the coordinates for the ðn; pÞ-th first order polynomial factor. As such wn;p represents the location of the complex zero of the first order polynomial factor. The orders of the reducible polynomials in the summation in Eq. (9) differ by 2, because a reducible Nth order polynomial contains as many degrees of freedom as there are for all the Nth order and ðN 1Þth order terms in a general Nth order complex-valued bivariate polynomial. 3. Propagation of polynomial astigmatic Gaussian beams In general the parameters in the expression of the bivariate polynomial prefactor in Eq. (9) depends on the position along the propagation direction. To find out how the prefactor changes during propagation one can follow a procedure such as the one provided in [31]. Here we shall adapt this procedure to the case of an astigmatic Gaussian beam. The polynomial prefactor of the astigmatic Gaussian beam is specified on an input plane at t ¼ t 0 . To find the expression for the prefactor at an output plane at t > t0 , one can use the Fresnel integral [32]. In terms of the normalized helical coordinates the Fresnel integral is formally expressed as
f ðw; w; tÞ ¼
Z Z i2ww exp f ðw0 ; w’; t 0 Þ t t0 t t0 D i2w’w’ exp t t0 i2 ½ww0 þ ww’ dw0 dw’; exp t t0
W
ð12Þ
ð6Þ
ð7Þ
f ðw0 ; w’; t0 Þ ¼ Pðw; wÞgðw0 ; w’; t0 Þ;
ð8Þ
and d0 is defined in Eq. (5). A polynomial astigmatic Gaussian beam consists of the astigmatic Gaussian beam in Eq. (1) or Eq. (6), multiplied by a complex-valued bivariate polynomial prefactor. A bivariate polynomial that can be factored into a product of first order polynomials is called fully reducible. Using matrix–vector notation and helical coordinates, one can represent any bivariate polynomial as a sum of fully reducible polynomials [30] Ndiv2 X N2p Y
nn;p
where W represents a t-dependent complex-valued constant; D represents the infinite two-dimensional integration domain in the input plane; the primed helical coordinates ðw0 ; w’Þ are defined on the input plane and the unprimed helical coordinates ðw; wÞ are defined on the output plane. We now substitute
!
where
Pðw; wÞ ¼ A0
W n;p ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ g2 þ iðt þ sÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g cos b 1 þ g2 þ iðt sÞ;
Q uu ¼ g cos b
1=2
ln;p ¼
"
where,
where A0 is an overall complex-valued factor, W is given in Eq. (7),
in Eq. (12), where Pðw; wÞ is the polynomial prefactor given in Eq. (9) and gðw0 ; w’; t 0 Þ is the astigmatic Gaussian beam given in Eq. (6) at t ¼ t 0 . The polynomial prefactor is now moved out of the Fresnel integral by replacing the transverse helical coordinates with partial derivatives
t t0 i2 t t0 w’ ! i2 w0 !
1 Here the term ‘cylindrical’ is meant in the paraxial sense. In actual fact the wavefront has a parabolic dependence along one transverse dimension and is constant along the orthogonal transverse dimension.
o ; ow o : ow
ð14Þ ð15Þ
The Fresnel integral now only contains the astigmatic Gaussian beam evaluated at t ¼ t 0 . This integral can be solved to obtain an expression for the field in the output plane given by
ð9Þ
n¼1
ð13Þ
f ðw; w; tÞ ¼
1=2 i2ww 2 exp t t0 t t0 pd0 ! t t0 o t t0 o W T QW ; exp P ; d0 i2 ow i2 ow
W
ð16Þ
F.S. Roux / Optics Communications 281 (2008) 4205–4210
where Pð Þ represents the differential operator that is obtained after the replacements of Eqs. (14) and (15) are made in the polynomial prefactor. It acts on the exponential function on its right-hand side, for which
Q¼
Q 11
Q 12
Q 21
Q 22
ð17Þ
;
with,
Q 11 ¼ is þ g expðibÞ;
ð18Þ
Q 22 ¼ is þ g expðibÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q 12 ¼ Q 21 ¼ i t þ i 1 þ g2 i
ð19Þ d0 ; ðt t 0 Þ
ð20Þ
and d0 is given in Eq. (5). The result after the differential operator acted on the exponential function is that the exponential function is multiplied by the polynomial prefactor at an arbitrary point along the propagation direction. The exponential function can be combined with the first exponential toward the left-hand side of the differential operator in Eq. (16) to reproduce the expression for the astigmatic Gaussian beam in Eq. (6). We are interested in the polynomial prefactor that is formed by the differential operator. The latter can, with the aid of Eq. (9), be expressed in matrix–vector notation as follows:
P ¼ A0
Ndiv2 X N2p Y p¼0
T n;p ðW
l
W n;p Þ;
ð21Þ
n¼1
4207
where ln and W n , respectively represent the morphology vectors and the shift vectors for the two first order polynomial factors. The prefactor that is generated with this differential operator is given by
P2 ¼ flT1 ½iðt t 0 ÞQW W 1 gflT2 ½iðt t 0 ÞQW W 2 g 1 þ ðt t 0 Þ2 lT1 Q l2 : 2
ð27Þ
Note that, in addition to the two first order polynomial factors, the propagation process also generates a coupling term, containing the morphology vectors of both first order polynomial factors. This is similar to the situation with polynomial stigmatic Gaussian beams [31]. The coupling term represents the coupling strength. In this case it is a function of t given by
1 2 ðt t0 Þ2 iðn2 n1 f2 f1 Þg sin b þ ðn2 n1 þ f2 f1 Þðg cos b þ isÞ ¼ 2d0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d0 : þ ðn2 f1 þ f2 n1 Þ it 1 þ g2 i t t0
j ¼ ðt t0 Þ2 lT1 Q l2
ð28Þ Note also that the coupling term is two orders lower than the order of the polynomial. For higher order polynomials more coupling terms are generated during propagation. However, the coupling terms are always of lower order than the order of the polynomial.
where,
" tt W¼
0 o i2 ow tt 0 o i2 ow
#
4. Global topological charge
:
ð22Þ
Note that when one applies the operators in Eq. (22) on the vector in Eq. (7) one obtains
WW T ¼
t t0 0 1 t t0 : ¼ i2 i2 1 0
ð23Þ
Next we consider a first order polynomial prefactor and a second order polynomial prefactor to see how the above process produce the desired polynomial prefactors. 3.1. First order polynomial prefactor For a first order polynomial prefactor the differential operator is given by
P1 ¼ lT0 ðW W 0 Þ
ð24Þ
where l0 is the morphology vector, defined as in Eq. (10) and W 0 is the shift vector, defined as in Eq. (11). The prefactor that is generated when this differential operator acts on the exponential function is given by
P1 ¼ lT0 ½iðt t 0 ÞQW W 0 ð25Þ
3.2. Second order polynomial prefactor Consider now a second order polynomial prefactor that is fully reducible in an input plane at t ¼ t0 . The differential operator that represents this prefactor is given by
P2 ¼ ½lT1 ðW W 1 Þ½lT2 ðW W 2 Þ;
m¼
1 2p
I
rT hðu; vÞ d^l;
ð26Þ
ð29Þ
C
where hðu; vÞ is the phase function of the beam on a cross-section perpendicular to the direction of propagation; C represents a circular contour with a radius that approaches infinity; rT is the twodimensional gradient operator on the cross-section; and d^l is the integration measure along the contour. One can compute the phase of any complex-valued function f ðu; vÞ with,
hðu; vÞ ¼ i ln½f ðu; vÞ þ i ln½jf ðu; vÞj:
¼ n0 ½iðt t0 ÞðQ 21 w þ Q 22 wÞ w0 þ f0 ½iðt t 0 ÞðQ 11 w Q 12 wÞ w0 :
Now we investigate the evolution of the global topological charge in the beam along the direction of propagation. It is known that for a polynomial stigmatic Gaussian beam (where the Gaussian envelope of the beam is rotationally symmetric) the global topological charge is conserved along the direction of propagation [31]. However, it is also known that this is not in general true for polynomial astigmatic Gaussian beams. In fact, when a Gaussian beam is sent through a cylindrical lens, the topological charge is inverted at same distance behind the lens [20]. Here we consider the most general polynomial astigmatic Gaussian beam to find the conditions for topological charge inversion. The global topological charge is given by
ð30Þ
However, since the second term in Eq. (30) will not make any contribution to the topological charge under the contour integral in Eq. (29), one only needs to use the first term in Eq. (30). Moreover, the astigmatic Gaussian envelope contains no topological defects in its phase and will therefore not contribute to the topological charge. So one only needs to substitute the prefactor into the first term in Eq. (30). Furthermore, since the contour integral in Eq. (29) is evaluated at points far from the origin, only the leading order terms in the prefactor need to be considered. Hence one can compute the global topological charge using,
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m¼
F.S. Roux / Optics Communications 281 (2008) 4205–4210
i 2p
Z
2p
0
o lnðPL Þd/; o/
ð31Þ
where PL is the prefactor in polar coordinates with only the leading order terms at an arbitrary point t along the propagation direction, which is given by, N
PL ðr; /Þ ¼ i ðt t0 Þ
N Y N
lTn QW
n¼1 N
¼ i ðt t0 ÞN
N Y
½nn ðQ 21 w þ Q 22 wÞ þ fn ðQ 11 w þ Q 12 wÞ;
ð32Þ
n¼1
with w ¼ r expði/Þ and w ¼ r expði/Þ. The resulting expression for the global topological charges after evaluating Eq. (31) is given by,
m ¼ N þ
N I 1 X 1 1 dz; þ i2p n¼1 C z zn z þ zn
ð33Þ
where z ¼ expði/Þ and
1=2 n Q þ fn Q 12 zn ¼ n 22 nn Q 21 þ fn Q 11
ð34Þ
and where we used contour integration with the contour of integration C going around the unit circle. Each term will give a value of i2p if j zn j< 1. So we see that the global topological charge can have values from N to N in steps of 2, depending on how many of the poles zn lie inside the unit circle. The leading order terms of the prefactor produce a fully reducible bivariate polynomial. An Nth order fully reducible bivariate polynomial has precisely N zeros, which represent N vortices, which in turn give rise to the global topological charge in the beam. Since each vortex has a topological charge of either 1 or 1, the global topological charge of an Nth order bivariate polynomial cannot be larger than N or smaller than N and it must change in steps of 2. If one includes all the lower order terms of the polynomial prefactor, it is in general not fully reducible. This implies that the total number of vortices can exceed N. However, those vortices in access of the global topological charge must appear in oppositely charged pairs, which are called vortex dipoles. These dipoles can appear or disappear during propagation without affecting the global topological charge of the beam. On the other hand, if one considers only the leading order terms of the prefactor, then one would not find all the vortex dipoles that are in the beam and one also looses the information about the lateral locations of the vortices in the beam, but one can obtain the correct value for the global topological charge. 5. Flip plane locations The global topological charge in a polynomial astigmatic Gaussian beam can change during propagation. The reason follows from the analysis done in Section 4. The global topological charge at any particular point along the propagation direction is given by the number of poles zn that lie inside the unit circle. However, from Eqs. (34), (18), (19) and (20) we see that the poles are functions of the propagation distance t. The poles can therefore move into, or out of, the unit circle during propagation. At any point along t where j zn j¼ 1 the topological charge of the associated vortex is inverted, which causes the global topological charge to change by a value of 2. The transverse plane at such a point is called a flip plane. One can find the locations for the flip planes along t by solving j zn j2 ¼ 1. The latter gives the following equation:
h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ 2sg sinðbÞ 2j2 s 1 þ g2 j3 ðs2 1Þ t 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffii þ 2g j1 ðs2 þ 1Þ sinðbÞ j2 ðs2 1Þ cosðbÞ 2j3 s cosðbÞ 1 þ g2 t j3 ½4g2 s2 cos2 ðbÞ þ ðs2 þ 1Þ2 ;
ð35Þ
where,
j1 ¼ nf þ n f ¼ sinðwÞ cosð/Þ; j2 ¼ iðnf n fÞ ¼ sinðwÞ sinð/Þ; j3 ¼ nn ff ¼ cosðwÞ:
ð36Þ ð37Þ ð38Þ
In Eq. (35) we picked the input plane at t ¼ t0 ¼ 0. The only real effect that this choice has is the particular values of the morphology parameters of the polynomial prefactor. The equation in Eq. (35) is second order in t, which means that there are potentially 2 flip planes for each vortex. The flip planes of the different vortices are in general located at different points. The locations depend on the beam parameters (s, g and b), which are the same for all vortices in the beam, but they also depend on the morphologies of the respective vortices, which are in general different for the different vortices. For general parameters the solutions of Eq. (35) are rather complicated expressions. Here we consider a few special cases, some of which serve as confirmation by being in agreement with previous results. 5.1. A symmetrical astigmatic beam When one sets g ¼ 0 one obtains a beam that is symmetric in the forward and backward directions apart from a 90° rotation around the propagation axis. The expression in Eq. (35) then simplifies to
0 ¼ ½2s sinðwÞ sinð/Þ cosðwÞðs2 1Þt 2 þ cosðwÞðs2 þ 1Þ2 :
ð39Þ
The resulting expression is now independent of b. The only remaining beam parameter is s. The flip plans are then located at
t¼
s2 þ 1 ½s2 2s tanðwÞ sinð/Þ 11=2
;
ð40Þ
for
s > sinð/Þ tanðwÞ þ ½sinð/Þ2 tanðwÞ2 þ 11=2 ;
ð41Þ
which is in agreement with [23]. Note that the two flip planes are located at equal distances on opposite sides of the central plane of the astigmatic Gaussian beam. 5.2. A non-astigmatic elliptical beam For s ¼ 0 the two focal planes are both located in the central plane. Although the resulting beam is not astigmatic it can still be elliptical as a result of a non-zero value of g. The expression in Eq. (35) then becomes
0 ¼ cosðwÞt 2 g sinð/ þ bÞ sinðwÞt þ cosðwÞ:
ð42Þ
It can be seen that with a rotation / ! / b one can remove the bdependence. Here the only remaining beam parameter is g. The flip planes are then located at
t¼
1=2 1 1 g sinð/Þ tanðwÞ g2 sin2 ð/Þ2 tan2 ðwÞ 1 ; 2 4
ð43Þ
2 ; sinð/Þ tanðwÞ
ð44Þ
for
g>
F.S. Roux / Optics Communications 281 (2008) 4205–4210
4209
which is also in agreement with [23]. Here both flip planes lie on the same side of the central plane. 5.3. Flip plane in the central plan One can force one of the flip planes to be located in the central plane by setting the morphology in the central plane equal to that of an edge dislocation ðw ¼ p=2Þ. The resulting expression for the equation for the flip planes, from Eq. (35), is
h i 0 ¼ s ð1 þ g2 Þ1=2 sinð/Þ g sinðbÞ t2 þ g½sinð/ bÞs2 sinð/ þ bÞt:
ð45Þ
The other flip plane is located at
t¼
g½sinð/ þ bÞ sinð/ bÞs2 : s½ð1 þ g2 Þ1=2 sinð/Þ g sinðbÞ
ð46Þ
6. Numerical example Here we consider an arbitrary case with the aid of a numerical simulation. We select the parameters of the astigmatic Gaussian beam to be s ¼ 1=2, g ¼ 7=24 and b ¼ p=3. We consider the propagation of a single non-canonical vortex2 located on the optical axis. The morphology of this vortex in the central plane ðt ¼ 0Þ is given by w ¼ 5p=8 and / ¼ 2p=3. The input beam in the central plane is represented by a 1024 1024 array of complex samples. The amplitude function and the phase function of the input beam are shown in Fig. 1a and b, respectively. The single non-canonical vortex located in the center can be clearly seen in both figures. The phase function in Fig. 1b has the typical saddle shape that one would expect from the phase function in the central plane of an astigmatic beam. With the aid of Eq. (35) one can determine that the flip plane for this case is located at t ¼ 1:435. We now use numerical beam propagation to investigate the behavior of the vortex. In Fig. 2 we show the amplitude functions and the phase functions obtained in three different planes: a small distance in front of the flip plane ðt ¼ 1:335Þ, at the location of the flip plane ðt ¼ 1:435Þ and a small distance behind the flip plane ðt ¼ 1:535Þ. The three amplitude functions, shown in Fig. 2a, c and e, show very little difference apart from a slight increase in size. They all have a thin dark line through them, which indicates the presence of a highly anisotropic vortex. In fact, in the flip plane the amplitude functions in Fig. 2c contains an edge dislocation. The phase function in front of the flip plane at t ¼ 1:335, shown in Fig. 2b, contains a vortex with the same topological charge as the vortex in the input plane, shown in Fig. 1b. At the flip plane ðt ¼ 1:435Þ the phase function contains an edge dislocation, as shown in Fig. 2d. Such an edge dislocation can be regarded as a pair of oppositely charged vortices going off to (or coming in from) infinity from opposite directions, thereby providing the mechanism for topological charge inversion. The phase function behind the flip plane at t ¼ 1:535, which is shown in Fig. 2f, contains a vortex with the opposite topological charge to that of the vortex in the input plane, shown in Fig. 1b. This confirms that topological charge inversion occurred at the flip plane and that the flip plane is located at its predicted location.
Fig. 1. The input beam in the central plane at t ¼ 0, showing (a) the amplitude function and (b) the phase function. One observes a single vortex on the optical axis.
7. Summary and conclusions The evolution of the global topological charge in a general polynomial astigmatic Gaussian beam is investigated. The global topo-
2
We could also have considered multiple vortices, but their behavior can become rather complicated in the region of the optical axis due to the subleading order terms, which are not considered in this analysis.
Fig. 2. The vortex bearing beam during propagation: (a) the amplitude function at t ¼ 1:335; (b) the phase function at t ¼ 1:335; (c) the amplitude function at t ¼ 1:435; (d) the phase function at t ¼ 1:435; (e) the amplitude function at t ¼ 1:535; and (f) the phase function at t ¼ 1:535.
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logical charge is determined by the optical vortices that exist in the beam as a result of a bivariate polynomial prefactor of finite order with which the astigmatic Gaussian beam is multiplied. Only the leading order terms of this polynomial is considered because they determine the global topological charge. The leading order terms can be factored into a product of first order polynomials, each of which represents an optical vortex with a specific morphology. The maximum value of the global topological charge is given by the order of the polynomial. During propagation the value of the global topological charge can change in steps of 2. Such a change would occur due to the charge inversion of an optical vortex at a specific point along the propagation direction, which is called a flip plane. The locations of these flip planes are determined by the parameters of the astigmatic Gaussian beam and the morphologies of the first order polynomial factors of the leading order terms of the polynomial prefactor. A numerical simulation is provided to confirm the predicted location of the flip plane, as well as that charge inversion does occur in the flip plane. References [1] [2] [3] [4] [5]
J.F. Nye, M.V. Berry, Proc. R. Soc. London, A 336 (1974) 165. P. Cullet, L. Gil, F. Rocca, Opt. Commun. 73 (1989) 403. G. Indebetouw, J. Mod. Opt. 40 (1993) 73. F.S. Roux, J. Opt. Soc. Am. B 12 (1995) 1215. D. Rozas, C.T. Law, G.G.A. Swartzlander Jr., J. Opt. Soc. Am. B 14 (1997) 3054.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
I. Freund, Opt. Commun. 181 (2000) 19. I. Freund, Opt. Commun. 187 (2001) 71. G. Molina-Terriza, E.M. Wright, L. Torner, Opt. Lett. 26 (2001) 163. F.S. Roux, J. Opt. Soc. Am. B 21 (2004) 655. F.S. Roux, J. Opt. Soc. Am. B 21 (2004) 664. F. Flossmann, U.T. Schwarz, M. Maier, Opt. Commun. 250 (2005) 218. L. Allen, M.W. Beijerbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 13 (1996) 967. M.S. Soskin, V.N. Gorshkov, M.V. Vasnetsov, J.T. Malos, N.R. Heckenberg, Phys. Rev. A 56 (1997) 4064. M.V. Berry, Proc. SPIE 3487 (1998) 6. M.J. Padgett, J. Courtial, Opt. Lett. 21 (1999) 430. A.T. O’Neil, I. MacVicar, L. Allen, M.J. Padgett, Phys. Rev. Lett. 88 (2002) 053601. F.S. Roux, Opt. Commun. 242 (2004) 45. J.E. Curtis, D.G. Grier, Phys. Rev. Lett. 90 (2003) 133901. G. Molina-Terriza, J. Recolons, J.P. Torres, L. Torner, E.M. Wright, Phys. Rev. Lett. 87 (2001) 023902. A.Y. Bekshaev, M.S. Soskin, M.V. Vasnetsov, J. Opt. Soc. Am. A 20 (2003) 1635. A. Wada, T. Ohtami, Y. Miyamoto, M. Takeda, J. Opt. Soc. Am. A 22 (2005) 2746. F.S. Roux, S. Afr. J. Sci. 102 (2006) 601. E. Abramochkin, V. Volostnikov, Opt. Commun. 102 (1993) 336. R. Piestun, Y.Y. Schechner, J. Shamir, J. Opt. Soc. Am. A 17 (2000) 294. F.S. Roux, Opt. Commun. 268 (2006) 196. W.D. Montgomery, J. Opt. Soc. Am. 57 (1967) 772. Z. Bouchal, R. Horak, J. Wagner, J. Mod. Opt. 43 (1996) 1905. M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, New York, 1980. F.S. Roux, Phys. Rev. A 77 (2008) 033837. F.S. Roux, Opt. Commun. 266 (2006) 433. J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw Hill, Boston, 1996.