Tight focusing of elliptically polarized vortex beams

Tight focusing of elliptically polarized vortex beams
Baosuan Chen and Jixiong Pu*
College of Information Science and Engineering, Huaqiao University, Quanzhou, Fujian 362021, China
*Corresponding author: [email protected]
Received 30 October 2008; revised 17 December 2008; accepted 26 January 2009;
posted 27 January 2009 (Doc. ID 103431); published 23 February 2009
We study the focusing properties of elliptically polarized vortex beams. Based on vectorial Debye theory,
some numerical calculations are given to illustrate the intensity and phase distribution properties of
tightly focused vortex beams. It is found that the spin angular momentum of the elliptically polarized
vortex beam will convert to orbital angular momentum by the focusing. The influence of corresponding
parameters on focusing properties is also investigated in great detail. It is shown that elliptical light
spots can be obtained in the focal plane. Moreover the elliptical spot may rotate and the spot shape
may change with the change of certain parameters. These properties are quite important for application
of this kind of elliptically polarized vortex beam. © 2009 Optical Society of America
OCIS codes:
050.1960, 110.1220, 260.5430.
1. Introduction
2.
Since the 1950s, tight focusing properties of laser
beams have been investigated in great detail [1–3].
It has been found that beams focused by a high
NA have unique properties, such as producing a
strong longitudinal component and focusing to a
tighter spot [4,5]. For its unique properties, tightly
focused light beams have wide potential applications
in optical data storage, microscopy, material processing, and optical trapping [6–10]. Therefore, behavior
of tightly focused beams has attracted much attention from researchers [10–13].
Recently, a new kind of beam with an optical vortex
became an issue of interest [14]. Different from common beams, this kind of beam has a screw wavefront
that generates orbital angular momentum [15,16].
Laser beams with such orbital angular momentum
have wide applications in micromanipulation, in
an optical spanner, and in quantum information,
and have been extensively studied in recent years
[17–20]. Here, based on vectorial Debye theory
[21], we extend the analysis to the tight focusing
of elliptically polarized vortex beams, which can
produce controllable elliptical beam spots.
Let us first consider the focusing of a linearly polarized beam by a high numerical aperture (NA). According to the classic paper by Wolf [1], the electric
field Eðr; φ; zÞ in the focal plane can be expressed as
0003-6935/09/071288-07$15.00/0
© 2009 Optical Society of America
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APPLIED OPTICS / Vol. 48, No. 7 / 1 March 2009
Theoretical Model
2
Ex
3
ikf
6 7
Eðr; φ; zÞ ¼ 4 Ey 5 ¼ −
2π
Ez
Z
α
0
Z
2π
0
Aðθ; ϕÞ
× exp½ikðz cos θ þ r sin θ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
× cosðϕ − φÞÞ sin θ cos θ
3
2
cos2 ϕ cos θ þ sin2 ϕ
7
6
× 4 cos ϕ sin ϕðcos θ − 1Þ 5dϕdθ:
ð1Þ
sin θ cosðϕÞ
where r, φ, and z are the cylindrical coordinates of an
observation point, shown in Fig. 1. k ¼ 2π=λ is the
wave vector, and f is the focal length of the high
NA objective. α ¼ arcsin NA is the maximal angle
determined by the NA of the objective. Aðθ; ϕÞ is
the pupil apodization function at the objective aperture surface, which is related to the electric field of
the incident beam.
Elliptically polarized light is simply the superposition of two orthogonal linearly polarized beams
focal length of the objective. Therefore, the pupil apodization function of the linearly polarized LG beams
can be expressed as
Fig. 1. Scheme of tight focusing.
pﬃﬃﬃ
2f sin θ jmj
Amj ðθ; ϕÞ ¼ Amj ðθÞ expðimϕÞ ¼ E0j
w0
2 2 f sin θ
expðimϕÞ; ðj ¼ x; yÞ: ð5Þ
× exp −
w0 2
with a certain retardation β between them. Thus the
electric field in the focal region when an elliptically
polarized light is tightly focused can be calculated as
On substituting Eq. (5) into Eq. (2) and after some
simplification, the x, y, and z components of the electric field in the focal region can be simplified as
2
Ex Ex 0
3
Z Z
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ikf α 2π
7
6
sin θ cos θ exp½ikðz cos θ þ r sin θ cosðϕ − φÞÞ
Eðr; φ; zÞ ¼ 4 Ey Ey 0 5 ¼ −
2π 0 0
Ez Ez 0
3
2
Ax ðθ; ϕÞðcos2 ϕ cos θ þ sin2 ϕÞ Ay ðθ; ϕÞeiβ cos ϕ sin ϕðcos θ − 1Þ
7
6
× 4 Ax ðθ; ϕÞ cos ϕ sin ϕðcos θ − 1Þ Ay ðθ; ϕÞeiβ ðcos2 ϕ þ sin2 ϕ cos θÞ 5dϕdθ:
ð2Þ
Ax ðθ; ϕÞ sin θ cosðϕÞ Ay ðθ; ϕÞeiβ sin θ sinðϕÞ
Laguerre–Gaussian (LG) beams have a helical
phase structure of expðimϕÞ, therefore they can be
treated as vortex beams. Here we consider the incident beam an elliptically polarized LG beam, the
electric field of which can be expressed as
E ðrÞ ¼ Ex ex Ey eiβ ey :
ð3Þ
Here, Eþ ðrÞ indicates the right-hand elliptical (RHE)
polarized beam, while E ðrÞ indicates the left-hand
elliptical (LHE) polarized beam. Ex and Ey are the
electric fields of two orthogonal linear components,
respectively. β is the retardation between the two
beams. ex and ey are unit vectors along the x and
the y directions, respectively. Considering p ¼ 0,
the electric field of a linearly polarized LG beam
in the source plane is given as
pﬃﬃﬃ jmj
r2
2r
Emj ðr; ϕÞ ¼ E0j
exp − 2 expðimϕÞ;
w0
w0
ð4Þ
ðj ¼ x; yÞ;
where E0j and w0 are the constants representing amplitude and beam size. m is the topological charge.
Since objectives are often designed to obey the sine
condition [21], we get r ¼ f sin θ, where f is the
E;x ðr; φ; zÞ ¼ −
ikf
2
Z
α
0
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
sin θ cos θ expðikz cos θÞ
× fAmx ðθÞð1 þ cos θÞim J m ðkr sin θÞ expðimφÞ
1
þ ðAmx ðθÞ ð−iÞAmy ðθÞeiβ Þðcos θ − 1Þimþ2
2
× J mþ2 ðkr sin θÞ exp½iðm þ 2Þφ
1
þ ðAmx ðθÞ iAmy ðθÞeiβ Þðcos θ − 1Þim−2
2
× J m−2 ðkr sin θÞ exp½iðm − 2Þφgdθ;
E;y ðr; φ; zÞ ¼ −
ikf
2
Z
0
α
ð6Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
sin θ cos θ expðikz cos θÞ
iβ
× fAmy ðθÞe ð1 þ cos θÞim J m ðkr sin θÞ expðimφÞ
1
þ ½ð−iÞAmx ðθÞ∓Amy ðθÞeiβ ðcos θ − 1Þimþ2
2
× J mþ2 ðkr sin θÞ exp½iðm þ 2Þφ
1
þ ðiAmx ðθÞ∓Amy ðθÞeiβ Þðcos θ − 1Þim−2
2
× J m−2 ðkr sin θÞ exp½iðm − 2Þφgdθ;
1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
ð7Þ
1289
ikf
E;z ðr; φ; zÞ ¼ −
2
Z
0
α
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
sin2 θ cos θ expðikz cos θÞ
× f½Amx ðθÞ ð−iÞAmy ðθÞeiβ imþ1 J mþ1 ðkr sin θÞ
× exp½iðm þ 1Þφ þ ðAmx ðθÞ
iAmy ðθÞeiβ Þim−1 J m−1 ðkr sin θÞ
× exp½iðm − 1Þφgdθ:
ð8Þ
From the above-derived equations we can see that,
after being focused by a high NA objective, both
E;x ðr; φ; zÞ and E;y ðr; φ; zÞ have three parts each
with topological charge of l ¼ m and l ¼ m 2 and
E;z ðr; φ; zÞ has two parts each with topological
charge of l ¼ m þ 1 and l ¼ m − 1, indicating that,
through focusing, the spin angular momentum
(SAM) of the incident beam will covert to the orbital
angular momentum (OAM) [17]. Based on Eqs. (6)–
(8), the total intensity distribution and its x, y, and z
components for the RHE- and LHE-polarized beam
can be obtained, respectively. Moreover the experiment can be realized in the laboratory by the system
shown in Fig. 1, which will be the subject of our
further study.
3. Results and Discussion
We performed some numerical calculations on the focusing properties for the elliptically polarized beams.
The fixed parameters for the calculations are
λ ¼ 632:8 nm, f ¼ 1 cm, and w0 ¼ 1 cm. Figure 2
shows the total intensity distribution and its x, y,
and z components in the focal plane for different topological charges and different polarization directions. All the components are normalized to their
total intensity. It is shown that elliptical focal spots
are obtained when the elliptically polarized beam is
focused by a high NA objective. That may be attributed to the different amplitudes of the incident fields
Emx ðr; ϕÞ and Emy ðr; ϕÞ, and the phase retardation
between them. Elliptical spots are quite important
when using laser beams to manipulate elliptical particles, for example, liquid crystalline molecules.
When topological charge m ¼ 0 (RHE-polarized
beams) [i.e., nonvortex; see Figs. 2(a), 2(d), 2(g),
and 2(j)], the total intensity (I t ) and its x and y components (I x and I y ) in the focal plane are nonzero at
the center, while the z component (I z ) has a zero central intensity. It is shown in Figs. 2(b), 2(e), 2(h), and
2(k) that, for topological charge m ¼ 1 (RHEpolarized beams), i.e., vortex beams, there is a dark
core for I t, I x , and I y . However, it is interesting to discover that the z component I z has two dark cores,
which indicates that there are two vortices in the intensity distribution of I z. The focusing properties for
LHE-polarized beams with m ¼ 1 are shown in
Figs. 2(c), 2(f), 2(i), and 2(l). Different from the
RHE-polarized beams, the central intensity of I z is
not hollow since the OAM is partly compensated
by the OAM converted from the SAM of the elliptically polarized beam.
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APPLIED OPTICS / Vol. 48, No. 7 / 1 March 2009
The phase contours of Ez ðr; ϕ; zÞ with different topological charges in the focal plane are shown in
Fig. 3. It is shown that the phase contour of the m ¼
0 beam exhibits a counterclockwise [Fig. 3(a), RHEpolarized beam] and a clockwise [Fig. 3(b), LHEpolarized beam] helical phase distribution, which
provides effective evidence that the SAM of the incident beam will covert to the OAM when the beam is
focused by a high NA objective. The phase contour in
Fig. 3(c) exhibits a combined vortex with each helical
phase changing from −π to π, which also explains why
the z-component intensity distribution has two dark
cores [shown in Fig. 2(k)]. From Fig. 3(d), it is seen
that the phase of the center is fixed, which indicates
that the wavefront of the LHE-polarized beam has no
screw in the focal plane. That is because the OAM is
partly compensated by the OAM converted from the
SAM of the elliptically polarized beam. The number
of vortices is directly correlated with the topological
charge (l) of the beam, and the OAM of the beam can
be calculated by lℏ [15], therefore, the change in the
number of vortices indicates the change of OAM of
the beam while focusing. It is shown that compared
with the incident beam, the OAM of the focused
RHE-polarized beam increases [see Fig. 3(a), 3(c),
and 3(e)], however, the OAM of the focused LHEpolarized beam was found to be decreased [see
Fig. 3(b), 3(d), and 3(f)]. That is because for the
RHE-polarized beam, the OAM converted from
SAM is in the same direction as the original OAM,
enhancing the total OAM, while for the LHE-polarized beam, the OAM converted from SAM is in the
opposition direction to the original OAM, reducing
the total OAM.
Phase contours of the x, y, and z components in the
focal plane with m ¼ 1 are shown in Fig. 4. The phase
contours present spiral structures in the focal plane,
which is called screw wavefront dislocation [14]. It is
shown that for both RHE- and LHE-polarized beams
there is only one vortex in the phase contours of
Ex ðr; ϕ; zÞ and Ey ðr; ϕ; zÞ, but the phase contours of
Ez ðr; ϕ; zÞ exhibit a combined vortex and no vortex
for RHE- and LHE-polarized beams, respectively,
which indicates that the SAM to OAM conversion
is less obvious in the phase distributions of Ex ðr; ϕ; zÞ
and Ey ðr; ϕ; zÞ than that in Ez ðr; ϕ; zÞ.
Then we investigate the influence of relative parameters on the focusing properties. Figures 5–8 are
plotted to illustrate the influence of varying E0y on
the total and three components of intensity distribution in the focal plane. Figure 5 shows the influence
of E0y on the total intensity I t in the focal plane with
m ¼ 1 (RHE-polarized beam). It is shown that when
E0y is comparable to E0x , E0y and the phase retardation β that determine the incident beam polarization
act together on the total intensity pattern in the focal
plane and induce the intensity pattern to rotate at an
angle that is correlated with the phase retardation,
shown in Fig. 5(a). And with the increase of E0y, the
intensity pattern rotates in a counterclockwise
direction. Moreover, when E0y ¼ 40, i.e., E0y ≫ E0x ,
Fig. 2. Intensity distributions in the focal plane for different topological charges. (a), (d), (g), (j) m ¼ 0 (RHE-polarized beam); (b), (e), (h),
(k) m ¼ 1 (RHE-polarized beam); (c), (f), (i), (l) m ¼ 1 (LHE-polarized beam). (a)–(c) total intensity It, (d)–(f) x component I x , (g)–(i) y component I y , (j)–(l) z component I z . The other parameters are chosen as NA ¼ 0:9, β ¼ π=4, E0x ¼ 1, E0y ¼ 1:5.
indicating that E0y becomes the main factor that influences the intensity distribution. In this case, the
incident beam can be taken as a y-linearly polarized
beam, the intensity pattern exhibits no rotation and
becomes symmetric to both the x axis and the y axis.
This phenomenon is also shown in the influence of
E0y on I z distribution; see Fig. 8. The influence of
varying E0y on I x distribution in the focal plane with
m ¼ 1 (RHE-polarized beam) is illustrated in Fig. 6.
It is found that, with the increase of E0y, the beam
spot in the focal plane is elongated in the diagonal
direction, and the intensity finally disperses on
two orthogonal ellipses. When E0y ¼ 40 (the incident
beam can be taken as y-linearly polarized), the intensity distribution is shown in Fig. 6(d). The research
finding indicates that E0y is an important parameter
in controlling the focused field properties. This result
is significant since it allows the possibility to control
the intensity pattern to match specific applications.
It can be seen from Fig. 7 that varying E0y has little
influence on the y-component intensity distribution.
Figure 9 shows the focused total intensity distribution with different phase retardation β. As β changes
from 0 to π=2, the intensity pattern gradually
1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
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Fig. 5. Influence of varying E0y on It in the focal plane with m ¼ 1
(RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 2, (c) E0y ¼ 4,
(d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
Fig. 3. Phase contours of Ez ðr; ϕ; zÞ in the focal plane. (a), (b)
m ¼ 0; (c), (d) m ¼ 1; (e), (f) m ¼ 2; (a), (c), (e) RHE-polarized beam;
(b), (d), (f) LHE-polarized beam. The other parameters are the
same as in Fig. 2.
to the elliptical distribution, but with the longer axis
perpendicular to that of the original ellipse.
The influence of varying NA on the total intensity
distribution in the focal plane is presented in Fig. 10.
It is shown that with the increase of NA, the
changes from elliptical distribution into circular
distribution. The central intensity gradually decreases and leads to a dark core distribution when
β ¼ π=2, i.e., when the incident beam is a circularly
polarized vortex beam. Then as the phase retardation β further increases, the intensity changes back
Fig. 4. Phase contours of the x, y, and z components in the focal
plane with m ¼ 1. (a), (b), (c) RHE-polarized beam; (d), (e), (f) LHEpolarized beam; (a), (d) Ex ðr; ϕ; zÞ; (b), (e) Ey ðr; ϕ; zÞ; (c), (f)
Ez ðr; ϕ; zÞ. The other parameters are the same as in Fig. 2.
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APPLIED OPTICS / Vol. 48, No. 7 / 1 March 2009
Fig. 6. Influence of varying E0y on Ix in the focal plane with m ¼ 1
(RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 4, (c) E0y ¼ 14,
(d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
Fig. 9. Influence of varying phase retardation β on I t in the focal
plane with m ¼ 1 (RHE-polarized beam). (a) β ¼ 0, (b) β ¼ π=6,
(c) β ¼ π=3, (d) β ¼ π=2, (e) β ¼ 2π=3, (f) β ¼ π. E0y ¼ 1. The other
parameters are the same as in Fig. 2.
Fig. 7. Influence of varying E0y on I y in the focal plane with m ¼ 1
(RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 4, (c) E0y ¼ 14,
(d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
characteristics of the intensity patterns stay unchanged except that the focal spot becomes smaller.
That is because the increase of NA leads to a tighter
focused transverse field.
Last but not least, the influence of varying phase
retardation β on the phase distributions of Ez ðr; ϕ; zÞ
Fig. 8. Influence of varying E0y on I z in the focal plane with m ¼ 1
(RHE-polarized beam). (a) E0y ¼ 1, (b) E0y ¼ 2, (c) E0y ¼ 4,
(d) E0y ¼ 40. The other parameters are the same as in Fig. 2.
in the focal plane is indicated in Fig. 11. It is found
that with β changing from 0 to π=2, the core of the two
vortices will gradually get closer and will be combined when β ¼ π=2, i.e., when the incident beam
is a circularly polarized vortex beam. Then by further
increases in β, the two cores of the combined vortex
separate again and return to the original phase distribution when β ¼ π, but with its direction perpendicular to that of the original distribution (when
β ¼ 0).
Fig. 10. Influence of varying NA on the total intensity in the focal
plane with m ¼ 1 (RHE-polarized beam). (a) NA ¼ 0:8,
(b) NA ¼ 0:85, (c) NA ¼ 0:9, (d) NA ¼ 0:95. E0y ¼ 1. The other
parameters are the same as in Fig. 2.
1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS
1293
Fig. 11. Influence of varying phase retardation β on phase distributions of Ez ðr; ϕ; zÞ with m ¼ 1 (RHE-polarized beam) in the focal
plane. (a) β ¼ 0, (b) β ¼ π=6, (c) β ¼ π=3, (d) β ¼ π=2, (e) β ¼ 2π=3,
(f) β ¼ π. E0y ¼ 1. The other parameters are the same as in Fig. 2.
4. Conclusions
Based on vectorial Debye theory, the tight focusing
properties of elliptically polarized vortex beams have
been analyzed. We have studied the intensity and
phase properties in the focal plane and the influence
of relative parameters on them. It is found that elliptical beam spots are obtained when the elliptically
polarized vortex beams are focused by a high NA objective. And the SAM of the elliptically polarized
beam will convert to OAM when the beam is tightly
focused. Moreover, by adjusting relative parameters,
the shape of the beam spot and the direction of the
longer axis can be controlled. This research finding is
important in the applications of tightly focused elliptical spots, especially the applications in elliptical
particle trapping, manipulation, and so on.
This research is supported by the Natural Science
Foundation of Fujian Province under grant
A0810012, and Key Project of Science and Technology of Fujian Province under grant 2007H0027.
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