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Optical levitation by means of two horizontal laser beams: A theoretical and experimental study
Volume
59A, number
OPTICAL
1
PHYSICS
LEVITATION
1 November
LETTERS
BY MEANS OF TWO HORIZONTAL
A THEORETICAL
AND EXPERIMENTAL
LASER
1976
BEAMS:
STUDY
G. ROOSEN and C. IMBERT Laboratoire d’Expe?iences Fondamentales en Optique, Institut d’optique, Bat. 503, Universite’de Paris-Sud, 91405 Orsay, France Received
9 July
1976
Stable optical levitation of transparent solid glass spheres has been demonstrated using two horizontal TEMOO laser beams. The experimental results agree with a theoretical calculation which gives the value of the lateral force pushing the sphere towards the region of maximum light intensity and maintaining it on the axis of the beam.
Ashkin’s [l] experimental demonstration of optical levitation of solid glass spheres by the radiation pressure exerted by a vertical laser beam with a gaussian intensity structure, shows that, in addition to a force along the beam axis, there is a lateral force drawing the sphere into the beam, which leads to stable equilibrium. We have observed this force and also calculated it, using the geometric optics approach, for spheres located near the beam waist where the curvature of the wave is negligible. The equation obtained is valid for spheres having a diameter 2p greater than or equal to 15 pm. With the reference axes and angles as shown in fig. 1, we obtain, for the force F, along the axis zz’ of the beam and for the orthogonal force Fy :
Fz=-f2d0
~~d~[E~sinBcosO] 0
0
X
2(6 -r) +R cos 201
R cos 20 + 1 - ___ 1 tR2
Fv = - TciO
1 ’
t2Rcos2r
/Xdtp[ssin28
0
x
(1) T2 (cos
sin p]
0
sin e _ T2 + R sin 20 1 __ {sin 2(6 ___ - r)__.~ 1tR2t2Rcos2r
1’
(2)
t
Fig. 1. The beam is parallel to the axis ZZ’. ‘p is the angle between the plane of the figure and the one which the ray propagates.
6
t Fig. 2. Top view of the experimental of two horizontal TEMOO beams.
4
device permitting
optical
levitation
by means
Volume FI IN)
59A, number
1
PHYSICS
I
R
3104
2
10-e-
01
LETTERS
0.5
,
2
1.5
Fig. 3. Variation of Fr when the ratio p/w varies. Various curves are obtained for different values of n. F, is always in the direction of the beam.
The light flux $ carried by the beam is connected the electric field E by:
The beam has a gaussian intensity
structure,
with
and so we
2(p2sin28 +pi - 2ppu sin 19sin cp) W2
-RII:R,
1 November
and
1976
TII+ Tl
T=----
2
where R ,, and T,, , R, and T, are respectively the intensity factors of reflection and transmission in M for the T.E. and T.M. electric field. We deduce these factors from the Fresnel coefficients. Fig. 3 shows the variation of F, when the ratio p/w between the radius of the sphere and the beam changes. Curves are given for various values of the sphere refractive index relative to the external medium. The incident flux is taken equal to 1 W. The curves indicate that it is possible to obtain a stable equilibrium under the waist for spheres having a diameter greater than the waist. Fig. 4 shows the variation of F,, when the ratio pa/w between the offset and the beam radius changes and curves are given for various values of the ratio p/w and the relative index n. When n is greater than 1, Fv keeps the sphere centered in the beam and therefore in stable equilibrium. To study F,, experimentally we produced optical levitation by means of two horizontal laser beams. A TEMOO beam from an Argon laser (fig. 2) is split up into two identical beams by the beam-splitter ST. The two beams are focused by two identical lenses LI
where p0 is the distance between the beam axis and the diameter zz’ of the sphere. For R and T we take the values
(a)
(b)
Fig. 4. Variation of Fy when the shift PC,varies. Curves A show the variation of Fy for various diameters of the sphere. F_,, is always directed to the axis of the beam. Curves B show that when the relative index is less than 1, F,, pushes the sphere out of the beam.
PHYSICS LETTERS
Volume 59A, number 1
Fig. 5. Refractive sphere levitated by two horizontal beams. and L2 into the cell E. Depending
on the focal length
of the lens, (50 or 35 mm), the waist diameter
is 21 or 15 pm. By dropping spheres from tube T, we are able to trap one or more of them in the horizontal beams. We observe the sphere through a microscope with horizon. tal axis orthogonal to the beam direction (axis xx’, fig. 2). Thus levitated the sphere is very stable. In our experiments we could not see any movement which means that the stability is greater than.1 pm. When the sphere is levitated by the two horizontal beams, its weight P is balanced by the sum of the forces Fy exerted by each beam. Measurement of the sphere diameter through the microscope gives its weight. We measure the distance z between the levitated sphere and each waist, and then we calculate the beam diameter 2w at the equilibrium plane. We then calculate F_,,inserting 2w and the measured values of p. obtained for various values of @in eq. (2) and we compare the results obtained for Fy with P. For instance at z = (100 * 50) pm of each waist where the beam diameter is 2w = (21.2 + 0.3) pm, a sphere having a diameter 2p = (18 f 1) ,um and then a weight P = (7
1 November 1976
+ 1) lo--l1 N is in a stable equilibrium at p. = (5 + 1) ,um under the common axis of the beams which each carries the flux $J= (140 + 7) mW. The calculated value for the sum of the forces F$ exerted by each beam is (7 t 2) IO-* 1 N which agrees with P. This procedure has been performed for spheres of different diameters placed at various distances z from each waist, thus confirming the relation (2). The relative uncertainty is 3% which is acceptable considering the smallness of po. We also tested the validity of the expression (1) for F, by repeating Ashkin’s experiment of levitation using a vertical beam [l] . Depending on the flux @,the sphere is levitated at a distance z of the waist where the beam diameter is 2w. We calculate Fz by inserting in eq. (1) the calculated value of 2w and the measured value of 4. For instance at z = (1000 + 50) pm above the waist, where the beam diameter is 2w = (38 + 1) ,um a sphere having a diameter 2p = (25 + 1) pm and then a weight P = (2 + 0.2) 1O-1o N is equilibrated by a beam carrying a flux 4 = (338 * 17) mW. The calculated value for the force F, is (2.1 * 0.3) lo-lo N which agrees with P. This procedure has been performed for various values of ~,6and for spheres of different diameters thus confirming the relation (1). The confirmation by the experiments of expressions (1) and (2) calculated using only the radiation pressure of light, permits us to ignore the intervention of thermal effects in levitation experiments at atmospheric pressure. The authors thank Dr. S. Slanski for his help in the computation.
References [l] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156; Appl. Phys. Lett. 19 (1971) 283.
59A, number
OPTICAL
1
PHYSICS
LEVITATION
1 November
LETTERS
BY MEANS OF TWO HORIZONTAL
A THEORETICAL
AND EXPERIMENTAL
LASER
1976
BEAMS:
STUDY
G. ROOSEN and C. IMBERT Laboratoire d’Expe?iences Fondamentales en Optique, Institut d’optique, Bat. 503, Universite’de Paris-Sud, 91405 Orsay, France Received
9 July
1976
Stable optical levitation of transparent solid glass spheres has been demonstrated using two horizontal TEMOO laser beams. The experimental results agree with a theoretical calculation which gives the value of the lateral force pushing the sphere towards the region of maximum light intensity and maintaining it on the axis of the beam.
Ashkin’s [l] experimental demonstration of optical levitation of solid glass spheres by the radiation pressure exerted by a vertical laser beam with a gaussian intensity structure, shows that, in addition to a force along the beam axis, there is a lateral force drawing the sphere into the beam, which leads to stable equilibrium. We have observed this force and also calculated it, using the geometric optics approach, for spheres located near the beam waist where the curvature of the wave is negligible. The equation obtained is valid for spheres having a diameter 2p greater than or equal to 15 pm. With the reference axes and angles as shown in fig. 1, we obtain, for the force F, along the axis zz’ of the beam and for the orthogonal force Fy :
Fz=-f2d0
~~d~[E~sinBcosO] 0
0
X
2(6 -r) +R cos 201
R cos 20 + 1 - ___ 1 tR2
Fv = - TciO
1 ’
t2Rcos2r
/Xdtp[ssin28
0
x
(1) T2 (cos
sin p]
0
sin e _ T2 + R sin 20 1 __ {sin 2(6 ___ - r)__.~ 1tR2t2Rcos2r
1’
(2)
t
Fig. 1. The beam is parallel to the axis ZZ’. ‘p is the angle between the plane of the figure and the one which the ray propagates.
6
t Fig. 2. Top view of the experimental of two horizontal TEMOO beams.
4
device permitting
optical
levitation
by means
Volume FI IN)
59A, number
1
PHYSICS
I
R
3104
2
10-e-
01
LETTERS
0.5
,
2
1.5
Fig. 3. Variation of Fr when the ratio p/w varies. Various curves are obtained for different values of n. F, is always in the direction of the beam.
The light flux $ carried by the beam is connected the electric field E by:
The beam has a gaussian intensity
structure,
with
and so we
2(p2sin28 +pi - 2ppu sin 19sin cp) W2
-RII:R,
1 November
and
1976
TII+ Tl
T=----
2
where R ,, and T,, , R, and T, are respectively the intensity factors of reflection and transmission in M for the T.E. and T.M. electric field. We deduce these factors from the Fresnel coefficients. Fig. 3 shows the variation of F, when the ratio p/w between the radius of the sphere and the beam changes. Curves are given for various values of the sphere refractive index relative to the external medium. The incident flux is taken equal to 1 W. The curves indicate that it is possible to obtain a stable equilibrium under the waist for spheres having a diameter greater than the waist. Fig. 4 shows the variation of F,, when the ratio pa/w between the offset and the beam radius changes and curves are given for various values of the ratio p/w and the relative index n. When n is greater than 1, Fv keeps the sphere centered in the beam and therefore in stable equilibrium. To study F,, experimentally we produced optical levitation by means of two horizontal laser beams. A TEMOO beam from an Argon laser (fig. 2) is split up into two identical beams by the beam-splitter ST. The two beams are focused by two identical lenses LI
where p0 is the distance between the beam axis and the diameter zz’ of the sphere. For R and T we take the values
(a)
(b)
Fig. 4. Variation of Fy when the shift PC,varies. Curves A show the variation of Fy for various diameters of the sphere. F_,, is always directed to the axis of the beam. Curves B show that when the relative index is less than 1, F,, pushes the sphere out of the beam.
PHYSICS LETTERS
Volume 59A, number 1
Fig. 5. Refractive sphere levitated by two horizontal beams. and L2 into the cell E. Depending
on the focal length
of the lens, (50 or 35 mm), the waist diameter
is 21 or 15 pm. By dropping spheres from tube T, we are able to trap one or more of them in the horizontal beams. We observe the sphere through a microscope with horizon. tal axis orthogonal to the beam direction (axis xx’, fig. 2). Thus levitated the sphere is very stable. In our experiments we could not see any movement which means that the stability is greater than.1 pm. When the sphere is levitated by the two horizontal beams, its weight P is balanced by the sum of the forces Fy exerted by each beam. Measurement of the sphere diameter through the microscope gives its weight. We measure the distance z between the levitated sphere and each waist, and then we calculate the beam diameter 2w at the equilibrium plane. We then calculate F_,,inserting 2w and the measured values of p. obtained for various values of @in eq. (2) and we compare the results obtained for Fy with P. For instance at z = (100 * 50) pm of each waist where the beam diameter is 2w = (21.2 + 0.3) pm, a sphere having a diameter 2p = (18 f 1) ,um and then a weight P = (7
1 November 1976
+ 1) lo--l1 N is in a stable equilibrium at p. = (5 + 1) ,um under the common axis of the beams which each carries the flux $J= (140 + 7) mW. The calculated value for the sum of the forces F$ exerted by each beam is (7 t 2) IO-* 1 N which agrees with P. This procedure has been performed for spheres of different diameters placed at various distances z from each waist, thus confirming the relation (2). The relative uncertainty is 3% which is acceptable considering the smallness of po. We also tested the validity of the expression (1) for F, by repeating Ashkin’s experiment of levitation using a vertical beam [l] . Depending on the flux @,the sphere is levitated at a distance z of the waist where the beam diameter is 2w. We calculate Fz by inserting in eq. (1) the calculated value of 2w and the measured value of 4. For instance at z = (1000 + 50) pm above the waist, where the beam diameter is 2w = (38 + 1) ,um a sphere having a diameter 2p = (25 + 1) pm and then a weight P = (2 + 0.2) 1O-1o N is equilibrated by a beam carrying a flux 4 = (338 * 17) mW. The calculated value for the force F, is (2.1 * 0.3) lo-lo N which agrees with P. This procedure has been performed for various values of ~,6and for spheres of different diameters thus confirming the relation (1). The confirmation by the experiments of expressions (1) and (2) calculated using only the radiation pressure of light, permits us to ignore the intervention of thermal effects in levitation experiments at atmospheric pressure. The authors thank Dr. S. Slanski for his help in the computation.
References [l] A. Ashkin, Phys. Rev. Lett. 24 (1970) 156; Appl. Phys. Lett. 19 (1971) 283.