1

S2: Tuning the beam-waist size.

In this document, we explain how the value of the beamwaist radius 0 in our setup can be

continuously varied, between 9 m and 1.25 m, approximately. The optical arrangement of

the experiment is sketched in Fig. S2.1. Essentially it is constituted of two couples of

objectives, (L1, L2) and (L3, L4). L1, L2 and L3 are simple lenses, with focal lengths f1=130

mm, f2=150 mm, f3=400 mm. L4 is a long-working-distance microscope objective (Zeiss Apo

Plan 50X, NA 0.5, ∞), f4 ≈ 3.3 mm.

The focal points of each lens are indicated in the figure as Fi, F’i, corresponding to the front

and image focal planes, for 1 ≤ i ≤ 4. In the “ideal” fully confocal configuration, each image

focal point is made to coincide with the next front focal point, i.e. F2=F’1, F3=F’2 and F4=F’3.

In this situation, (L1, L2) and (L3, L4) are Keplerian telescopes, and F2 (=F’1) is conjugate to

the centre of L4 pupil. The setup axis at this location is folded with a mirror. Rotating the

mirror allows us to translate the beam in the sample volume, as in standard designs of optical

tweezers (we use a pair of galvanometric mirrors, denoted GM in the figure).

The beam from the laser source (Coherent Genesis, wavelength in air = 514 nm) is close

to Gaussian, with a beam-waist radius 1 945 m (defined at 1/e2 from the on-axis

intensity). The corresponding Rayleigh length is 2

1 1z 5.5 m. If we suppose that the

laser beam-waist position (bw1) coincides with F1, the following bwi are located in the

successive focal planes, with:

31 2 42 3 4 0

1 2 3 4

, , ,ff f f

(S2.1)

In Eq. (S2.1) and in Fig. S2.1, index 2 denotes the region between L1 and L2, index 3 that

between L2 and L3 , index 4 that between L3 and L4 . 0 is the ultimate beam-waist radius, in

the sample volume. The corresponding Rayleigh lengths are given by:

22 2 2

31 2 42 3 4 0

1 2 3 4

, , ,ff f f

z z z zz z z z

(S2.2)

The fully confocal configuration then yields 0 1 2 4 1 3f f f f 9 m, and 0z 494 m.

Note that our supposition that bw1 coincides with F’1 is not essential, because of the very large

value of 1z . In practice Eqs (S2.1, 2) are close to correct even if bw1 is far from F1 (the

position mismatch is < 0.5 m, at worst).

2

FIG. S2.1: Scheme of the optical levitator setup, from the laser source up to the sample volume.

We now examine the effect of changing the distances between the lenses, in other words of

moving the optical arrangement out of the fully confocal configuration. Let be x the abscissa

along the optical axis and let us denote as , 1 1i i i id x L x L the distance between lenses

Li, Li+1. We now define , 1i i , the distance mismatch from the confocal configuration:

, 1 1 , 1i i i i i id f f (S2.3)

We obviously have , 1 0i i everywhere in the confocal configuration. A direct

consequence of the mismatch is that the successive beam-waist planes will no more coincide

with the lenses focal planes. Let us define:

i i ix F x bw (S2.4)

i is the shift between bwi and Fi. Calculating the successive shifts and Rayleigh lengths is

a straightforward task, using the ABCD transfer matrix formalism for the transformation of

Gaussian beam by lenses [1]. For each lens, we have:

1 0

1 1i

A B

fC D

(S2.5)

Here we use the transfer matrix for simple lenses, for simplicity. We checked that the results

to be arrived at below were the same using general forms for thick lenses.

Let us define the complex wave curvature radius at ix L as:

F2++ + + +

+

F’1

+F3

F’2 F’3 F’4F4

L1

L2 L3 L4

Laser

GM

+ F1

Sample volume

(2) (4)(3) (0)

3

i i i iq x L x bw j z , (S2.6)

where 1j . When the beam passes through lens Li , iq is transformed into

1 1i i i iq x L x bw j z (S2.7)

given by :

ii

i

Aq Bq

Cq D

(S2.8)

Application of Eq. (S2.7) then yields the position 1ix bw of bwi+1 and the new Rayleigh

length 1iz . From 1ix bw we deduce 1i i ix F x bw

. We thus find:

2

1 2

21

ii

ii

i

fz

zz

, (S2.9)

and

1

i i

i iz z

(S2.10)

The above equation indicates that the error in the beam-waist position, scaled by the local

beam diffraction length, is an invariant within the change in sign. Also note that Eq. (S2.9) is

similar to the basic Eq. (S2.2), as it may be cast as 2

1i i iz f z , now with an effective focal

length given by

1 22

21 i

i i

i

f fz

(S2.12)

The new beam-waist position mismatch is:

11 , 1 , 1

ii i i i i i i

i

z

z

(S2.12)

Calculating the value of the beam-waist in the sample volume and its position then amounts

to cascading Eqs S2.9 and S2.11 through the four lenses of the setup. We obtain 0 as a

function of 12 23 34, , . Below we show maps of 0 values versus inter-lens distances.

4

-40 -20 0 20 40-20

-15

-10

-5

0

5

10

15

20w

0 [m]

12

= 0

34

[mm]

23 [

mm

]

3.6

4.1

4.6

5.1

5.6

6.1

6.6

7.1

7.6

8.1

8.6

9.0

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

34

= 0

w0 [m]

23

[mm]

12 [

mm

]

1.4

2.1

2.8

3.5

4.2

4.9

5.6

6.3

7.1

7.8

8.5

9.0

5

-40 -20 0 20 40-20

-15

-10

-5

0

5

10

15

20

23

= 0

34

[mm]

12 [

mm

]

1.0

1.8

2.5

3.3

4.0

4.7

5.5

6.2

7.0

7.7

8.5

9.0

w0 [m]

FIG. S2.2: Color-coded maps of 0 [m] as a function of inter-lens distances 12 23 34, , . The parameter

12 seems to be the most sensitive, as explained in the text. For non zero values of 34 , the beam-waist radius

goes through a maximum that is shifted in 12 .

As can be seen from Eqs. S2.9 and S2.11, the ratio i iz is the relevant parameter to

estimate the impact of a change of the distance between iL and 1iL . The change is most

effective when iz is smallest. In practice 12 is the variable to be played with, because of the

tight focusing of the beam by 1L . We estimate the effect of 12 by setting 23 340, 0 . We

obtain:

1 2

2 4 10 2

1 3 1 11

f f z

f f z f

, (S2.13)

with 12 , to simplify the notation.

Below we compare the values obtained with Eq. (S2.12) (which corresponds to the dashed

vertical line in Fig. S2.2 with 23=0) to experimental measurements. The measurements have

been performed not directly in the sample volume but on the microscope image of the sample

6

-20 -15 -10 -5 0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Theory

Beam

wais

t ra

diu

s [m

]

12

[mm]

w0 (X axis)

w0 (Y axis)

plane, using an automated beam profiler (Thorlabs BP209-VIS); see Fig. S2.3 below.

Essentially the microscope is a simple confocal arrangement, made of the microscope

objective 5L (identical to 4L ) and of a tube lens ( TLf = 300 mm). The beam profiler entrance

plane is located in the image focal plane of TL. For each value of , we move the whole

microscope axially, and lock to the position where the beam profile is narrowest. We thus

obtain a magnified view of the beam-waist in the sample volume, with a magnification

5TLG f f 91.2 .

FIG. S2.3: Sketch of the microscope, including the slit beam profiler (SBP).

FIG. S2.4: Theoretical values (blue curve) of the beam-waist in the sample volume and measured values

(open and solid symbols). The real profile turns out slightly elliptical, with an ellipticity that depends on . The

profilometer then yields a couple of 0 values for each .

+++

SBPTLL5L4

Microscope

7

Results are displayed in Fig. S2.4. Note that the real profiles are not axially symmetrical,

meaning that diameters measured along two perpendicular axes are slightly different. The

beam profiler then provides 2 values of the beam-waist, indicated as open circles and black

squares in Fig. S2.4. Overall, the graphs show that measurements and theoretical predictions

match together well; the agreement is almost perfect for the short radius (open circles). The

maximum value of 0 is 9 m, in agreement with the prediction for the fully confocal

configuration.

Lenses 1L and 2L in the setup are mounted on translation stages. In practice we can

vary 12 between about 0 and 22 mm. The corresponding range for the beam-waist is then 1.25

0 9 m.

References

[1] A. E. Siegman, An introduction to lasers and masers, (McGraw-Hill, New York, 1971),

chapt. 8 .