Quasi-Bessel beam generated by oblate-tip axicon

Oto Brzobohaty, Tomas Cizmar, and Pavel Zemanek

Institute of Scientific Instruments of the ASCR, v.v.i.,

Academy of Sciences of the Czech Republic

Kralovopolska 147, 612 64 Brno, Czech Republic

ABSTRACT

We focused here on the real shape of the tip of the axicon - which is not sharp but rather oblate. We simulatednumerically and verified experimentally that tiny deviations of the tip shape from the ideal sharp profile inducesignificant oscillations of the beam intensity along its propagation. Such unwanted intensity modulation disturbsthe unique properties of the quasi-Bessel beam - constant shape of the lateral intensity profile and especiallyslow variation of the on-axis intensity along the beam propagation. We demonstrate how the spatial filtration ofthe beam in the Fourier plane removes such undesired modulation and restores the properties of the quasi-Besselbeam.

Keywords: Bessel beam, non-diffracting beam, Fourier optics

1. INTRODUCTION

Non-diffracting beams received their names due to the invariance of their lateral and axial intensity profilesalong their propagation.1 A prominent example of such a beam is zero-order Bessel beam whose lateral intensityvariation is described by the zero-order Bessel function of the first kind. Idealized Bessel beams are of infinitetransverse extent and carry infinite amount of energy and, therefore, they cannot be generated experimentally.However, over a limited spatial range, an approximation to such idealized beam can be obtained experimentally- so called quasi-non-diffracting beam or quasi-Bessel beam (QBB).2–4 Its transverse spatial extent is confined,the beam keeps its transversal profile but intensity varies over the axial range of the beam existence. Pioneeringways of QBB generation used an annular slit illuminated with a collimated light wave and placed at the backfocal plane of a lens.1 However, the efficiency of beam intensity transfer from the illuminating beam to the QBBwas very low and, moreover, the amplitude of the generated QBB was modulated by the diffraction envelope ofthe slit.1, 5 Holograms can generate QBB with higher diffraction efficiency of ∼ 40.5%6–8 but the axial profile ofthe optical intensity is still diffraction modulated. Ideal conical lens (axicon)9 provides a promising option forobtaining high-intensity QBB.

There is an increasing number of papers dealing with theoretical aspects of QBBs and attempts to get theirproperties closer to the properties of the ideal non-diffracting beams (see for example Ref.10 and referencestherein). Let us focus here only on axicon generated beams. Simpler treatments ignore diffraction effects on theaxicon edges, consider ideally sharp axicon tip and – within the scalar description – deal with different spatialprofiles of beams (i.e. Gaussian, Laguerre-Gaussian) incident on the axicon.11–14 They show that the beam keepsits lateral shape and size but its intensity varies smoothly along the propagation axis. Extensions to vectorialdescriptions for linearly, radially, or azimuthally polarized beams are known, too.15–19 More complex modelsshow that diffraction from the axicon edges causes noticeable modulation of the on-axis optical intensity along thebeam propagation.4, 10, 20 Fortunately this effect can be neglected if the waist of the Gaussian beam illuminatingthe axicon is at least two-times smaller than the axicon radius.21, 22 However the influence of imperfect tip ofthe axicon in the region of QBB existence is rarely addressed in the literature23–25 even though it also causesundesired axial modulation of the QBB.

Numerous practical applications of the QBB are based on invariable lateral profile of optical intensity alongthe beam propagation, for example simultaneous micromanipulation or guiding of microparticles or atoms,16, 26–33

Further author information: (Send correspondence to P.Z.)P.Z.: E-mail: zemanek@isibrno.cz, Telephone: 00420 541 514 202

16th Polish-Slovak-Czech Optical Conference on Wave and Quantum Aspects of Contemporary Optics, edited by Agnieszka Popiolek-Masajada, Elzbieta Jankowska, Waclaw Urbanczyk, Proc. of SPIE Vol. 7141,

714126 · © 2008 SPIE · CCC code: 0277-786X/08/$18 · doi: 10.1117/12.822425

Proc. of SPIE Vol. 7141 714126-1

second harmonic generation,34, 35 optical coherence tomography over a large depth range,23 generation of waveg-uides,36, 37 and optoporation.38 Therefore the suppression of the on-axial intensity oscillations and beam widthvariations is a crucial step to obtain the desired QBB properties.

2. INFLUENCE OF THE TIP CURVATURE ON THE QUASI-BESSEL BEAMAXIAL INTENSITY PROFILE

2.1 Quasi-Bessel beam

An ideal Bessel beam is a product of interference of plane waves of the same intensity with their wavevectorscovering the surface of a cone with semi-apex angle α0 (see Fig. 1). Its intensity does not change as the beampropagate along z-axis. However this condition can be fulfilled only with an unreal beam incident on the axicon.Its electric field intensity should decrease as 1/

√ρ in front of the axicon.

Consequently, the angular spectrum of such an ideal Bessel beam is described by a delta function δ(α − α0)where the α is the polar angle and α0 is related to the axicon properties as

α0 = arcsin

(n

n0

cosτ

2

)+

τ − π

2≈ n − n0

n0

π − τ

2. (1)

We denote n and n0 as refractive index of the axicon and medium around the axicon, respectively, and τ asthe apex angle of the axicon (see Fig. 1). However if an axicon with sharp tip (further called perfect axicon) isilluminated with a Gaussian beam, the plane wave spectrum behind the axicon is no longer a delta function andits extent increases with decreasing Gaussian beam waist w0.

13, 39 This causes that the non-diffracting beambehind the axicon does not have a uniform intensity profile along its axis of propagation and exists only over alimited range zmax = w0 cosα0/ sinα0. Using the scalar description of the electric component of the beam onecan obtain39

E(ρ, z) = E0

√2πkzw0 sin α0

zmax

exp

(− z2

z2max

− πi

4

)J0(kρ sin α0) exp(ikz cosα0), (2)

where k is an angular wavenumber and ρ denotes the radial distance from the optical axis z. The optical intensityprofile can be obtained using I(ρ, z) = cε0|E(ρ, z)|2/2:

I(ρ, z) =4Pk sin α0

w0

z

zmax

J20 (kρ sin α0) exp

(− 2z2

z2max

)≡ I0(z)J2

0

(2.4048

ρ

ρ0

), (3)

where P is the total power of the incident Gaussian beam and ρ0 = 2.4048/(k sinα0) is the radius of the highintensity core of the QBB. An example of the axial and radial intensity profiles of such QBB is shown in Fig.1A.

In the following discussion, we will omit the diffraction from the axicon outer edges (which is justified forw0 < A/2, where A is the axicon radius21, 22) and focus on the influence of the oblate-tip on the final beamprofile. Let us assume that the shape of the axicon surface is a hyperboloid of revolution of two sheets (see insetin Fig. 1B). If we put the centre of coordinate system to the tip of the perfect axicon, the surface of hyperboloidof interest (for z < 0) is described by the following formula:

z2

a2− ρ2

b2= 1 giving z = −a

b

√b2 + ρ2. (4)

Since b/a = tan(τ/2), where τ is the apex angle of the perfect axicon (angle between asymptotes of the hyper-boloid), we can rewrite Eq. (4):

z = −√

a2 +ρ2

tan2(τ/2). (5)

Obviously, the smaller the parameter a the better the axicon approaches the ideal sharp shape.

Proc. of SPIE Vol. 7141 714126-2

Intuitively one can expect that parts of the axicon, that are placed radially far from the axis, contributeto QBB formed farther from the tip. However the oblate-tip will focus part of the incident Gaussian beampropagating closer to the optical axis and create a convergent (divergent) nearly-spherical wave behind the axicon.This wave interferes with the QBB behind the axicon which results not only in the significant modification ofthe field distribution near the axicon tip but also farther from the axicon. The axial component of the nearly-spherical wavevector is equal to k on the optical axis, whereas the axial component of the QBB is equal tokz = k cosα0. Therefore, the two co-propagating waves with different wavevector lengths interfere on the opticalaxis and create a periodic modulation of the axial beam intensity with a period λ/(1 − cosα0). Depth of thismodulation decreases at axial positions placed farther from the axicon because of the decreasing intensity of thediverging spherical wave (see Fig. 1B).

Figure 1. A: Formation of a quasi-Bessel beam (QBB) by a perfect axicon illuminated by a Gaussian beam with a beamwaist placed on the axicon front surface. Wavevectors k of the plane waves forming the QBB lie on the surface of acone with semi-apex angle α0. Intensity profile along the propagation axis is shown together with invariant shape of theradial profile at two axial positions z1 and z2. zmax is the maximum propagation distance where the QBB exists and τis the apex angle of the axicon. B: Influence of the oblate-tip axicon. New wave refracted by the oblate-tip propagatesbehind the axicon and interferes with the QBB. This results in axial modulation of the optical intensity with a periodλ/(1 − cos α0), where λ is the wave wavelength in the medium. Due to the interference the radial intensity profile is nolonger invariant (see examples at z1 and z2). Inset: Approximation of the oblate-tip of the axicon by a hyperboloid ofrevolution of two sheets and the meaning of its parameters a and b.

The beam propagation behind the oblate-tip axicon was studied numerically using the free-space propagationmethod based on decomposition of the electric field component E(ρ, 0) in plane z = 0 into a spectrum of planewaves (spatial-frequency spectrum done by the Fourier transform).40 To express this field we assumed that theoblate-tip axicon is illuminated with a Gaussian beam having its waist of width w0 at the front axicon surface.

Proc. of SPIE Vol. 7141 714126-3

We further approximated the axicon as a thin lens and used the cylindrical system of coordinates (ρ, φ, z) havingits origin at the tip of the perfect axicon with z axis following the beam propagation. Following40 we obtain forthe electric field behind the axicon

E(ρ, 0) = E0 exp

(− ρ2

w20

)exp(ikn∆0) exp

[ik(n0 − n)

√a2 +

ρ2

tan2(τ/2)

], (6)

where n is the refractive index of the axicon material, n0 is refractive index of the surrounding medium (air),∆0 is the maximum thickness of the axicon (on its axis), and k is the wavenumber. Since this is a rotationallysymmetrical field, the two-dimensional Fourier transform reduces to the form of the zero order Hankel transformthat provides time-efficient computation of the field propagation.41 This algorithm was applied in the followingstudies (Figs. 4-8) of the beam propagation.

2.2 QBB transformation by a telescope

The QBB obtained by an axicon is usually too wide to be used for fine experiments at micro-level. Thereforea demagnifying telescope is often used (see Fig. 2). The lenses forming the telescope were considered as phasemasks and the field immediately behind the lens Li was obtained from

E(ρ, zLi) = E0(ρ, zLi

) exp

(−i

k

2fi

ρ2

), (7)

where fi was the focal length of the first (i = 1) and second (i = 2) lens in the telescope and zLiis its axial

position. The presence of such a telescope in optical set-up generating QBB is highly advantageous for possiblespatial filtration of the beam. The back focal plane of the first lens L1 represents the Fourier plane where thespatial-frequency spectrum (SFS) of the QBB is formed. The oblate-tip axicon creates low frequency componentsin SFS and therefore an opaque circular obstacle (spatial filter) of proper radius placed into the back focal planeof the lens L1 would block them. This brings the beam spatial intensity profile closer to QBB produced by theperfect axicon. Radius of this spatial filter Rf can be predicted from the simultion of the beam propagation.41

. As we demonstrate below this simple modification supresses the axial intensity oscillations and establishes theQBB propagating with unvarying width and slowly varying intensity envelope. Therefore the beam parametersare closer to the QBB generated by the perfect axicon.

3. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS

3.1 Experimental set-up

Experimental setup for the QBB generation is depicted in Fig. 2. The incident Gaussian beam is transformed bya commercially available axicon EKSPLA into a QBB. This beam is further demagnified by a telescope formedfrom lenses L1 and L2 to get a narrow QBB. This QBB is imaged on a CCD camera by an objective. The spatialfilter (opaque circular obstacle placed in the back focal plane of L1) was used to suppress the oscillations causedby the interference of the unwanted waves with the QBB.

3.2 The axicon shape

We used axicon EKSPLA 130-0270 with apex angle τ = 170◦. Shape of its tip was measured by opticalprofilometer (MicroProf FRT, Fries Research & Technology GmbH) and it is shown in Fig. 3. We fitted themeasured profile by a hyperboloid surface using Eq. (5) and we found τ = (169.81 ± 0.01)◦ and aexp = (33.7 ±0.1) µm. Therefore τ coincides reasonably well with the value presented by the manufacturer.

Proc. of SPIE Vol. 7141 714126-4

Figure 2. Experimental set-up. Laser: IPG, YLM-10-1064-LP, wavelength 1064 nm, maximal power 10 W, beam-waistof incident Gaussian beam w0 = 2140 µm; axicon: EKSPLA 130-0270, apex angle τ = 170◦; lenses L1, L2: focal lengthsf1 = 50 mm and f2 = 11 mm; objective: Mitutoyo M Plan Apo SL 80X; CCD camera: IDT X Stream VISION XS-3.

5

5

5

55

5

5

510

10

10

10

10

10

1015

15

1515

15

15 20

20

20

20

20

20

25

25

25

25

25 30

30

30

30

30

35

35

35

35

40

40

40 45

45

x [µ

m]

y [µm]−500 0 500

−500

0

500

Figure 3. Measured shape of the axicon and its contour plot.

3.3 Measurement of the beam parameters

In order to compare the theoretical predictions with the experimental beam profile we measured the lateralintensity profile of the QBB behind the axicon and demagnified QBB behind the telescope. We imaged anumber of lateral planes zi (z′i) satisfying zi+1 − zi = 300 µm behind the axicon (z′i+1 − z′i = 10 µm behind thetelescope) with a CCD camera. At each position zi ten subsequent beam profiles were recorded. Each lateralprofile was fitted by a formula based on Eq. (3):

I(ρ, zi) = I0(zi)J20

⎧⎨⎩2.4048

√[x − x0(zi)]

2+ [y − y0(zi)]

2

ρ0(zi)

⎫⎬⎭ + O(zi), (8)

where I0(zi), x0(zi), y0(zi), ρ0(zi), and O(zi) are parameters of the fit at each zi. The parameter I0(zi) for allzi give the axial profile I0 of the QBB demonstrated in the following figures.

3.4 Comparison of the theory with the measured original disturbed beam

Experimental profiles of the axial intensity I0 (see Eq. (8)) behind the axicon (measurement region A) andbehind the demagnifying telescope (measurement region B) are shown in Fig. 4 as thin curves with plus marks.Intensity oscillations due to the oblate-tip of the axicon are clearly visible in the high-intensity part of the beam.In order to calculate the expected axial intensity profiles, we applied the theoretical procedure described inSection 2 with the experimentally determined value of the axicon tip parameter aexp = (33.7 ± 0.1)µm. Thetheoretical intensity profiles are shown in Fig. 4 as full curves for both measured regions A and B. Figure 5

Proc. of SPIE Vol. 7141 714126-5

0 30 600

0.2

0.4

0.6

0.8

1

z [mm]

I 0 [a.u

.]

measurement region A

experimentsimulation

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

z, [µm]

I 0 [a.u

.]

measurement region B

experimentsimulation

Figure 4. Comparison of measured and calculated axial intensity profile I0 of the QBB generated by the oblate-tip axiconmeasured directly behind the axicon (left - measurement region A in Fig. 2) and behind the demagnifying telescope (right- measurement region B).

compares the measured and calculated 2D intensity profiles for the same parameters as in Fig. 4. The resultsprove that the chosen theoretical description of the problem is sufficient to obtain excellent coincidence withthe measured results. Presented figures demonstrate serious variations of the beam generated behind oblate-tipaxicon from the expected QBB profile.

0 15 30 45 60

−60

−40

−20

0

20

40

60

z [mm]

ρ [µ

m]

0

0.2

0.4

0.6

0.8

1

0 15 30 45 60

−60

−40

−20

0

20

40

60

z [mm]

ρ [µ

m]

0.2

0.4

0.6

0.8

1

Figure 5. The measured (left) and calculated (right) spatial intensity profile of the QBB generated behind the oblate-tipaxicon for the same parameters as in Fig. 4.

3.5 Comparison of the theory with the measured filtered beam

In order to optimize the size of the spatial filter we first simulated how the radius of the filter influences thefinal axial shape of the QBB generated by the oblate-tip axicon. Identically to the previous comparison we useda = aexp = 33.7µm and the theoretical results are demonstrated in Fig. 6. It can be seen that the filtrationdoes not give the same axial intensity profile as the ideal QBB; however filter radii larger than 2138 µm provideslowly varying axial intensity profiles without any oscillations over a distance of 2/3 zmax. Figure 7 presents theexperimental axial intensity profile (I0 in Eq. (8)) if a filter of radius Rexp = (1925±5) µm was used (full curve).To compare this result with the theoretical simulations we again assumed a = aexp = 33.7 µm and searched forRfth giving the best overlap with the measured axial intensity profile. The results are shown in Fig. 7 as a fullblue curves. Even though we found Rfth = (2115 ± 5) µm which is by 10% larger than Rexp, we are persuadedthat the agreement is very good considering the sensitivity of the intensity profile to filter radius and its positionbetween the lenses of relatively small focal lengths.

The measured and calculated 2D spatial intensity profiles of the filtered QBB generated behind the oblate-tip axicon are shown in Fig. 8. They clearly demonstrate that the spatial filtration suppresses the unwanted

Proc. of SPIE Vol. 7141 714126-6

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

z [µm]

I 0 [a.u

.]

ideal QBB

non−filtered

1800 2000 22000

2

4

6

|S(R

)|2 [a

.u.]

ρ [µm]

Figure 6. Calculated axial intensity profile for perfect axicon (open circle with full line), and oblate-tip axicon witha = 33.7 µm. Different radii Rf of the spatial filter placed in the Fourier plane were considered: 0 µm (full), 1973 µm(dashed), 2064 µm (dotted), and 2138 µm (dot-dashed) (see the inset for corresponding spatial frequency spectrumcut-off).

0 500 1000 1500 2000 2500

0.2

0.4

0.6

0.8

1

z, [µm]

I 0 [a.u

.]

experimentsimulation

Figure 7. Comparison of measured (full with plus marks) and best fit calculated (full blue) axial intensity profiles I0 forfiltered QBB.

oscillations and that the lateral shape of the beam does not vary along the beam propagation - similarly as inQBB generated by the perfect axicon.

4. CONCLUSIONS

We considered an axicon with oblate-tip approximated by a hyperboloid of revolution of two sheets and weanalyzed theoretically and experimentally the properties of the beam generated behind such an axicon. Wedemonstrated that if the axicon tip deviates in its apex from the ideal sharp tip in the range of tens of micrometers,the beam generated behind it does not posses the invariant beam properties expected for quasi-Bessel beam.Significant axial oscillations of the optical intensity occur due to the interference between the quasi-Bessel beam,

Proc. of SPIE Vol. 7141 714126-7

500 1000 1500 2000

−10

−5

0

5

10

z, [µm]

ρ [µ

m]

0

0.2

0.4

0.6

0.8

1

500 1000 1500 2000

−10

−5

0

5

10

z, [µm]

ρ [µ

m]

0.2

0.4

0.6

0.8

1

Figure 8. The measured (left) and calculated (right) 2D spatial intensity profiles of filtered QBB generated behind theoblate-tip axicon. The parameters were the same as in Fig. 7.

formed by off-axis part of the axicon, and the wave refracted by the oblate-tip of the axicon. Such an intensityprofile can significantly influence experimental results expected for ideal quasi-Bessel beam and causes artifactsfor example in the measurement of weak inter-particle interactions in optical binding experiments.42 We measuredthe real shape of the axicon, approximated the axicon tip by a hyperboloid, and found persuading coincidencebetween optical intensity profile measured and theoretically simulated using Hankel transform. We furtherdemagnified the beam by a telescope and demonstrated how the spatial filtration of the beam in the telescopehelps to remove the undesired modulation and establish the original invariant intensity beam profile.

Authors highly appreciate the critical comments Dr. A. Jonas and the help of Prof. M. Ohlıdal and Dr. T.Fort with the axicon profile measurements and acknowledge the support from the 6FP EC NEST ADVENTUREActivity (ATOM3D, project no. 508952), MEYS CR (LC06007, OC08034), ISI IRP (AV0Z20650511) and MCT(FT-TA2/059).

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