Collinear non-diffracting beams: classification and properties

$: Collinear non-diffracting beams: classification and properties$
Collinear non-diffracting beams: classification and properties Balpreet Singh Ahluwalia*a and Woei Ming Leeb

aDepartment of Physics and Technology, University of Tromsø, Tromsø 9037, Norway bSUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, Fife KY16

9SS, United Kingdom

ABSTRACT

Non-diffracting laser modes and interfering non-diffracting beams have been extensively studied. Interfering non-diffracting beams generate novel laser modes. In this paper we accumulate various interfering conditions for non-diffracting beams and discuss the properties of the resultant beam. Contrasting intensity profiles and topological charge distribution are obtained on varying interfering conditions. Collinear propagation of non-diffracting laser beams is also reported.

Keywords: Non-diffracting beams, Bessel Beams, Helicon beams, Collinear Bessel beams

1. INTRODUCTION The first experimental generation of a beam with ‘limited-diffraction length’ dates back to 1954 by McLeod et. al. with an optical element he coined as ‘axicon’ (meaning axis image) [1, 2]. Axicon is used to produce long and narrow focal lines along the optical axis. The transverse beam intensity generated by an axicon can be approximated to a Bessel-like function [3, 4]. The non-diffracting beams are defined as an optical field, whose transverse intensity profile remains unchanged for a finite distance during free-space propagation. There exists a family of non-diffracting laser beams or propagation-invariant beams (also referred as Bessel beams) with a finite propagation distance over which the core of the beam do not diffract [3, 4]. J. Durnin et al reported the theory and the experimental generation of a Bessel beam as it is widely accepted today. The non-diffracting beams are defined as an optical field, whose transverse intensity profile remains unchanged for a finite distance during free-space propagation. The non-diffracting beams have a conical wavevector distribution and its core is immune to diffraction within the non-diffraction distance. The inception papers on non-diffracting laser beams generated considerable interest as well as debate. A comment to J. Durnin et. al. paper was pointed by D. Debeer et al [5, 6] whereby they compared the ‘non-diffracting beams’ as a line image and equivalent to a Poisson’s spot. Since then the non-diffracting beams have been studied and employed for various applications [7-16]. Bessel beams found applications in non-linear studies [7], interferometry [9], laser Compton Scattering system [10]. Due to its longer depth of focus, Bessel beams found application in optical interconnect [11] and metrology for scanning optical systems [12]. Recently Bessel beams have found tremendous application in an optical tweezers system [15].

Interfering non-diffracting beams were employed for the generation of various novel laser modes [16-23]. Initially researchers reported the interference of two zero-order Bessel beams (azimuthal indices l1=l2=0) with varying radial wavevector (kr1 ≠ kr2) in order to generate smaller central core of the resultant beam [20]. The self-imaging phenomena by interfering two zero-order Bessel beams (l1=l2=0) with varying radial wavevectors (kr1 ≠ kr2) were also reported. The similar concept was recently exploited by tailoring the radial wavevector of superimposing zero-order Bessel beams to obtain self-imaged optical bottle beams (three-dimensional intensity null points) [23]. Interfering higher-order Bessel beams (l1, l2>0) (singular Bessel beams) with same radial wavevector (kr1 = kr2) was reported to generate much richer vortex content during the free space propagation [16]. Interference of Bessel beams with varying radial wavevector (kr1 ≠ kr2) and varying topological charges (l1 ≠ l2) generated a novel rotating propagation-invariant field coined as helicon beams or ‘periodical-non-diffracting beams’ [17, 22]. Each of above-mentioned conditions for interfering non-diffracting beams have been separately studied. In this paper, we report different interfering conditions for Bessel beams. The properties of the resultant laser mode are discussed. Contrasting intensity profile and topological charge distribution of the resultant beam is obtained on varying interfering conditions. We further report a ‘collinear’ propagation of non-interfering and interfering Bessel beams. Collinear Bessel beams propagation was obtained by interfering Bessel beams *[email protected] phone +47 776451162

Complex Light and Optical Forces IV, edited by Enrique J. Galvez, David L. Andrews, Jesper Glückstad, Proc. of SPIE Vol. 7613, 76130U · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.840390

Proc. of SPIE Vol. 7613 76130U-1

Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/09/2014 Terms of Use: http://spiedl.org/terms

with a big difference in the topological charges. Under these conditions it is possible to have collinear propagation of a non-interfered Bessel beam at the centre and the interfering Bessel beams at the outer rings.

2. INTERFERING PROPAGATION INVARIANT BEAMS 2. 1 Theory of non-diffraction beams

Propagation-invariance property of a beam is easily understood in a Fourier domain. The diffraction of a beam can be analyzed employing angular spectrum approach. Diffraction of a beam is linked with the spread of the angular spectrum in the frequency domain. The spread of angular spectrum is associated with the relative phase change of the Fourier components. So to have an ideal non-diffracting beam it should have no spread in its angular spectrum. Based on the above criteria the necessary and sufficient condition on the angular spectrum of a beam to possess non-diffracting property is to have a single radial frequency. Thus the angular spectrum of a non-diffracting beam possesses a single ring in the frequency domain. J. Durnin et. al. [3] experimental set-up to generate a non-diffracting beam by employing an annular aperture at the back focal length of a lens is in principle selecting a single radial frequency in the angular spectrum.

The general solution for the amplitude of a non-diffracting wave is given as [17]: -

[ ] φρρθφρλπφρδ

πθ

ρ

π

ϕ

ddirziilkzrU r )cos(exp2exp)exp()(21),,( 2

2

0

2

0

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛Δ−= ∫ ∫

∞

= = (1)

Integrating Equation (1) over the entire limits of radial frequency (ρ) and azimuthal angle (φ ) the equation can be further simplified to: -

[ ] φθφφρλπ

πθ

π

φ

dikilzikzrU rr )cos(exp)exp(2exp21),,(

2

0

22

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛Δ= ∫

= (2)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛= 2

22exp)exp()()(),,( ρλπθθ izilrkJikzrU rl

lr

(3)

222 ρ

λπ

−⎟⎠⎞

⎜⎝⎛=zk

(4)

Jl denotes the lth order Bessel beam, and for a zero-order non-diffracting beam l=0. Substituting Eq. (4) in Eq. (3), the non-diffracting field can be expressed as: -

[ ]zrll

r izkilrkJikzrU exp)exp()()(),,( θθ = (5)

The intensity profile of a non-diffracting beam contains Bessel function as shown in Eq. (5), thus such beams are also commonly known as a Bessel beam, as mentioned above. The longitudinal distance over which the core of a Bessel beam is non-diffracting (Zmax) depend on radial wavevector (kr) and is given as: -

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ro k

kwZ max

(6)

where wo is the beam spot size. Non-diffracting or Bessel beams can be interfered based on azimuthal indices (l) and radial wavevectors (kr). Starting with the intensity of the two Bessel beams given as: -



[ ]11111 exp)exp()(),,( zrl izkilrkJzrI θθ = (7)

[ ]22222 exp)exp()(),,( zrl izkilrkJzrI θθ = (8)

The amplitude of the resulting interference of the beam is therefore can be obtained as:-

).exp()()().exp()()(),,( 222221111121 zkilrkJikzkilrkJikzrU zrl

lzrl

lr +++= θθθ (9)

The intensity distribution of the resultant beam can be determined by squaring the amplitude:-

[ ])).()(cos)()(2)()(),,( 12122211212

222

22

112

1 ψθθ +−+−++= zkkllrkJrkJkkrkJkrkJkzrI zzrlrlrrrlrrlr (10)

Where ψ is arbitrary additional phase change picked in the optical set-up. Eq. (10) represents generalized equation for interfering Bessel beams. 2.2 Interfering Bessel beams with different kr (kr1 ≠ kr2) and same l Interfering two zero-order Bessel beams (l1 = l2=0) with different radial wavevector will simplify Eq. (10) as:-

[ ])).(cos)()(2)()(),,( 122010212

202

22

102

1 ψθ +−++= zkkrkJrkJkkrkJkrkJkzrI zzrrrrrrrr (11)

Due to the presence of the cosine term and its dependency on z-axis in Eq. (11), it is evident that the longitudinal intensity will oscillate along the z-axis. Thus interfering Bessel beams under this condition will yield a beam whose transverse intensity profile will possess self-imaging affect. By carefully opting correct parameters it is also possible to generate self-imaged beam with embedded optical bottles (or three dimensional null-intensity points) where two Bessel beam interfere destructively [23]. Such beams were coined as self-imaged optical bottle beams and have found applications in optical tweezers for micro-manipulation [23, 24]. Figure 1(a) shows the longitudinal intensity profile of the generated self-imaged optical bottle beam by interfering two zero order Bessel beams (l1 = l2 =0) with different radial wavevectors (kr1/kr2 =1.6). Due to the destructive interference there are four embedded intensity null points or bottles present along the propagations-axis.

However when interfering two higher order Bessel beams (l1 = l2 ≠0) with different radial wavevectors (kr1 ≠ kr2) the bottling effect cease. Figure 1(b) shows the longitudinal intensity profile of the self-imaged beam obtained by interfering Bessel beams with kr1/kr2=1.6 and l1=l2=2. The intensity of the beam is still given by Eq. (11). As the interfering Bessel beams are of higher order (l>0), the resultant beam have central dark core throughout the propagation distance ceasing any bottling affect (three dimensional dark spot), as shown in Fig 1(b). Furthermore, it is worth highlighting that the resultant topological charge in Fig. 1(b) is zero and thus beam does not carry any default orbital angular momentum [26]. The interfering non-diffracting beams shown in Figure 1(a, b) have similar radial wavevectors (kr1/kr2 = 1.6) thus the periodicity of constructive and destructive interference pattern are identical. Figure 1(a, b) shows a longitudinal distance of min(Zmax1, Zmax2), where Zmax1 and Zmax2 are the maximum non-diffracting distance (Zmax) of the interfering Bessel beams. The distance between two bottles is given as 2π/(kz1-kz2). It was noticed that bottling effects are obtained within a distance where both the interfering Bessel beams are non-diffracting. If either of the interfering Bessel beams crosses Zmax distance, the interference ceased, this is true for all interfering Bessel beams discussed later as well.

By tailoring the parameters of the interfering Bessel beams, the location, number and the dimension of the central dark region (optical bottle) can be altered. In optical and atom trapping, three-dimensional dark spots are employed for trapping of low-index micro-particle and atoms, respectively. Investigation of tailoring the three-dimensional dark spots is thus useful. Figure 2 shows the feasibility to tailor both the axial and transverse dimension of the embedded optical bottle. By altering the parameters of radial wavevectors of interfering Bessel beams (kr1 and kr2) it is possible to change both axial and transverse dimension of the embedded optical bottle. It can be noticed from Figure 2 that by increasing the ratio of radial wavevectors of interfering Bessel beams, it is possible to shrink both the axial and transverse dimensions of the embedded optical bottle. Furthermore, the location and the number of embedded optical bottles can also be altered. Table 1 shows the number of optical bottles obtained for different conditions shown in Figure 2. It is worth highlighting that the number and the dimension of embedded optical bottle do not only depend on the ratio of



wavevectors (kr2/kr1) but also on the actual non-diffracting distance of interfering Bessel beams (Zmax). This is further dependant of radial wavevectors, wavelength of light and spot size of the beam as shown in Eq. (6).

Figure 1 Longitudinal intensity profile of a) self-imaged optical bottle beam generated by interfering Bessel beams with l1= l2=0 and kr1/kr2 = 1.6 and b) self-imaged beam generated by interfering Bessel beams with l1= l2=2 and kr1/kr2 = 1.6.

0

2

4

6

8

10

1 2 3 4 5 6

Transverse Dimension (1 Pixel = 1mm)

Axia

l Dim

ensi

on

(1 P

ixel

=Zm

ax1/

100)

kr2/kr1 =1.1kr2/kr1 =1.2kr2/kr1=1.6kr2/kr1=2kr2/kr1=3

Figure 2:- Tailoring axial and transverse dimensions of embedded optical bottle, where Zmax1=(wo*k/kr1), is maximum

non-diffracting distance of a Bessel beam.

Table 1:- Number of optical bottles obtained for different radial wavevectors of interfering Bessel beam.

kr2/kr1 1.1 1.2 1.6 2 3

No. of Bottles 1 1 4 6 11

2.3 Interfering Bessel beams with same kr (kr1= kr2) and different l

Interfering Bessel beams with same radial wavevector (kr1= kr2) and different topological (l1 ≠ l2 ≠0) charge generates non-diffracting beams with intensity modulations of remaining topological charges l2-1l. The intensity of the resultant beam under this condition is given as: -



[ ]))(cos)()()(2)()(),,( 122122

222

12 ψθθ +−++= llrkJrkJkrkJkkJkzrI rlrlrrlrrlr (12)

As kr1= kr2 (kz1 = kz2) and thus there is no cosine dependence of intensity along the longitudinal axis, ceasing self-imaging. However as the interfering Bessel beams possess different topological charges thus the resultant beam is richer in topological charges distribution. The topological charges at the center and at the outer rings of the resultant beam are l2-l1. Figure 3 (a, b) show transverse intensity profile of the interfering non-diffracting beams with kr1=kr2, l1=1 and l2=4 and kr1=kr2 and l1= 1 and l2=-1, respectively. It is evident from Fig. 3(a, b) that under these conditions the transverse intensity is modulated by remaining topological charges of l2-l1. Interfering Bessel beams under condition l1 ≠ l2 and l1 or l2 = 0 generates beam with contrasting topological charge distribution as compared to Fig. 3(a, b). The intensity of the resultant beam is still given by Eq. (17). Under this condition (l1 or l2 = 0) the central core of the resultant beam does not carry l2-l1 charges but outer rings does. The central core of the resultant beam is not broken and carries zero topological charge which is surrounded by outer rings carrying l2-l1 charges. Figure 3(c, d) shows the transverse intensity profile of the interfering Bessel beams with kr1=kr2 and l1=0 and l2=3 and kr1=kr2 and l1=0 and l2=2, respectively. Comparing Fig. 3(a, b) to Fig. 3(c, d) the contrasting topological charge distribution at the central core can be observed. The outer rings in both cases carry identical l2-l1 topological charges as shown in Fig 2. However, the resultant beams in Fig 3(c, d), possess unmodulated central core contrary to Fig 3 (a, b). The resultant beam with zero topological charge at the centre and higher topological charges at the outer rings can be explored in optical tweezers applications. When used in conventional optical tweezers the outer rings of such beams can impart orbital angular momentum [26] and concurrently perform optical levitation from the central core to the trapped micro-particles [15]. The central core of the resultant beam also inherits self-reconstruction and non-diffraction property of a Bessel beam [21].

Figure 3 Transverse intensity profile of the resultant beam obtained by interfering Bessel beams with kr1=kr2 a) l1= 1 and

l2=4, b) l1= 1 and l2=-1, c) l1= 0 and l2=3 and d) l1= 0 and l2=2

2.4 Interfering Bessel beams with different kr (kr1 ≠ kr2) and different l (l1 ≠ l2)

Interfering higher-order Bessel beams with different radial wavevectors (kr1 ≠ kr2) and azimuthal indices (l1 ≠ l2) generates a unique family of ‘non-diffracting-rotating-beams’ [17, 22]. Such fields have also been previously termed as helicon beams or periodic-non-diffracting. As the interfering non-diffracting beams under these conditions possess



different radial wavevectors and topological charges, the resultant beam possess both self-imaging, rotation and intensity modulation by the topological charges. The intensity of the resultant beam is similar to Eq. (10) and substituting the value of θ’ as given below in Eq. (10) it can be further simplified as: -

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=12

12'llkk

z zzθθ (13)

[ ])')((cos)()(2)()(),,( 122211212

222

22

112

1 ψθθ +−++= llrkJrkJkkrkJkrkJkzrI rlrlrrrlrrlr (14)

)0,,()0,',(),,(12

21⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+==llkkzrIrIzrI zzθθθ

(15)

Just like a Bessel beam it can be noticed from Eq. (15) that the transverse intensity of the resultant beam is propagation-invariant in the free-space except the rotation of the intensity profile around its optical axis. The rate of rotation of the helicon beams are proportional to the difference of the squared radial wavevectors of the interfering Bessel beams and is inversely proportional to the difference of the topological charge of interfering Bessel beams : -

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

≈⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=∂∂

12

21

22

12

12

21

llkk

llkk

zrrzz

πθ

(16)

The rotation sign depends on the sign z∂∂θ

if it is positive than the beam will rotate in clockwise direction and if it is negative the beam will rotate in anti-clockwise direction. The longitudinal distance required to complete 2π rotation is given as: -

( )2

12

2

122 4

/2

rr kkllk

zz

−−

=∂∂

=Δ πθπ

π

(17)

It is the topological charge distribution which is most interesting for such beams. The topological charge distributions for such beams depend on the value of radial wavevectors and azimuthal indices of interfering Bessel beams. The following cases are possible:-

(i) For kr2> kr1 and mod (l2) > mod (l1), the charge at the center and at the outer ring is (l2-l1) with the central dark core. Figure 4(a) shows the transverse and longitudinal intensity profile of the interfering Bessel beams with kr2/kr1 = 1.6 and l1= 1 and l2=3. The transverse intensity profile is modulated with topological charges of l2-l1 both at the central and outer rings. The intensity modulation can also be observed from the longitudinal intensity profile of the resultant beam. Figure 5 shows the transverse intensity profile of the resultant beam during free space propagation. As mentioned in Eq. (16, 17) the transverse intensity profile of the beam rotates in clockwise direction during free

space propagation as z∂∂θ

is positive.

(ii) If kr2 > kr1 and mod (l1) > mod (l2), the charge at the center will be l2 and at the outer ring is (l2-l1). If the sign of radial wavevector changes (kr2< kr1) then the charge distribution changes respectively. Figure 4(b) shows the longitudinal intensity profile of the interfering Bessel beams with kr2/kr1 = 1.6 and l1= 3 and l2=1. Figure 6 shows the transverse intensity profile of the resultant beam, outer rings possess topological charges of l2-l1 and the central core of the resultant beam carries l2 charge without any intensity modulation. The transverse intensity profile of the resultant beam rotates in anti-clockwise direction during free space propagation. On comparing the longitudinal intensity profiles of Fig. 4(a) and Fig. 4(b), the difference of topological charge distribution at the central core of two cases is evident. It is worth highlight that by just interchanging the topological charges (l1 and l2); the beams with contrasting intensity profile can be generated.

(iii) When mod (l1) = mod (l2) (example l1= 2 and l2=-2) for both the cases kr2> kr1 and kr2< kr1, the charge at the center and at the outer ring is (l2-l1).



Figure 4: Longitudinal intensity profile of the resultant beam obtained by interfering Bessel beams with kr2/kr1 = 1.6 and

a) l1= 1 and l2=3 and b) l1= 3 and l2=1.

Figure 5:- Transverse intensity profile of the resultant beam obtained by interfering Bessel beams with kr2/kr1 = 1.6 and

l1= 1 and l2=3. The transverse intensity profile rotated in clockwise direction during free space propagation.

Figure 6:- Transverse intensity profile of the resultant beam obtained by interfering Bessel beams with kr2/kr1 = 1.6 and

l1 = 3 and l2 = 1. The transverse intensity profile rotated in anti-clockwise direction during free space propagation.

3. COLLINEAR PROPAGATION OF BESSEL BEAMS Collinear propagation of coherent vortex beams or Laguerre-Gaussian beams (LG) have been previously studied [27-29]. Collinear propagation of two coherent Bessel beams, such that either one or both collinear propagating beams maintain their respective properties will be useful in applications like optical tweezers [15] and optical communications



[30]. Sections 2.5 and 2.6 showed that on interfering coherent Bessel beams under special conditions, the central core of the beam can carry un-modulated topological charge and the outer rings carry modulated topological charges, as shown in Fig 3(c, d) and Fig. 4(b). Thus it was foreseen that exploiting this condition and by interfering Bessel beams with big difference in their topological charges, it could be possible to have collinear propagation of Bessel beams. The aim is to have one a collinear propagation of non-interfered and interfered Bessel beams.

On interfering Bessel beams with big difference in their topological charges, we obtained collinear propagation of Bessel beams. Fig. 7(a, b) shows the transverse and the longitudinal intensity profile on interfering Bessel beams with kr2 = kr1 and l1= 25 and l2=0. The intensity of the resultant beam is given by Eq. (12). The collinear propagation of non-interfered zero-order Bessel beam at the centre and interfering Bessel beams pattern at the outer rings are evident in Fig. 7(a, b). The non-interfering region is highlighted in Fig. 7(a) and Fig. 7(c) plots the transverse intensity along the dotted line shown in Fig. 7(a). Figure 7(d) shows the intensity plot of pure zero-order Bessel beam with same parameters and at same propagation distance as Fig. 7(c). The intensity plot (power distribution in the rings) in the non-interfering region of Fig. 7(c) is similar to power distribution of pure zero-order Bessel beam as shown in Fig. 7(d). Moreover, the power distribution in the outer rings (outside the non-diffracting region) in Fig. 7(c) is irregular and different that from Fig. 7(d). This is due to the interference pattern of interfering Bessel beams. Thus the collinear propagation of pure zero-order Bessel beam and interfering Bessel beams is obtained. Due to equal radial wavevectors, the self-imaging and intensity rotation is not observed.

Figure 7: a) Transverse and b) longitudinal intensity profile obtained on interfering Bessel beams with kr2 =kr1 and a) l1 = 25 and l2 = 0, c) Intensity plot along the dotted lines shown in (a) and d) Intensity plot of pure zero-order Bessel beam

with same parameters.



In Fig. 7 a central core and five outer rings of a non-interfered Bessel beam is obtained at the centre. By increasing or decreasing the difference in the topological charges of interfering Bessel beams, the number of outer rings of the non-interfered Bessel beam can be tailored. For example, Fig. 8(a, b) show the transverse intensity profile obtained on interfering Bessel beams with identical radial wavevector (kr2 = kr1) and l1 = 0, l2 = 15 and l1 = 1, l2 = 15, respectively. The central core and two outer rings of the non-interfered Bessel beam at the centre are obtained. Figure 9 shows the longitudinal intensity profile of the beams shown in Fig. 8. The collinear propagation of non-interfering Bessel beam at the centre and interfering Bessel beams at the outer rings is evident from longitudinal intensity profile (Fig. 9) and transverse intensity profile (Fig. 8) respectively. At the centre, the propagation of non-interfered Bessel beam with l1 = 0 and l1 = 1 can be noticed for Fig. 8(a) and Fig 8(b) respectively.

It is worth highlighting that for all cases discussed in section 2, collinear propagation of Bessel beams are obtained whenever interfering Bessel beams have big difference in the topological charges.

Figure 8: Transverse intensity profile of the resultant beam on interfering Bessel beams with kr2 = kr1 and a) l 1= 0 and l2

= 15 and b) l1 = 1 and l2 = 15.

Figure 9: Longitudinal intensity profile of the resultant beam on interfering Bessel beams with kr2 = kr1 and a) l1 = 0 and l2 = 15 and b) l1= 1 and l2 = 15.

4. CONCLUSION Classification of interfering non-diffracting beams is reported. Emphasize on the resultant beam and their intensity profiles with varying parameters are discussed. Interfering Bessel beams generate various interesting beam profiles. Generation of a self-imaged beam, self-imaged bottle beam and rotating beam are obtained by interfering Bessel beam



under different conditions. Furthermore, coherent collinear propagation of Bessel beams is also reported. Collinear propagation of non-interfered Bessel beam at the centre and interfering Bessel beams at the outer rings are generated. Such collinearly propagating Bessel beams will be useful in optical tweezers and in multiplexing the information in optical communication [30].

One of the significant applications of laser beams with topological charges is its use in single molecule fluorescence microscopy [31-32]. In STED microscopy, an LG beam is used to “clean up” a single molecule fluorescence signal by imposing stimulated emission of the fluorophores. The spatial beam pattern of the LG beam (doughnut shaped) provides a small region at its centre core that does not get depleted by the STED beam. Hence the excess fluorescent signal at the edges of the molecule is removed, thus providing a clear fluorescent image. Using conventional Gaussian beam, it is possible to obtain a three dimensional high resolution image by z-scanning processing. By using non-diffracting light field (with and without topological charges) at different intervals along the axial position of the beam, it would be possible to provide direct STED and excitation light beam simultaneously across a large portion of the axial imaging depth. Using the technique describe in this paper, it might be possible to engineer an axial varying intensity pattern as a STED beam for high resolution single molecule three dimensional imaging, as shown in Fig. 1-3. This adds to the possible advantage of using these non-diffracting high order laser beam modes in STED microscopy.

Acknowledgements

This work is supported by the Research Council of Norway under FRINAT-programme

REFERENCES

[1] J. H. McLeod, “The Axicons: A new type of optical element”, J. Opt. Soc Am., 44, 592 (1954). [2] J. H. MclLeod, “Axicons and their uses”, J. Opt. Soc Am., 50, 166 (1960). [3] J. Durnin, Jr. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams”, Phys. Rv. Lett. 58, 1499 (1987). [4] J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory”, J. Opt. Soc. Am. A 4, 651 (1987). [5] D. Debeer, S.R. Hartmann and R. Friedberg, “Comment on Diffraction-Free beams”, Phys. Rev. Lett., 59, 2611

(1987). [6] J. Durnin, J.J. Miceli and J.H. Eberly, “Reply”, Phys. Rev. Lett., 59, 2612 (1987). [7] T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams”, Phy. Rev. Lett., 70, 1401 (1993). [8] J. Arlt, K. Dholakia, L. Allen and M. J Padgett, “Parametric down-conversion for light beams possessing orbital

angular momentum”, Phy. Rev. A 60, 2438 (1999). [9] Mathieu Fortin, Michel Piché and Ermanno F. Borra, “Optical tests with Bessel beam interferometry”, Opt. Exp.,

12, 5887 (2004). [10] Dazhi Li, Kazuo Imasaki, Masnori Aoki, Shuji Miyamoto, Sho Amano and Takayasu Mochizuki, “Application of

Nondiffracting Laser Beam to Laser Compton Scattering”, J. Nucl. Sci. & Tech., 40, 579 (2003). [11] Changyuan Yu, Michael R. Wang, Alberto J. Varela, Bing Chen, “High-density non- diffracting beam array for

optical interconnection,” Opt. Commun., 177, 369 (2000). [12] R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system”,

Appl. Opt., 31, 6653 (1992). [13] S.P. Tewari, H. Huang, and R.W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase-

matching”, Phys. Rev. A, 54, 2314 (1996). [14] S. Klewitz, F. Brinkmann, S. Herminghaus, and P. Leiderer “Bessel-beam-pumped tunable distributed-feedback

laser”, Appl. Opt., 34, 7670 (1995). [15] V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in

multiple planes using a self-reconstructing light beam,” Nature 419, 145 (2002). [16] S. Orlov, K. Regelskis, V. Smilgevičius, A. Stabinis, “Propagation of Bessel beams carrying optical vortices”, Opt.

Commun. 209, 155 (2002). [17] Carl Paterson and Robin Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system”, Opt.

Commnu., 124, 131 (1996). [18] S. Chávez-Cerda, E. Tepichin, A. Memeses-Nava, and G. Ramirez, “Experimental observation of interfering Bessel

beams”, Opt. Exp., 3, 524 (1998).



[19] S. Chávez-Cerda, A. Memeses-Nava, and J. Miguel Hickmann, “Interference of traveling nondiffracting beams”, Opt. Lett. 23, 1871 (1998).

[20] Z. Jaroszewicz A. Kolodziejczyk, A. Kujawski and C. Gomez-Reino, “Diffractive patterns of small cores generated by interference of Bessel beams”, Opt. Letts. 21, 839 (1996).

[21] Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: Theory and experiment”, J. Mod. Opt., 43, 1905 (1996).

[22] Rafael Piestun and Joseph Shamir, “Generalized propagation-invariant wave fields”, J. Opt. Soc. Am. A., 15, 3039 (1998).

[23] B. P. S. Ahluwalia, X.-C. Yuan and S. H. Tao, “Generation of self-imaged optical bottle beam”, Opt. Commun. 238, 177 (2004).

[24] B. P. S. Ahluwalia, X.-C. Yuan and S. H. Tao, “Transfer of ‘pure’ on-axis spin orbital angular momentum to the absorptive particle using self-imaged optical bottle beam”, Opt. Exp., 12, 5172 (2004).

[25] B. P. S. Ahluwalia, X.-C. Yuan, S. H. Tao, W.C. Cheong, L.S. Zhang and H. Wang, "Micro-manipulation of high and low indices microparticles using a microfabricated double-axicon", J. Appl. Phys., 99, 113104 (2006).

[26] K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt and K. Dholakia “Orbital angular momentum of a high-order Bessel light beam”, J. Opt. B-Quantum Semicl. Opt. 4, S82 (2002).

[27] I. Maleev and G. Swartzlander, “Composite optical vortices”, J. Opt. Soc. Am. B, 20, 1169 (2003). [28] D. Rozas, C. T. Law and G. A. Swartzlander, Jr., “Propagation dynamics of optical vortices”, J. Opt. Soc. Am. B 14,

3054 (1997). [29] J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, "Collinear superposition of multiple helical beams generated by a

single azimuthally modulated phase-only element," Opt. Lett. 30, 3266 (2005). [30] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space

information transfer using light beams carrying orbital angular momentum” Opt. Exp.12, 5448 (2004). [31] Stefan W. Hell and Jan Wichmann, "Breaking the diffraction resolution limit by stimulated emission: stimulated-

emission-depletion fluorescence microscopy," Opt. Lett. 19, 780 (1994). [32] B. Harke, K. I. Willig, G. Donnert, and S. W. Hell, "Recent Developments in STED-Microscopy," in Frontiers in

Optics, OSA Technical Digest (CD), paper FThB4 (2007).



Collinear non-diffracting beams: classification and properties

Documents

Transcript of Collinear non-diffracting beams: classification and properties