1. Introduction

In the last two decades, an increasing attention has been focused on physical properties, experimental realization and applications of nondiffracting (ND) beams. In an exact mathematical treatment, the monochromatic ND beams can be established as solutions of the Helmholtz equation. The complex amplitude of the ND beam can then be described by the well-known mathematical functions obtained applying the separation of variables in the convenient coordinate systems [1, 2, 3, 4, 5]. The Fourier optics provides an illustrative explanation of the creation of the ideal ND beam. In this case, the ND beam can be comprehended as an interference field created by plane waves whose wave vectors k ≡ (k_x,k_y,k_z) have the same projection to the direction of the beam axis, k_z = const. The condition of the ND propagation puts a restriction on propagation directions of the plane-wave components of the spatial spectrum of the ND beam but admits an arbitrary amplitude and phase offset of each component. A targeted amplitude and phase modulation of the spatial spectrum enables the beam transformation important for its practical use. For example, the intensity profile of the ND beam can be shaped to a predetermined form or an array of the ND beams can be created [6, 7]. In an ideal case, the intensity profile of the ND beam is independent of the propagation coordinate and remains unchanged from -∞ to +∞. Its spot size can be adjusted in a wide interval by changing the angle, which the wave vectors of the plane-wave components make with the beam axis. A conception of the ND propagation of light has only a theoretical meaning and cannot be directly connected with real experiments. The reason is an infinite energy covered by the cross-section of the ideal ND beam. In experiments realized by means of the annular amplitude mask, the axicon, the diffractive optical element or the computer generated hologram sent to the spatial light modulator [8, 9], the so-called pseudo-nondiffracting (PND) beam can be generated. In its mathematical model, the transverse amplitude profile of the beam is bounded by a square integrable envelope so that the energy carried by the PND beam is finite [10]. Under convenient experimental conditions the realizable PND beams represent a good approximation of the theoretical ND beams.

In recent years, the physical properties of the PND beams have been successfully utilized for optical manipulation of particles. For such experiments, an extremely long propagation range of the narrow PND beams is of particular importance. The beams serve as light filaments enabling guiding particles, trapping long thin objects or simultaneous manipulation of both high and low refractive index particles [11]. In the paper, the PND beams with variable parameters useful for both metrology and optical trapping are examined. The main attention is focused on design, realization and testing of the laser convertor performing transformation of the laser diode beam or the fiber mode to the movable beam of the Bessel-Gauss (B-G) type. The B-G beam generated by the set-up has a narrow spot whose position can be continuously relocated across the plane perpendicular to the propagation direction of the beam. The movement of the beam spot is achieved by an appropriate linear phase modulation of the beam spatial spectrum performed inside the laser convertor. By the convertor connected to the laser tweezer the dynamical manipulation of particles can be realized. In the paper, this possibility is demonstrated on the transport of microparticles along a predetermined trajectory.

2. Conception of nondiffracting propagation of light

2.1. Theoretical nondiffracting beam

In a scalar approximation, the ideal monochromatic ND beam can be comprehended as a modelike field propagating along the well defined direction (z-axis). Its complex amplitude U_N can be written as a product of the transverse amplitude profile u_N and the phase oscillating term appointed by the propagation constant k_z,

Intensity profile of the ND beam I = ∣u_N∣² remains unchanged while it propagates and its dimensional scale depends on the parameter k_t, that can be expressed by means of the wave number k as k_t ² = k ² - k_z ². The spatial spectrum of the ND beam is reduced to a single radial spatial frequency so that it can be expressed by the Dirac delta-function δ(k_x ² + k_y ² - k_t ²) in the k-space. If Eq. (1) is substituted into the Helmholtz equation, the admissible profiles of the nondiffracting beams u_N can be found applying the separation of variables in the Cartesian, circular cylindrical, elliptical cylindrical or parabolic cylindrical coordinates [12]. The non-diffracting fields known as the Bessel, Hermite-Bessel, Mathieu and parabolic beams can be obtained in this manner.

The ND propagation of light can be explained and clearly demonstrated if the ND beam is considered to be an interference field of plane waves with specially chosen propagation directions. The complex amplitude of the separate plane waves is given as U_PW(r) = a exp(-i k _t ∙ r _t - ik_zz + iΨ), where k _t ≡ (k_x,k_y), r ≡ (r _t,z), r _t ≡ (x,y), a is a wave amplitude and Ψ denotes a phase offset. The ND propagation is ensured only if the longitudinal component of the wave vector k_z is the same for all plane-wave components of the beam. If we assume that the ND field rises from a continuous superposition of plane waves, their wave vectors create a conical surface whose axis coincides with the propagation direction of the ND field. The vertex angle of the cone 2θ is simply related to the transverse and longitudinal components of the wave vector, k_t = k sin θ, k_z = k cos θ. Change of the angle θ causes dimensional scaling of the transverse intensity spot of the ND beam. If θ increases, the ND beam becomes narrower.

If the geometry of the interfering waves is described in the circular cylindrical coordinates, x = ρ cos ϑ, y = ρ sin ϑ, k_x = k_t cos ψ and k_y = k_t sin ψ, the ND field U_N is obtained by integration of the plane waves U_PW along the azimuthal angle,

where A(ψ) = a(ψ) exp[i Ψ(ψ)]. The dependence of a and Ψ on ψ means that the the amplitude and the phase offset can differ for separate plane waves. In general, A(ψ) is an arbitrary periodical complex function. Its choice can be utilized to shape the beam intensity but it does not disturb the ND propagation of the beam. The azimuthal modulation of interfering plane waves is exploitable for generation of very complex ND fields including Bessel, Mathieu and parabolic beams, single or composed vortex fields, arrays of ND beams or ND fields with a predetermined shape [6, 7]. An admissible variability of A(ψ) can also be used for a controllable transverse relocation of the beam spot [9]. It is achieved if the linear phase Ψ_L is inserted in Eq. (2) as A(ψ) →A(ψ)exp[iΨ_L(Δx,Δy)], where

In this case, the obtained ND beam preserves its shape and propagation direction but its center is transversely shifted to the position given by coordinates [Δx, Δy]. The complex amplitude of the beam is then given by U_N(x - Δx,y - Δy, z). The required linear phase change of the angular spectrum Eq. (3) can be simply realized optically by means of the diasporometer. It was use to design of the laser convertor enabling generation of the B-G beam with an adjustable optical axis.

2.2. Realizable pseudo-nondiffracting beams

The ND beam whose transverse intensity profile remains exactly invariable during propagation is an unattainable idealization. To simulate the experimentally realizable beam, the theoretical ND beam must be transversely bounded. In this case, the intensity asymptotically vanishes faster than 1/ρ ², where ρ is the transversal distance from the center of the beam. As a result, the PND beam carrying finite energy is obtained. A simple and practically applicable model of the PND beam is obtained if the bounding envelope has a Gaussian form. In this case, description of the PND beam follows from the Helmholtz equation solved in the paraxial approximation. In the circular cylindrical coordinates, the well-known B-G beam is obtained as a realizable approximation of the theoretical Bessel beam. The PND beam with a general intensity profile can be introduced applying the integral representation. In this case, the PND beam can be comprehended as an interference field of inclined monochromatic Gaussian beams G whose axes create a conical surface with the vertex angle 2θ [10].The axes of the separate Gaussian beams are specified by the angles θ and ψ and their amplitudes are given by A(ψ). The complex amplitude of the PND field can then be expressed as

The Gaussian beams under the integral can be specified by the complex parameter q defined by the confocal parameter q ₀ as q = z+iq ₀. The complex amplitude U_PN then becomes [13]

where U_G denotes the Gaussian envelope and U is given as

As is obvious, U has a form of the ND beam given by Eq. (2), where the propagation constant k_z is replaced by its paraxial approximation

and the variable ρ is replaced by a scaled variable ρQ. The field intensity is not propagation invariant as the complex scaling factor Q depends on the propagation coordinate,

The envelope U_G has the form of the standard Gaussian beam whose parameters w ₀, R and Ω representing the beam waist radius, the wavefront radius and the Gouy phase shift, respectively, depend on the angle θ due to the inclination of the beam axis. Their exact form is given in [13].

3. Properties of minisized pseudo-nondiffracting beams

The ND and PND beams represent a very wide group of optical fields with different intensity profiles and various physical properties. In the paper, the beams of the Bessel and B-G type are examined. Such beams were successfully applied to various laser tweezer experiments demonstrating simultaneous micromanipulation in multiple planes [14] or trapping and subsequent precise delivery of several submicron particles over a distance of hundreds of micrometers [15]. The B-G beams were also verified to be well suited to rotationally align rod-like particles along the beam direction and to build and manipulate stacks of particles [16]. In the paper, attention is focused on geometrical parameters and energetics of the B-G beam and on optical means enabling relocation of the beam intensity spot across the transverse plane. An expediency of the B-G beams for their utilization in optical manipulation is discussed and demonstrated comparing the parameters of the B-G and Gaussian beams.

3.1. Geometrical parameters

The complex amplitude of the ND and PND beams can be expressed by the integrals Eq. (2) and Eq. (6) in which an arbitrary periodic function A(ψ) appears. If it represents the phase modulation given as A(ψ) = exp(imψ), m = 0,±1,±2,⋯, the integral Eq. (2) results in

where J_m denotes the m-th order Bessel function of the first kind. If the same azimuthal phase modulation A(ψ) is applied to the integral providing the PND beam, its complex amplitude UPN can be expressed by Eq. (5) with U given by

In this case, the B-G beam is obtained. Its transverse amplitude profile is described by the Bessel function J_m bounded by the Gaussian function U_G. For m ≠ 0, the Bessel and B-G beams are dark at the axis. They have a helical wavefront and belong to the group of optical vortices. In the paper, the bright zero-order Bessel and B-G beams (m = 0) are examined. In the case of the zero-order Bessel beam, the normalized intensity can be written as

As a member of the group of the theoretical ND beams, the zero-order Bessel beam reaches from -∞ to +∞. The beam is unbounded and carries infinite energy. In the case of the B-G beam with the complex amplitude U_BG, the normalized intensity is defined by

By means of Eq. (5), it can be rewritten to the form

At the plane z = 0, the beam is expressed as the ideal Bessel beam bounded by the Gaussian envelope,

The Bessel function J ₀ defines intensity profile resembling the well-known Airy diffraction pattern. Its central disk has the radius given by ρ ₀ = 2.4/k_t. If ρ ₀ is small in comparison with the waist radius of the Gaussian envelope w ₀, the B-G beam well approximates the ideal Bessel beam in a finite propagation region. Its length z_BG depends on w ₀ and in experiments it can be changed for beams with the same ρ ₀. The length of the region where the B-G beam exists can be examined by means of the axial intensity. Its normalized form can be written as

For propagation distances of interest which are considerably shorter than the Rayleigh distance of the Gaussian envelope, z ≪ q ₀, the axial intensity can be simplified to the form

The maximal propagation distance z_BG can be defined by means of an admissible decrease of the normalized axial intensity, I_BG(0,z_BG) = 1/e ². The length of the usable propagation region of the B-G beam then can be written as

As is obvious, the length of the B-G beam with the fixed size of the intensity spot ρ ₀ is enlarged if the waist radius w ₀ of the Gaussian envelope expands. This is a unique property of the PND beams unattainable with common laser beams. It can be successfully utilized to convey microparticles in optical manipulation where the narrow optical beams with a long propagation range are required. To demonstrate preferences of the PND beams, the attainable parameters of the B-G and strongly focused Gaussian beams are compared under similar demands on optical elements used in the optical set-up (Fig. 1).

 

Fig. 1. Comparison of (a) the focused Gaussian beam and (b) the PND beam of the B-G type.

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If the collimated Gaussian beam with the waist radius w ₀ impinges on the lens with the focal length f and numerical aperture NA (Fig. 1(a)), it is focused to the spot with the waist radius w _0F given by

where NA ≈ w _0F/f. The transformed beam can be treated as well focused inside a region whose length z_G is given as double Rayleigh distance,

The angular divergence of the focused Gaussian beam θ_G can be written as

To examine propagation properties of the B-G beam, a simple experiment illustrated in Fig. 1(b) is considered. In this case, the annular light source is placed at the front focal plane of the lens with the Gaussian transparency defined by the waist radius w ₀. The spherical waves emitted by the source are transformed to the Gaussian beams whose propagation axes create a conical surface with the vertex angle 2θ. Behind the lens, the B-G beam is created by interference of the Gaussian beams. Its transverse amplitude profile can be approximated by the Bessel function J ₀ bounded by the Gaussian envelope. The angle θ is connected with the radius of the central spot ρ ₀ of the generated beam,

As is obvious, the focused Gaussian beam and the B-G beam have the same spot size (ρ ₀ = w _0F) just when the angular divergence and the superposition angle of the B-G beam are matched, θ_G = θ = NA. It seems reasonable to compare the length of the propagation range of both types of beams just for this case. The length of the region where the B-G beam maintains its intensity profile can be approximated by

The ratio of the propagation lengths of the B-G beam and the focused Gaussian beam of a comparable transverse size can be written as

Important geometrical parameters of the examined beams can be approximately expressed by means of the focal length f and the numerical aperture of the lens NA,

As an example, the light beams with the wavelength λ = 0.632μm transformed by the lens with f = 250 mm and NA = 0.025 can be compared. In this case, the waist radius of the Gaussian beam and the radius of the central spot of the B-G beam are given as w _0F = ρ ₀ ≈ 8 μm. While the Gaussian beam remains focused in the range of length only z_G ≈ 0.65 mm, the B-G beam propagates with the central spot unchanged through the distance z_BG ≈ 250 mm. In this case, the propagation distance is approximately 385 times longer for the B-G beam than for the Gaussian beam. The theoretical prediction was experimentally verified by means of the set-up shown in Fig. 1(b). The annular amplitude mask was placed at the front focal plane of the lens with f = 250 mm and NA = 0.025 and illuminated by the collimated beam of the He-Ne laser. The created PND beam was imaged by the microscope objective to the CCD camera and analyzed by means of the Beam View Analyzer. In an agreement with the theory, the PND beam with the spot radius 8 μm appeared inside the region of the length of 250 mm without apparent changes of the intensity profile. The beam intensity spots detected at the distances 15, 125 and 250 mm behind the lens are illustrated in Fig. 2. Applying the microscope objective, even narrower beams can be generated. The ratio K then exceeds several times the demonstrated value.

 

Fig. 2. The B-G beam with the central spot radius 8 μm generated by the set-up in Fig. 1(b). Intensity spots at distances (a) 15 mm, (b) 125 mm and (c) 250 mm behind the lens.

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3.2. Efficiency of the power capture inside the beam propagation region

To convey the microparticles in experiments with the optical tweezers, narrow optical beams with a long propagation distance are desirable. To compare various types of beams, their spatial localization of the electromagnetic energy must be quantified. As a convenient measure, the fraction of the power captured by the detector of a given size and position can be accepted. To evaluate changes of the efficiency of the power detection inside the propagation region of the beam, the power captured by the detector with the radius R_D is explored in dependence on its position along the beam propagation direction. In this connection, the power capture coefficient η is defined as

where P_Z is the optical power received by the detector at the position specified by the longitudinal coordinate z and P ₀ denotes the maximal power captured by the detector at the plane of the best transverse localization of the beam. In the case of the Gaussian beam, the power P ₀ is related to the waist plane of the focused beam and P_Z denotes the power captured at the distance z from the waist plane. In this case, the power capture coefficient η_G is defined by

where

For the B-G beam, the power capture coefficient can be obtained in an acceptable approximation. It is based on the assumption that the detector with the radius R_D is comparable in size with the central spot of the beam. In the propagation range of the B-G beam, the size and shape of the central spot remain unchanged and only the axial intensity must be scaled due to redistribution of energy in the side lobes. If the detector is placed inside the propagation region (z < z_BG), the captured power can be approximated by

where I_BG(0,z) is the normalized axial intensity of the B-G beam given by Eq. (16) and P ₀ denotes the optical power captured by the detector at the initial plane of the beam propagation region. In the used approximation, the power capture efficiency of the B-G beam can be estimated as

To compare energetics of the Gaussian and B-G beams, it is useful to express a distance at which the power transfer efficiency falls to the given value η_G and η_BG, respectively. For the Gaussian beam it can be written as

where

For the B-G beam we obtain

Changes of the efficiency of the power capture inside the propagation region of the Gaussian and B-G beams can be examined by Eq. (33) and Eq. (35). Similarly as in the example demonstrating the geometrical parameters of the beams we assume that the angles of the Gaussian and the B-G beams are matched, θ_G = θ = NA. If the parameters λ = 0.632 μm, f = 10 mm and NA = 0.5 are adopted, the waist radius of the Gaussian beam is equal to the radius of the central spot of the B-G beam, w _0F = ρ ₀ = 0.4 μm. In the analysis of the captured power, the detector with the radius comparable to the beam spot, R_D = 0.4 μm, is placed at the plane where the maximal power is detected. This choice was motivated by the fact that the B-G beams can manipulate particles whose size is comparable to the beam core. If the object is moved along the axis of the beam, some energy escapes due to diffraction effects and the captured power goes down. In the case of the focused Gaussian beam, the power falls to 80% on a very short distance of about 1.5 μm. For the B-G beam, the power captured by the detector remains over 80% on a very long distance of more than 3 mm.

3.3. Optical transverse relocation of the beam spot

As follows from Eq. (2), the ND beam is created when plane waves with the specified propagation directions and arbitrary amplitudes and phases interfere. The amplitude and phase modulation A(ψ) of the plane wave spectrum can be used to shape the beam intensity profile or to perform the targeted transverse relocation of the beam spot. If the amplitude of the separate plane waves is constant and their phase is modulated by

the complex amplitude of the created ND beam is proportional to the expression

In this case, the beam amplitude profile is described by the zero-order Bessel function. The beam spot remains unchanged in free propagation but its axis is transversely shifted to the position given by [Δx,Δy]. If the phase modulation Eq. (36) is used in Eq. (4), the B-G beam with the transversely relocated intensity spot is described. Unlike the theoretical Bessel beam, the spot of the B-G beam is nonsymmetrically bounded by the Gaussian envelope at the off-axis position. The intensity profiles of the B-G beam at on-axis and off-axis positions are illustrated in Fig. 3. In experiments, the phase shift of the plane wave components Eq. (36) can be realized at the Fourier plane of the lens in the 4-f optical system illustrated in Fig. 4. The transformation can be simply explained for the case of the zero-order input Bessel beam whose axis coincides with the lens axis. The spatial spectrum of the beam created by the first Fourier lens has the shape of a bright circle with the radius ρ_S inversely proportional to the spot size of the input beam ρ ₀. Alternatively, it can be expressed by the beam parameter k_t, ρ_S ≈ fk_t/k, where f is the focal length of the lens. To perform required phase shift A(ψ) of the separate plane wave components localized at the circle it is sufficient to place a wedge prism with the vertex angle α near the focal plane of the lens. If the wedge prism is oriented along the x-axis as shown in Fig. 4, the phase modulation of the transmitted light can be approximately expressed by

 

Fig. 3. The intensity spot of the B-G beam at (a) on-axis and (b) off-axis positions.

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where n is the refractive index of the wedge prism and the insignificant constant phase shift was omitted. Interference of the light transmitted through the wedge prism creates the inclined divergent B-G beam [9]. By the second Fourier lens, it is transformed to the off-axis B-G beam. In the assumed theoretical case its complex amplitude is given by Eq. (37). From comparison of Eq. (36) with Eq. (38) it is clear that the wedge prism in the assumed orientation causes the transverse relocation of the beam spot to the position (Δx,0). The shift along the x-axis is given by

It is simple to show that the relocation of the beam spot to the arbitrary position [Δx,Δy] can be ensured by means of the wedge prism with a variable vertex angle whose position can be changed by rotation around the lens axis. In experiments it can be realized by means of the diasporometer.

 

Fig. 4. Illustration of the phase modulation of the spatial spectrum of the B-G beam resulting in the transverse relocation of the beam spot.

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4. Design of the laser beam converter

As was shown, the PND beams possess unique propagation features interesting for applications. Their main advantage consists in (i) realization of an extremely narrow beam with a controllable length of the propagation range, (ii) achievement of the nearly constant efficiency of the power capture inside the propagation region, (iii) possibility to relocate the beam spot by phase modulation of the spatial spectrum. In the paper, the design and realization of the laser convertor enabling generation of the adjustable PND beam of the B-G type is described. The convertor is designed as a compact optical system illuminated by laser radiation guided by the optical fiber. Alternatively, the laser diode with the beam collimator can be connected to the convertor. As the output, the B-G beam of specified parameters is obtained. It has an intensity spot of the required size whose shape remains unchanged in a long propagation region. The core axis of the beam is adjustable, it can be transversely relocated in a well defined area without changing direction. This action can be performed by a working movement of the optical diasporometer. Such beam is suitable for adjusting optical elements as its directionality is well maintained for all transverse positions of the beam spot. By means of a simple auxiliary optical system, the B-G beam can be reduced to a minisized beam with the core of several micrometers. In this configuration, the laser convertor works as the laser tweezer enabling conveying and transport of microparticles along the desired trajectories. The movement of the optical traps is controlled by a simple action of the diasporometer.

 

Fig. 5. (a) Laser convertor C used in the set-up for conveying and transfer of microparticles along desired trajectory (F-fiber, L_C-collimating lens, A-axicon, D-diasporometer, L ₁, L ₂, L ₃-lenses, M-mirror, L_M-microscope objective, S-sample). (b) Photo of the laser convertor with fiber optics illumination.

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An optical scheme of the designed laser convertor is illustrated in Fig. 5(a). The laser radiation leaving the optical fiber F is collimated by the lens L_C and directed to the axicon A. Behind the axicon, the B-G beam whose intensity profile can be approximated by the zero-order Bessel function bounded by the Gaussian envelope is created. The subsequent lenses L ₁ and L ₂ are placed so that they work as a telescope. The role of the telescope is dual - it scales the input beam and enables its relocation by the phase modulation introduced by the diasporometer placed at the back focal plane of the lens L ₁. The diasporometer is designed in a way that can operate as a wedge prism with variable vertex angle rotating around the beam axis.

Though the B-G beam entering the telescope can be realized by several ways, the method using the axicon was chosen owing to its very good energetic efficiency. The radius ρ ₁ of the spot size of the created beam depends on the axicon vertex angle τ and can be estimated by

where λ and n denote the wavelength of the used radiation and the refractive index of the axicon, respectively. The telescope composed of the lenses L ₁ and L ₂ scales the input beam in a measure depending on the magnification of the telescope given by the focal length of the lenses f ₁ and f ₂, Γ₁ = -f ₁/f ₂. The core radius of the beam behind the telescope can then be written as

The length of the propagation region of the beam created by the axicon z ₁ depends on the axicon vertex angle and on its diameter. The telescope changes not only the spot size of the input beam but also its range. It can be expressed by means of the magnification as

The diasporometer acting as the wedge prism with a variable angle α causes the off-axis shift of the beam spot whose magnitude is given by

The direction of the shift is given by the azimuthal orientation of the wedge prism. To reduce the size of the beam to microscale, an additional telescope must be used. If the beam is to be exploited in optical manipulation, the telescope is composed of the auxiliary lens L ₃ with the focal length f ₃ and the microscope objective L_M with the focal length f_M. Changes of the beam spot size and the beam propagation range caused by the additional telescope are also given by (41) and (42) but the magnification Γ₁ must be replaced by Γ₂ = -f ₃/f_M. The off-axis shift of the beam core h is reduced by the additional telescope to the value h/Γ₂.

5. Realization of the beam convertor and experimental results

After a basic design of dimensions, the laser convertor shown in Fig. 5 was optimized by the standard optical software and parameters of its components were specified. In the convertor testing, the radiation of the He-Ne laser (632 nm, 15 mW) was brought by the optical fiber with the numerical aperture NA = 0.13 and mode field diameter MFD = 3.3 μm. The collimation of the input light was performed by the lens L_C with the focal length f_c = 25.4 mm. The B-G beam was created by the axicon with the vertex angle τ = 178° (Eksma 130-0278). For the transverse relocation of the beam spot the diasporometer with the range of deviation angles ±0.5° was used. The transformation of the B-G beam was realized by the telescope composed of the lenses with the focal lengths f ₁ = 35 mm and f ₂ = 250 mm. The photo of the compact laser convertor is shown in Fig. 5(b). The beam appearing behind the laser convertor has the core radius 200 μm and the length of existence about 14 m. It is applicable to the centering or adjusting of optical elements. Alternatively, a collimating lens with the larger focal length can be used. In this case the propagation range of the beam can be enlarged up to 20 m. The transverse relocation of the beam core can be performed across the area of 4.5 × 4.5 mm².

In the performed experiment, the theoretical efficiency of the power capture inside the propagation region of the generated beam was also examined. Dimensions of the B-G beam created by the laser convertor were ten times reduced by a telescope so that the radius of its core was approximately ρ ₀ = 20 μm. The power captured by the detection area with the radius R_D = ρ ₀ = 20 μm was measured for various positions inside the beam propagation region. It was performed in such a way that the reduced beam was imaged to the CCD camera and the accepted power was determined by means of the image processing. The obtained data enabled evaluation of the dependence of the coefficient η_BG defined by Eq. (32) and Eq. (16) on the propagation coordinate z. In Fig. 6, a very good agreement of theoretical and experimental results is illustrated.

 

Fig. 6. Efficiency of the power capture inside the propagation region of the B-G and the Gauss beams.

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6. Application of the laser convertor to the optical manipulation

The laser convertor was also tested in the set-up for the manipulation of microparticles Fig. 5(a). In this case, the Verdi V2 laser (532 nm, maximal power 2 W) was used. The B-G beam leaving the convertor was dimensionally reduced by the additional telescope composed of the lens L3 of the focal length f ₃ = 200 mm and the microscope objective f_M = 1.8 mm (Olympus, UPLFLN 100×O2). The lowest power of the laser enabling optical manipulation was 60 mW, the power delivered to the created optical trap was approximately 50% of the light power leaving the fiber. The radius of the core of the beam in the optical tweezer was around 2μm. By this beam we were able to catch, convey and shift the polystyrene particles (Duke Scientific) with diameter 5μm across the area of 40×40μm². Results of the experiment are shown in Fig. 7. By means of the diasporometer, the particles were continuously transposed along the illustrated path. The snapshots illustrate the caught particles at the positions (a)-(f) of the defined trajectory. The small particles with the diameter around 1μm could be conveyed, but not manipulated. After their catching and conveying, they were stuck on the cover glass.

 

Fig. 7. Catching and movement of the polystyrene bead along a required trajectory by means of the set-up with the laser convertor.

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7. Conclusion

In the paper, the geometrical and energetic parameters of the PND beams of the B-G type were examined theoretically and experimentally. Their unique properties were utilized for realization of the experiment enabling conversion of the laser diode beam or the fiber mode to the long and narrow B-G beam whose axis can be transversely relocated across the defined area of the plane perpendicular to the propagation direction of the beam. The experimental set-up was optimized and produced as a compact laser convertor in the Meopta - optika company. The laser convertor was tested in real experimental conditions aimed for centering and adjusting of optical components. The particular attention was focused on a verification of the possibility to utilize the laser convertor as a device working as the laser tweezer enabling dynamical optical manipulation. In this case, the transverse relocation of the beam spot is controlled by a simple working movement of the diasporometer placed inside the laser convertor. By this way the particles with the diameter of several micrometers can be caught, conveyed and moved along the required trajectory.

The realized set-up represents a simpler and cheaper alternative of the holographical tweezers based on utilization of the spatial light modulator [17]. Some of the applications of the B-G beams demonstrated in the biophotonic workstation based on the use of the spatial light modulator [18] can be ensured with the set-up utilizing the laser convertor. Furthermore, an advanced application of the laser convertor is enabled by inclusion of simple optical components into the set-up. For example, by means of the spiral phase mask placed near the diasporometer the laser convertor can be used as a generator of movable optical vortices. They can be applied to the promising experiments on the transfer of the orbital angular momentum to microparticles. If the spatial spectrum of the input beam localized at the plane of diasporometer is influenced by an appropriate diffractive mask [6], the array of collinear B-G beams can be detected behind the laser convertor. Such structures of PND beams are useful for free-space optical interconnects [19].