Bessel beams from semiconductor light sources

Author's Accepted Manuscript

Bessel beams from semiconductor light sources G.S. Sokolovskii, V.V. Dudelev, S.N. Losev, K.K. Soboleva, A.G. Deryagin, K.A. Fedorova, V.I. Kuchinskii, W. Sibbett, E.U. Rafailov

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S0079-6727(14)00031-7 http://dx.doi.org/10.1016/j.pquantelec.2014.07.001 JPQE179

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Progress in Quantum Electronics

Cite this article as: G.S. Sokolovskii, V.V. Dudelev, S.N. Losev, K.K. Soboleva, A.G. Deryagin, K.A. Fedorova, V.I. Kuchinskii, W. Sibbett, E.U. Rafailov, Bessel beams from semiconductor light sources, Progress in Quantum Electronics, http://dx.doi.org/ 10.1016/j.pquantelec.2014.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Bessel beams from semiconductor light sources

G.S.Sokolovskiia,*, V.V.Dudeleva, S.N.Loseva, K.K.Sobolevaa,b, A.G.Deryagina, K.A.Fedorovac, V.I.Kuchinskiia,d, W.Sibbette, E.U.Rafailovc a

Ioffe Physical-Technical Institute, 26 Polytechnicheskaya st., St Petersburg 194021, Russia b Saint-Petersburg State Polytechnical University, St Petersburg 195251, Russia; c Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, UK; d Saint-Petersburg State Electrotechnical University “LETI”, St Petersburg 197376, Russia; e School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK

Abstract We report on recent progress in the generation of non-diffracting (Bessel) beams from semiconductor light sources including both edge-emitting and surface-emitting semiconductor lasers as well as light-emitting diodes (LEDs). Bessel beams at the power level of Watts with central lobe diameters of a few to tens of micrometers were achieved from compact and highly efficient lasers. The practicality of reducing the central lobe size of the Bessel beam generated with high-power broad-stripe semiconductor lasers and LEDs to a level unachievable by means of traditional focusing has been demonstrated. We also discuss an approach to exceed the limit of power density for the focusing of radiation with high beam propagation parameter M2. Finally, we consider the potential of the semiconductor lasers for applications in optical trapping/tweezing and the perspectives to replace their gas and solidstate laser counterparts for a range of implementations in optical manipulation towards labon-chip configurations. © 2014 Elsevier Ltd. All rights reserved. Keywords: Bessel beams; Semiconductor lasers; Light-emitting diodes

Contents Abstract ....................................................................................................................................................................... 1 1. Introduction............................................................................................................................................................. 2 2. Basic concepts of non-diffracting (Bessel) beams................................................................................................. 2 2.1 Non-diffracting solution of the wave equation ............................................................................................ 2 2.2 Generation and properties of Bessel beams ................................................................................................. 3 3. Generation of non-diffracting light beams from semiconductor-based light sources........................................... 7 3.1 Generation of Bessel beams with VCSELs and LEDs ................................................................................ 7 3.2. Study of Bessel beams from broad-stripe edge-emitting semiconductor lasers ........................................ 9 3.3. Generation of Bessel beams from VECSELs ........................................................................................... 12 4. Effects of beam propagation parameter and axicon tolerance............................................................................. 17 4.1. Influence of high M2 on generation of Bessel beams from semiconductor lasers ................................. 17 4.2 Effect of a round-tip axicon on the transverse size of the central lobe of the Bessel beam ..................... 20 4.3. Experimental study of the effects of high M2 and rounded axicon tip on the generation of Bessel beams from semiconductor lasers ............................................................................................................................... 22 5. Superfocusing of multimode semiconductor lasers and light emitting diodes ................................................... 23 6. Optical trapping with Bessel beams generated from semiconductor lasers ........................................................ 26 Conclusion ................................................................................................................................................................ 28 Acknowledgments .................................................................................................................................................... 28 References................................................................................................................................................................. 28

*

Corresponding author. E-mail address: [email protected] (G.S. Sokolovskii)

2

1. Introduction Propagation-invariant (non-diffracting) light beams, that are capable of retaining their intensity during propagation, were reported in 1987 by Durnin [1] (and by Zel’dovich et al. and by McLeod as early as in the 1950s-60s [2,3]) and called Bessel beams because their amplitude profiles can be described by a zero-order Bessel function of the first kind as illustrated in Fig.1. In the projection onto a transverse plane (perpendicular to the propagation direction) such a beam appears as a bright spot surrounded by concentric fringes. The Bessel beams are generated through the interference of convergent beams taking place when a collimated Gaussian beam transits a cone-shaped lens (axicon). The central spot diameter of the Bessel beam is determined by the axicon angle and can be of the order of the optical wavelength. In practice, Bessel beams exhibit a finite propagation distance that depends on the cross-sectional diameter of the initial collimated beam. Another remarkable property of Bessel beams is the ability of the central ray to self-reconstruct its profile when disturbed by an obstacle [4]. Non-diffracting beams are extremely important in biomedical research because of applications involving optical trapping and tweezing [5,6]. The use of Bessel beams allows the distance between the focusing optics and manipulated objects to be significantly increased and eliminates the need for fine adjustment, thus making micromanipulation systems more flexible and attractive for practical implementations. The utilization of Bessel beams also opens new horizons in microporation [7], Doppler velocity measurement and colloid research [8], manipulation of micro-machines [9] and microfabrication [10,11], as well as for frequency doubling and for other nonlinear optical effects [12,13]. In optical coherence tomography (OCT), Bessel beams enable an order of magnitude increase of focusing depth into a sample [14,15], whereas in optical microscopy, these ‘needle’ beams are capable of producing in-depth 3D images with only 2D scan [16] to faciltiate reduced scattering artifacts with increased image quality and penetration depth in dense media [17]. Interestingly, in projection tomography, Bessel beams can be used to image at much greater depths than for confocal microscopy [18]. 2. Basic concepts of non-diffracting (Bessel) beams 2.1 Non-diffracting solution of the wave equation Non-diffracting light beams come as the exact solutions of the wave equation in free space: 1 ∂2 ΔE ( x, y, z , t ) − 2 2 E ( x, y, z , t ) = 0 , (1) c ∂t such that they are not subject to transverse spreading, i.e., retain their intensity during propagation. By restricting parameters to a monochromatic field with axial symmetry, propagation in the z direction can then be written in the cylindrical form, E(r,φ,z,t)=A(r)exp(imφ+ikzz-iωt), to give the equation for the complex amplitude of this field as: ∂ 2 A(r ) 1 ∂ A(r ) ⎛ 2 m 2 ⎞ (2) + + ⎜ kr − 2 ⎟ A(r ) = 0 , ∂ r2 r ∂r r ⎠ ⎝ where kz and kr are the longitudinal and transverse propagation constants of the field E(r,φ,z,t), correspondingly, so that kz2+kr2=ω2/c2. With a real kz and kr solution of (2) yields a class of fields that do not diffract, i.e., their time-averaged transverse intensity profile at z=0 is exactly reproduced in any transverse plane along the z axis. It can therefore be appreciated that the finite solution of (2) is an m-th order Bessel function of the first kind for which the non-diffracting field takes the form:

3

(3) E (r , ϕ , z ) = J m ( kr r )eimϕ +ikz z −iωt , where m is an integer number. In the simplest (and the most important) case of m=0 such a field has the transverse intensity profile of J02(krr) and appears as a bright spot surrounded by concentric fringes in the projection onto a transverse plane as shown in Fig.1.

a)

b)

Fig. 1. (a) Transverse profile of an ‘ideal’ Bessel beam intensity (i.e. the square of the zero-order Bessel function). (b) Transverse profile of intensity of the first-order Bessel beam.

Higher orders of Bessel beams are also possible. They appear as a doughnut-shaped bright spot surrounded by concentric rings. Fig.1b shows such a tubular Bessel beam of the first order (its transverse profile is defined by the first-order Bessel function of the first kind). From the practical viewpoint, these tubular beams are important for optical manipulation either of particles having low refractive index or single atoms [19-21]. It should be noted here that because the transverse intensity distribution of the Bessel beam Jm2(krr) decays as (krr)-1, each ring in Fig.1 carries approximately the same amount of energy. This means that diffraction-free propagating ‘ideal’ Bessel beams of infinite aperture are not possible. Therefore, all practical realizations of Bessel beams have a finite propagation length. 2.2 Generation and properties of Bessel beams Based on the theoretical analysis above, Durnin et al. demonstrated the generation of Bessel beams in 1987 [22] with a circular slit located in the focal plane of a lens. Unfortunately, such a method suffers extremely high losses of the collimated laser beam illuminating the circular slit in its central part. Another method for generatiing Bessel beams was demonstrated by Zel’dovich et al. as early as in 1966 [2]. This was based on the utilization of a cone-shaped lens (so-called axicon) that was first proposed by McLeod in 1954 [3]. The utilization of an axicon enables a much higher efficiency to be achieved compared to any other method for the generation of Bessel beams. For this reason, only axicon-generated Bessel beams will be considered here. Importantly, the option of exploiting spatial light modulators (SLMs) that are capable of mimicing most optical elements (including axicons of any kind) is adopted widely in the generation of Bessel beams using vibronic lasers. However, the utilization of SLMs with semiconductor laser diodes can be unattractive because of their cost and size. A discussion of the axicon parameters suitable for practical applications can be started from a definition of the physical parameters of a Bessel beam. Firstly, there is the diameter of the central lobe d0 and its propagation length zB0. A few definitions of d0 are possible, one of which is simply the first zero of the Bessel function krr ≈ 2.4. However, it is more practical to take the 1/e2 level definition of the central lobe size that yields krr ≈ 1.75 (see Fig. 2b). Another option is the half-width at half-maximum (HWHM) which defines the smallest size of the central lobe krr ≈ 1.13 when compared to the options above. For the considerations below, the dimensionless coefficient κ that takes the value that is appropriate for the chosen

4

definition of the transverse size of the central lobe of the Bessel beam (e.g. κ = 1.75 for 1/e2defined central lobe size) is taken. 1

b)

1

0.8

0.8

0.6

0.6

J02(krr)

J02(krr)

a)

0.4

0.2

0.2 0

-20

0.5 0.4

0

0

20

0

2

1/e 1.13

1.75

2.4

3

kr r kr r Fig. 2. (a) Radial distribution of the Bessel beam intensity (i.e. the square of the zero-order Bessel function). (b) The central lobe of Bessel beam. Lines show the half-maximum, 1/e2 and zero-intensity-defined size of the central lobe.

The propagation length of the Bessel beam zB0 is obviously limited by the initial aperture and by the cone angle (i.e., the angle of inclination of the conically converging rays whose interference constitutes the Bessel beam). The cone angle is also responsible for the transverse size of the central lobe d0 that means direct relation between the later and the propagation length of the Bessel beam zB0 as will be discussed below.

Fig. 3. Generation of the Bessel beam from a plane wave for an axicon with refractive index n, apex angle α, additional angle β=½(π-α) and aperture D0.

For this, the relation between the apex angle of the axicon α and the transverse size of the central lobe d0, which is defined by the angle of inclination of the rays behind the axicon γ as can be seen from Fig.3 and (3) is: 2κ 2κ d0 = = , (4) kr k sin γ where k is the wavevector and κ is dimensionless coefficient discussed above. The angle of inclination γ is readily deduced from the refraction of the incident wave on the axicon (see Fig.2):

(

)

γ = arcsin n cos α 2 − π 2 + α 2 , where n is the refractive index of the axicon.

(5)

5

Substitution of (5) into (4) produces the expression for the transverse size of the central lobe of the axicon-generated Bessel beam: d0 =

κλ π sin arcsin n cos α 2 − π 2 + α 2

(

(

)

(6)

)

This can be simplified for the axicon apex angles α close to 1800 to: κλ (7) d0 ≈ π ( n − 1) cos α 2 Fig.4 shows the transverse size of the central lobe of the Bessel beam as a function of the apex angle of the axicon made of BK-7 glass when illuminated by a plane wave of wavelength λ = 1.06 μm calculated with the exact formula (6) and for the approximation of a blunt apex angle (7). The graph shows the size of the central lobe for all three of the definitions mentioned above: zero intensity (κ=2.4, red lines), 1/e2-defined (κ=1.75, blue lines) and the full-width at half maximum (FWHM) of the central lobe (κ=1.13, green lines). It can be seen that the approximate expression (7) can be used for most of the practical values of apex angle down to 1400. The propagation length of the finite-aperture Bessel beam zB0 in a paraxial approximation can be deduced from Fig.3. Similar to the transverse size of the central lobe d0, it is defined by the aperture D0 and the angle of inclination of the rays behind the axicon γ: D0 D0 (8) zB0 = = 2 tan γ 2 tan arcsin n cos α − π + α 2 2 2 In Fig.5 the propagation length of the Bessel beam is shown as a function of the apex angle of the axicon made of BK-7 glass when illuminated by a plane wave of aperture D0 = 1 mm (solid line), 3 mm (dotted line) and 0.3 mm (dashed line).

(

)

)

Central lobe size, μm

(

Axicon apex angle, deg. Fig. 4. The size of the central lobe of the Bessel beam d0 vs. the apex angle α of the axicon made of BK-7 glass when illuminated by the plane wave of wavelength λ = 1.06 μm calculated with the exact formula (solid lines) and in approximation of blunt apex angle (dotted lines). Red lines show the zero-intensity-defined size of the central lobe (κ=2.4), blue lines show the 1/e2-defined size of the central lobe (κ=1.75) and green lines show the FWHM of the central lobe (κ=1.13).

6

Fig. 5. Propagation length zB0 for a Bessel beam vs. the apex angle α of the axicon made of BK-7 glass and illuminated by the plane wave of aperture D0 = 1 mm (solid line), 3 mm (dotted line) and 0.3 mm (dashed line).

There is a striking similarity between (8) and (6) and this yields the approximate relation between the propagation length of the Bessel beam zB0 and the transverse size of the central lobe d0 valid for the apex angle values close to 1800 as: π D0 (9) zB 0 ≈ d0 2κ λ This illustrates the difference between propagation of a finite-aperture Bessel beam and the diffraction (transverse spreading) of a Gaussian beam that is given by its Rayleigh length z0=πω02/4λ, where ω0 is the beam waist size. Comparing zB0 (9), which is a linear function of the central lobe size, with z0 being the squared function of the Gaussian beam waist, one can see that Bessel beams offer notable advantages for those applications that require tightly focused light fields.

Fig. 6. Illustration of the remarkable property of Bessel beams that are capable of self-reconstructing behind an obstacle [4]. This property allows for simultaneous manipulation of multiple particles with one Bessel beam [6].

Another remarkable property of Bessel beams, and the last one to be highlighted in this short overview, is their ability of the self-reconstruction behind the obstacle [4] as illustrated in Fig.6. The length of the shadow caused by an obstacle can be estimated in a way similar to (8) and (9) by the substitution of the size of the obstacle in place of the aperture D0 (obviously, the size of the obstacle should be small compared to the aperture). This property is very important for many practical applications as it allows for simultaneous manipulation of multiple particles with one Bessel beam [6].

7

3. Generation of non-diffracting light beams from semiconductor light sources Following the first demonstration, Bessel beams have generally been produced by reconfiguring the output beams from solid-state or gas lasers, but nowadays most of the practical applications require the sources to be more compact, efficient and cost-effective. One of the main impediments to developing light sources that produce non-diffracting beams is the coherence of the generated light. However, recently it was shown that Bessel beams can be formed from temporally incoherent light sources [23]. This opens up new avenues for the generation of Bessel beams from semiconductor light sources which are the most compact, reliable and efficient light sources available today.

3.1 Generation of Bessel beams with VCSELs and LEDs This discussion begins with the consideration of the possibility of generating Bessel beams with semiconductor light sources based around experiments where vertical-cavity surfaceemitting lasers (VCSELs) have been employed [24]. This type of semiconductor laser is well known for the circular symmetry of radiation and this therefore avoids and difficulties arising from astigmatism in the light source [25]. The experimental arrangement involved the collimated light from a semiconductor light source that was projected by a microscope lens onto an axicon having an apex angle of 178°. The resulting light pattern behind the axicon was monitored by a CCD camera with (where necessary) a magnifying lens [24]. The expectation is for a, observed very strong dependence of the generated Bessel beams on the coherence of the light source. Numerically, the coherence of the light source is characterized by the coherence length lc given by the ratio of the squared wavelength λ to the linewidth ∆λ (i.e., FWHM of the radiation spectrum), so that lc=λ2/∆λ. For this reason, the coherence length is shown in Table 1 together with other important parameters of the semiconductor light sources used [24]. Table 1. Parameters of VCSELs and LEDs used for the generation of Bessel beams. No.

Light Source

I, mA

∆λ, nm

λ0, nm

lc, µm

1

VCSEL (DFO-701d, Innolume GmbH)

1

0.1

988.5

9771

2

VCSEL (DFO-701d, Innolume GmbH)

3

0.9

988.5

1086

3

High-power red LED (LXHL-MD1D, Lumileds Co.)

36

19

642

22

4

High-power blue LED (LXHL-LB3C, Lumileds Co.); parabolic case, diameter of emitting surface 27 mm (Belyi Svet Ltd.)

36

20

473

11

5

Green LED (LY517PPG1-70, COTCO)

2

30

528

9.2

6

Red LED (KIPD 21 K-K, Planeta Corp.); parabolic case, diameter of emitting surface 20 mm (Belyi Svet Ltd.)

8

20

652

21

Fig. 7a shows the transverse pattern of the Bessel beam generated using a quasi-single-mode VCSEL and the corresponding radiation spectrum at a pump current I of 1 mA. As the pump current was increased to 3 mA (see Fig.7b), the laser switched to a multimode operation and this led to a drop in the coherence length (see Table 1) and the deterioration of the spatial coherence, which was manifested by the decrease of the contrast and the number of the concentric fringes and an increase in the central lobe size. Following the successful demonstration of Bessel beams generated from mid-infrared VCSELs, the authors of [24] proceeded to explore more demanding experimental arrangements involving red, green and blue light-emitting diodes (LEDs) that featured domeshaped and parabolic-shaped cases. The notable parameter difference with the previous assessment was the extremely low coherence length of LEDs which was two orders of magnitude lower than that of the VCSELs.

8

a)

b)

Fig. 7. The transverse profiles of a Bessel beam generated using a VCSEL (DFO-701d, Innolume GmbH) with the 1700 axicon and the corresponding radiation spectra of this laser at pump current I=1 mA (a) and I=3 mA (b).

a)

b)

Fig. 8. (a) Transverse profile of a Bessel beam generated using a red LED (KIPD 21 K-K) in a parabolic case with an emitting surface diameter of 20 mm (Belyi Svet Ltd.) with aperture of 5 mm diameter and radiation spectrum of this LED (I=8 mA). (b) Transverse profile of Bessel beam generated using a green LED (LY517PPG1-70, COTCO) in a dome case and the corresponding radiation spectrum of this LED (I=2 mA). The apex angle of the axicon is 1780.

9

The generation of Bessel beams from LEDs was demonstrated successfully and Fig. 8а shows the characteristic pattern of the Bessel beam generated using a red LED having a parabolic case (see the 6th line in Table 1) as a radiation source with the 1780 axicon and the correspondingradiation spectrum of this LED. Fig. 8b shows the Bessel beam pattern generated from the green LED with a dome case (see the 5th line in Table 1) with the 1780 axicon together with the radiation spectrum of this LED. The generation of Bessel beams with other LEDs using 1700 and 1780 axicons produced similar results. Given the very low coherence length of LEDs, the experiments on the generation of Bessel beams were started with placing the adjustable aperture between the light source and the axicon. However, it was found that the effect of the aperture could be mimicked through the adjustment of the distance between the LED and the axicon. This distance was varied depending on the light-emitting crystal size and the LED case shape, which confirmed the influence of spatial coherence on the formation of the Bessel beams. For example, this distance differed by a few-fold for the LEDs #3 and #6 which exhibit almost identical coherence lengths. Such a difference can be attributed to the different spatial coherence rather than the temporal coherence of these light sources. It should be emphasized that the spatial coherence plays an important role in the formation of Bessel beams. From these data [24], the maximum spatial coherence among the LEDs used in this study was inherent in that having small sizes (300 ×300 µm) of the light-emitting crystal and a parabolic shape of the LED case. It must also be noted that the typical size of the features of the Bessel beam in Fig.8 exceeds the coherence length of the LED by an order of magnitude. 3.2. Study of Bessel beams from broad-stripe edge-emitting semiconductor lasers Based on the experimental results described in the previous section, the authors of [24] also studied the possibility of generating non-diffracting light beams from broad-stripe edgeemitting laser diodes. Such lasers are capable of generating 10s-W output powers in the CW regime [26], and therefore they are of the prime interest as an alternative to vibronic lasers for the generation of non-diffracting light beams. The stripe width of the edge-emitting laser diodes used in these experiments was 100 µm. Bessel beams were generated using an axicon with an apex angle of 1700, thus enabling a central lobe diameter of about 10 µm. The radiation of the semiconductor laser was collimated by an optical system utilizing a cylindrical microlens that collimated radiation in the fast axis. Fig. 9 shows the transverse patterns of the generated Bessel beam at a pump current of I = 400 mA (close to the lasing threshold) and at I = 1000 mA (3-fold threshold) with the corresponding spectra of the radiation. One can see that an increase in the pump current leads to a significant change in the transverse distribution of the Bessel beam intensity, where several bright spots (rather than one) can be observed at the center of the pattern. The observed corruption of the central lobe of the Bessel beam can be explained either by multimode radiation (that is clearly seen from the spectrum at Fig.9b) or filamentation that is typical for almost all types of the broad-stripe edge-emitting laser diodes. In [27] it was suggested that the spatial homogeneity of the emission from a semiconductor laser plays a much greater role in the Bessel beam formation than does the temporal coherence. In order to confirm this hypothesis, Bessel beams were generated using the radiation of a broad-stripe edge-emitting external-cavity laser with a diffraction grating in the Littrow configuration [28]. This system design makes it possible to vary the width of the radiation spectrum (and, hence, the temporal coherence) within wide limits at a relatively small detuning of the cavity without altering the spatial parameters of the output beam.

10

a)

b)

Fig. 9. Transverse profiles of the Bessel beams generated from the edge-emitting laser diode and radiation spectra of this laser diode at pump current of (a) I=400 mA and (b) I=1000 mA. The axicon apex angle is 1780.

Fig. 10 shows the patterns of intensity distribution in the cross section of the Bessel beam generated from the broad-stripe external-cavity laser at various distances from the starting point of beam formation. The FWHM of the laser spectrum varied two-fold from ~0.6nm (Fig.10a) to ~1.2 nm (Fig.10b). The radiation powers in the central lobe was measured to be 7mW and 5.5mW, respectively.

Fig. 10. Experimental patterns of the intensity distribution in the cross section of the Bessel beam at various distances from the starting point of the beam formation from the broad-stripe (100 μm) external-cavity laser with the radiation spectrum full width at half-maximum (FWHM) of (a) ~0.6 nm and (b) ~1.2 nm. The insets show the corresponding radiation spectra [27].

11

The total output power was 450 mW and the aperture of the Gaussian beam was set to 2 mm. As can be seen from Fig. 10, the length of Bessel beam propagation without significant distortions was approximately the same (3–4 mm) in both cases. This value is significantly smaller than the maximum propagation length (23 mm) corresponding to the case of the plane monochromatic wave of the same aperture incident on the axicon. Given the two-fold difference of the temporal coherence in Figs.10a and 10b and very similar Bessel beam propagation lengths in both cases, the observed corruption of the central lobe of the Bessel beam can be explained by a lack of spatial coherence rather than poor temporal coherence [28]. Summarizing these results, it should be stated that the direct application of the conventional semiconductor lasers for the generation of practical Bessel beams is complicated because of the low symmetry of the beam, insufficient power level of the narrow-stripe diode lasers and due to the low spatial quality of the beam of the broad-area devices. It was found [29] that the main difficulties in generating non-diffracting beams from high-power laser diodes can be classified into three main categories:

Fig. 11. Schematic diagrams of Bessel beam profiles and the corresponding typical experimental radial intensity distributions in Bessel beams formed using broad-stripe semiconductor lasers in cases of (a) multimode radiation, (b) filamented radiation, and (c) oblique irradiation of the axicon.

(i) The multimode generation prevents collimation of the laser beam (due to different curvature of the wavefronts for different modes) and causes ‘washout’ of the fringes of Bessel beam as illustrated schematically in Fig. 11a. It should be noted that this may not affect the central lobe of the Bessel beam. (ii) The astigmatism of the laser diode radiation (as well as the filamentation) leads to the degradation of the Bessel beam (Fig. 11b) with the central spot taking a more and more elongated shape with propagation of the beam (or transforming to the line of several bright spots in case of the filamentation).

12

Fig. 12. Intensity distributions in various transverse cross sections of a Bessel beam formed using a c-DBR laser.

(iii) The oblique irradiation of the axicon caused by the misalignment of the optical scheme can cause degradation of the central spot of the beam into the diamond-shaped caustic (Fig. 11c. The size of this asteroid is proportional to the angle of the misalignment and the beam propagation distance, as earlier reported for the oblique irradiation of the axicon with gas and solid-state lasers [30,31]. It should be also noted that at a high astigmatism of the laser radiation, the emission from side edges of the broad stripe can be incident on the axicon at a significant angle, thus leading to the caustic formation even in a perfectly aligned optical scheme. These issues were investigated in greater detail [27] in an experimental study of the generation of Bessel beams using a broad-stripe curved-grating distributed Bragg reflector (cDBR) laser [32,33]. The grating grooves in this DBR laser represented the arcs of concentric circles, which corresponded to the cylindrical symmetry of the cavity and provided a common geometric origin for all the generated modes [34]. This served to decrease the negative influence of astigmatism and the multimode nature of the generation by ensuring a better spatial homogeneity in the collimated divergent laser beam used to form the Bessel beam. Indeed, the use of a broad-stripe c-DBR laser enabled the generation of a Bessel beam that exhibited the maximum propagation length, in which the onset of distortion of the central ray was observed only at the distance of 15 mm from the axicon tip (Fig. 12)! The comparison of the generation of Bessel beams in the case of the broad-stripe semiconductor lasers with the linear (Fig. 10) and cylindrical (Fig. 12) symmetry of the cavity demonstrates clearly that the astigmatism and the filamented radiation in these lasers had the maximum negative influence on the Bessel beam formation. This influence can thus be reduced significantly using broadstripe semiconductor lasers having a cylindrical symmetry of the cavity, in particular, the cDBR lasers [32,34]. 3.3. Generation of Bessel beams from VECSELs On the basis of the work described in this paper, it can be stated that the generation of the Bessel-type beams from broad-area vertical external-cavity surface-emitting lasers (VECSELs) [also known as semiconductor disk lasers (SDLs)], that offer Watt-level powers with the good beam quality [35] could represent a practical source option for applications in optical manipulation [36]. Two VECSEL types were used for the assessments reported here, where one was optically pumped and the other was electrically pumped.

13

Fig. 13. Simplified optical scheme of the EP-VECSEL for generation of Bessel beam.

The electrically-pumped VECSELs (EP-VECSELs) used in this work were InGaAs quantumwell (QW)-based laser sources emitting at 980 nm [37] (Novalux, USA). The active region of the EP-VECSEL was composed of several strain compensated InGaAs/GaAsP multiple quantum-wells (MQWs), grown by organometallic vapor deposition atop an n-type multilayer GaAs/AlGaAs Bragg reflector (DBR, R~0.7) and n–GaAs substrate. A high reflectivity ptype DBR completed the epitaxial structure that was bonded through a dielectric layer with an aperture providing electrical contact to a heat sink. The n-contact defined the optical aperture within the device to constrain the output in the TEM00 mode. P = 3 mW

P = 40 mW

b)

P = 288 mW

c)

z = 120 mm

a)

e)

f)

z = 180 mm

d)

Fig. 14. Propagation features of EP-VECSEL-generated Bessel beams at power levels of 3 mW (a,d), 40 mW (b,e), and 288 mW (c,f). The distance of propagation is 120 mm (a,b,c) and 180 mm (d,e,f). These Bessel beams were produced using a 1780 axicon.

14

100 μm

a)

f)

1000 μm

b)

g)

2000 μm

c)

h)

3000 μm

d)

i)

4000 μm

e)

j)

Fig. 15. Propagation of Bessel beams generated from a VECSEL in a quasi-single-mode regime with an axicon apex angle 1400 (a-e), and 1600 (f-j). The full width of the images is 100 μm.

15 100 μm

a)

f)

500 μm

b)

g)

1000 μm

c)

h)

1500 μm

d)

i)

2000 μm

e)

j)

Fig. 16. Propagation characteristics of Bessel beams generated from the VECSEL at the power level of 2.4 W with an axicon apex angle 1400 (a-e), and 1600 (f-j). The full width of the images is 100 μm.

16

The n-DBR was used to partly offset finite absorption losses for light travelling through the nGaAs substrate, as well as to stabilize the frequency output of the VECSEL. The device had a circular aperture of 150-µm diameter. The straight cavity configuration was formed by using the device as an end mirror. The cavity was completed by an output coupler with the radius of curvature RoC = -75 mm with 10% transmission. The simplified optical scheme of the EPVECSEL is shown in Fig.13. In the linear cavity as well as monolithic configurations, these lasers represent a technology capable of producing high brightness 980 nm CW beams. These lasers have produced up to 1 W of output power in the CW at 980 nm in low order multimode and 0.5W CW single-transverse mode and single-frequency operation. In the CW regime, these lasers operated with FWHM spectral bandwidths of <0.2 nm limited by the spectral resolution of the spectrometer. In experiments on the generation of Bessel beams from EP-VECSELs, good quality beams were obtained at a sub-Watt power level and central lobe diameters of 10-100 μm. With the 1780 axicon (and with a broader central lobe of the Bessel beam) in the single-mode operation, an aperture-defined propagation length of tens of centimeters could be demonstrated at power levels of up to a few hundred milliwatts [36] as illustrated in Fig.14. The optically-pumped VECSELs were based on novel InGaAs quantum-dot (QD) structure grown on GaAs/AlAs distributed Bragg reflector (DBR) [38]. The semiconductor gain chip was bonded to the intracavity diamond heat spreader and mounted on a copper mount (attached to a water-cooled holder) for heat dissipation to facilitate high output power operation. The gain chip was optically pumped by a 20 W, 808 nm fiber-coupled diode laser with the pump beam focused to a 120-µm diameter spot. The semiconductor chip served as a cavity end mirror for the external V-shape cavity configuration formed by using a RoC = -75 mm folding mirror and a plane output coupler with 0.6% transmission. The laser operated at 1040 nm. In the quasi-single-mode regime, Bessel beams with the central lobe diameter of approximately 4 µm and 8 µm were generated with the 1400 and 1600 axicons respectively. Essentially, no beam degeneracy was observed over the entire aperture-defined propagation length for the Bessel beam with the central lobe size 4 µm, as can be seen in Fig.15. Indeed, in these assessments, good quality Bessel beams were generated with power levels ranging up to a few Watts and it is believed that these are the best results yet reported for Bessel beams derived from any semiconductor light source. Importantly also, this type of performance is comparable to that achieved using the vibronic laser counterparts. At a higher power level, as the number of lasing modes increases (but M 2 still in singledigits) the divergence-defined propagation length of the Bessel beam originating from the quasi-Gaussian beam with the 5-mm waist reduces to only few millimeters both for the narrower and for the broader central lobes. This is illustrated in Fig.16 where the propagation of Bessel beams with the central lobe sizes of 4 and 8 µm is shown for the power level of 2.4 W. Most of the experiments on laser-diode-generated Bessel beams reported to date consider only zero-order beams. However, it was reported recently, for the first time to the best of our knowledge, the generation of higher-order Bessel beams from the semiconductor diode lasers [36]. It is well known that the higher-order Bessel beams have applications for manipulation of the low-refractive index particles and atom guiding and offer advantages over other hollow light beams due to their non-diffractive nature [39]. Therefore, the generation of such beams from the semiconductor lasers is highly desirable.

17

a)

b)

Fig. 17. The high-order Bessel beams generated from the EP-VECSEL at the power level of 200 mW with the 1600 (a) and 1400 (b) axicons. The full width of the images is 100 μm.

In Fig.17 it can be seen that higher-order Bessel beams with the central light-hole diameter of approximately 5 µm and 2 µm can be generated by the electrically-pumped VECSELs with the 1600 and 1400 axicons [36]. The output power of the VECSEL for these images was 200 mW. This direct axicon-assisted generation of these higher-order Bessel beams at such power levels with near-100% conversion efficiency (due to the absence of any additional beam-forming optical element such as a hologram [39,40]) was made possible by operating the VECSEL in a higher-order Laguerre-Gaussian mode. This can be achieved by the alignment of the external resonator due to the inherent doughnut-shaped electrical pumping distribution that is typical for all broad-aperture vertical-cavity surface-emitting lasers. 4. Effects of beam propagation parameter and axicon tolerance

4.1. Influence of high M2 on generation of Bessel beams from semiconductor lasers  In general, the quality of the laser beam is usually described by the beam propagation parameter M2 [41,42] (also termed as beam ‘quality’ parameter), which is defined as the ratio of the beam divergence to the divergence of an ‘ideal’ Gaussian beam (i.e., a beam with M2=1), corresponding to the diffraction limit. Similarly, the parameter M2 determines the ratio of the focal spot of the quasi-Gaussian beam to that produced by focusing the ‘ideal’ Gaussian beam by the same optical system. The parameter M2 is useful because it allows one to describe the quasi-Gaussian beams using the mathematical description developed for Gaussian beams. In this case, use is made of a simple replacement λ → М2λ, i.e., the wavelength is increased numerically М2-fold. When such a collimated quasi-Gaussian beam with high M2 is used for generating a Bessel beam [43] the transverse size of the central lobe of the Bessel beam gradually increases during propagation due to the divergence of the original quasi-Gaussian beam. Obviously, this effect compromises the ‘non-diffracting’ nature of the generated quasi-Bessel beam and limits its propagation distance zB (see Fig. 18). To evaluate the effect of the beam divergence, it is necessary to consider it in the expression for the transverse size of the central lobe of the Bessel beam (6) with the quasi-Gaussian beam divergence taken into account:

18

Fig. 18. Propagation of the Bessel beam formed from a quasi-Gaussian beam with high parameter M2: zB is the Bessel beam propagation distance due to the beam divergence; zB0 is the geometric propagation distance; z0 is the Rayleigh length.

dM 2 =

κλ

(10) α ⎛α π ⎞ π sin ⎜ − + arcsin(n cos ) − x( z ) ⎟ 2 ⎝2 2 ⎠, where x(z) is the beam divergence angle, which depends on the longitudinal coordinate z. Here, the quasi-Gaussian beam is assumed to be coaxial with the axicon and collimated in the plane passing through its apex. A paraxial approximation is also made and the axicon is assumed to be thin. Therefore, after determining (Fig. 18) the angle of divergence as the arctangent of the ratio of transverse coordinate of the original ray on the axicon to the wavefront curvature of the quasi-Gaussian beam R(z) when the beam reaches the symmetry axis, ⎛ z ⋅ tg(γ ) ⎞ (11) x( z ) = arctg ⎜ ⎟ ⎝ R( z ) ⎠ and using the well-known expression for the wavefront curvature of the quasi-Gaussian beam [44]: ⎛ ⎛ πω 2 ⎞2 ⎞ R( z ) = z ⎜1 + ⎜ 2 0 ⎟ ⎟ ⎜ ⎝ M λz ⎠ ⎟ ⎝ ⎠,

(12)

where ω0 is the quasi-Gaussian beam aperture, after obvious transformations and taking into account Snell’s law γ=(n–1)β in the paraxial approximation, then: (n − 1) β (13) x( z ) = 2 ⎛ πω02 ⎞ 1+ ⎜ 2 ⎟ ⎝ M λz ⎠ , where β=900–α/2 is the additional angle of the axicon. Substitution of (13) into (10), after simple trigonometric transformations gives: ⎛ ⎡ ⎤ ⎡ ⎤⎞ ⎜ ⎢ ⎥ ⎢ ⎥⎟ κλ ⎜ n −1 n −1 ⎢ ⎥ ⎢ ⎥⎟ dM 2 ( z) = β ⎥ − n sin ⎢ β ⎥⎟ ⎜ ( n − 1)sin ⎢ β + 2 2 2 π ⎜ ⎛ πω 2 ⎞ ⎢ ⎢1 + ⎛ πω0 ⎞ ⎥ ⎟ 1+ ⎜ 2 0 ⎟ ⎥ ⎜ 2 ⎜ ⎢⎣ ⎢⎣ ⎝ M λ z ⎟⎠ ⎥⎦ ⎟ ⎝ M λ z ⎠ ⎥⎦ ⎝ ⎠

−1

This cumbersome expression can be greatly simplified in the paraxial approximation to:

(14)

19

dM 2 ( z) ≈

⎡ ⎛ M 2λ z ⎞2 ⎤ ⎢1 + ⎜ ⎟ ⎥ π (n − 1)sin β ⎢ ⎝ πω02 ⎠ ⎥ ⎣ ⎦,

κλ

(15)

wherein the difference between the results of calculations using expressions (14) and (15) does not exceed 5% in the entire range of practically important apex angles of the axicon. Equation (15) also provides a simple calculation of the propagation distance of the Bessel beam, defined by the divergence of the forming beam [43]. Defining the propagation distance zB of the quasi-Bessel beam as the distance at which the transverse size of its core increases ⎯ times (similar to the definition of the Rayleigh length), and utilizing (15), then: by √2 2

⎛ M 2 λz B ⎞ ⎜⎜ ⎟ +1 = 2 (16) 2 ⎟ ⎝ πω0 ⎠ After completing the obvious transformations, the expression for the propagation length of the quasi-Bessel beam becomes: 2ω 2 πω 2 (17) zB = 2 − 1 20 ≈ 20 M λ M λ, which is fully consistent with the result obtained directly from the analysis of the divergence of the forming beam [43]. It should be noted that when the forming beam divergence is not taken into account, and only the geometric parameters of the optical scheme are considered, the Bessel beam propagation distance is found from the known equation [29]: π ω 0 d0 z B0 = (18)

κλ

(cf. (9) with D0=2ω0), which in view of (15) in the geometrical optics approximation takes the form:

zB0 ≈

ω0

(19) ( n − 1) sin β From the comparison of expressions (17) and (19) it can be easily seen that the reduction of the beam forming aperture ω0 in comparison with M 2λ (20) ω0< 2(n − 1) sin β leads to the restriction of the propagation distance of the Bessel beam because of the divergence of the forming beam, while the geometrical parameters of the optical scheme have no influence. Obviously, this effect is particularly important for the formation of Bessel beams from the quasi-Gaussian beams with large values of M2. Fig.19 shows the calculated dependence of the transverse size of the central lobe of the Bessel beam on the longitudinal coordinate z for different values of the beam propagation parameter M2 of the forming quasiGaussian beam and its aperture ω0.

20 10

d, um

8 6 4 2 0

0

1000

2000

3000

4000

z, um

Fig. 19 Dependences of the Bessel beam core diameter on the longitudinal coordinate z at ω0 = 100 mm, M2= 1 (dashed lines), ω0 = 50 mm, M2= 1 (solid lines), ω0 = 100 mm, M2= 5 (dotted lines). The calculation was performed for the axicon with an apex angle of α = 160° (the upper family of curves) and α = 140° (the lower family of curves).

4.2 Effect of a round-tip axicon on the transverse size of the central lobe of a Bessel beam The rounding of the axicon tip is a highly undesirable defect occurring during the manufacturing of the axicon [45]. It happens because of inevitable technological difficulties in the final polishing of the conical surface of the axicon. The effect of the round-tip axicon can be neglected in the study of the formation of Bessel beams with a large beam-forming aperture and significant propagation distance. However, when the beam-forming aperture is reduced to hundreds of micrometers and the Bessel beam propagation distance is correspondingly reduced, the accounting of the effect of the axicon tip rounding on the transverse dimension of the Bessel beam core is absolutely necessary. In [43], the axicon with a tip rounded to the radius R and the transverse size H of the rounded region was considered (see Fig. 20).

Fig. 20 Calculation of the effect of the axicon tip rounding on the formation of a Bessel beam.

The radius of rounding is related to the size of the rounded area by the obvious relationship:

21

H = R sin β (21) From this expression follows that the region of formation of the Bessel beam by the round-tip axicon is shifted with respect to the ‘ideally’ sharp axicon tip by the distance of R/n. With respect to the round tip, it is shifted by the distance of z0+δ (Fig. 20) where 1 − cos β (22) δ =R cos β As shown schematically in Fig. 20, the rounded tip of the axicon acts on the forming beam as a plano-convex lens with the focal length f, described by the matrix [44]: ⎡ 1 0⎤ ⎡ 1 0⎤ ⎢1 ⎥=⎢ (23) n − 1 ⎥⎥ ⎢ 1⎥ ⎢ 1 ⎦ ⎣⎢ f ⎦⎥ ⎣ R Therefore, in the z0+δ–long region of ‘geometrical shadow’ the central part of the forming beam is focused, and in the region of the Bessel beam propagation focused radiation may interfere with the conically converging rays, distorting the transverse profile of the Bessel beam [43]. From (14), the focal length of the ‘extra lens’ is: R (24) f = n −1 and with n≈1.5 it is a few-fold the length of the ‘geometric shadow’ z0+δ ≈ R/n. The numerical aperture of the lens of radius R and aperture H with the small focal length f taken into account is given by: 2 2 ⎡ ⎛ M c2λ ⎞ ⎤⎥ ( n − 1) ⎢ 2 2 4 ⎟ (25) NAR = sin β ± sin β − 4⎜⎜ 2 ⎢ πR(n − 1) ⎟⎠ ⎥ ⎝ ⎦, ⎣ 2 where Мс is the propagation parameter of the central part of the forming beam incident on the round-tip axicon. In the paraxial approximation using (21) and (24), the numerical aperture (25) can be written in the form 2

⎛ M c2λ ⎞ ⎟⎟ (26) NAR ≈ (n − 1) sin β − 4⎜⎜ ⎝ πR sin β ⎠ It should be noted that given the small size of the round-tip region, in most cases one can approximately assume Мс2≈ 1. The diameter of the focal spot of the forming beam with the given parameter Мс2 is 2M c2 λ (27) dR ≈ π (n − 1) sin β and the diameter of the focused beam as a function of z is given by the expression: 2

2

2

⎛ ⎞ M c2 λ ⎟⎟ + ( z (n − 1) − R )2 sin 2 β (28) d R ( z ) ≈ 2 ⎜⎜ − π ( n 1 ) sin β ⎝ ⎠ The comparison of (15) and (27) shows that the minimum transverse size of the beam focused by the lens formed by the round-tip of the axicon corresponds to the transverse size of the central lobe of the Bessel beam with good accuracy. From this, it is evident that in an experiment one should expect a relatively drastic visual increase in the transverse size of the central lobe of the Bessel beam at the distance shorter than the axicon ‘focal length’ (24). The effect of the axicon tip rounding on the transverse size of the central lobe of the propagating Bessel beam at different radii of rounding is shown in Fig.21. It can be seen that the round tip of the axicon can lead to the significant increase in the visible size of the central lobe of the Bessel beam, even when it is formed by an ideal Gaussian beam.

22 40

d, um

30 20 10 0

0

50

100

150

200

z, um Fig. 21. Calculated dependences of the transverse size of the central lobe of the Bessel beam on the distance from the axicon tip z. The parameters used in the calculation are: α = 140°, R = 50 mm, М2= 1(red solid line); α = 140°, R = 100 mm, М2= 1 (red dashed line); α = 140°, R = 50 mm, М2= 4 (blue dotted line); α = 160°, R = 100 mm, М2= 1 (blue dash-dotted line).

4.3. Experimental study of the effects of high M2 and rounded axicon tip on generation of Bessel beams from semiconductor lasers For experimental verification of expressions (15) and (28) that determine the variation of the transverse size of the central lobe of the Bessel beam generated with an axicon with a roundtip radius R from the quasi-Gaussian beam with the beam propagation parameter М2>1, an optically-pumped VECSEL with the wide active area described above was employed. The generation wavelength was 1040 nm [36]. 15

d, um

a) 10

5

0

0

1000

2000

3000

4000

3000

4000

z, um 15

d, um

b) 10

5

0

0

1000

2000

z, um

Fig. 22. The transverse size of the central lobe of the Bessel beam vs. the distance from the axicon tip z. The experimental values are indicated by black squares. The axicon apex angle is 1600. The beam propagation parameter of the VECSEL radiation is M2=2.5 at the power level of P=36mW (a) and M2=4 at the power level of P=2.4W (b). For calculations, the radius of the axicon tip rounding R is assumed to be 60 μm. The dashed line indicates the mean value of the central lobe size d0 in the case of a Bessel beam that is formed from a plane wave.

23

The quality parameter of the output beam in these experiments varied from 2 at the lasing threshold to 4 at 2.4W output power. Bessel beams were formed by the axicons with apex angle of 1400 and 1600 and recorded using a telescopic projection system and a CCD detector (see Figs. 15 and 16). The intensity distribution of the Bessel beams at different distances from the axicon was measured by moving the detection system on a micropositioner stage. The parameters of the detection system were chosen so as to ensure the width of the field of view was 100 mm. 10

a) d, um

8 6 4 2 0

0

1000

2000

3000

4000

3000

4000

z, um 15

d, um

b)

10

5

0

0

1000

2000

z, um

Fig. 23. The transverse size of the central lobe of the Bessel beam vs. the distance from the axicon tip z. The experimental values are indicated by black squares. Axicon apex angle is 1400. Beam propagation parameter of VECSEL radiation is M2=2.5 at power level of P=36mW (a) and M2=4 at power level of P=2.4W (b). For calculations, the radius of the axicon tip rounding R is assumed to be 60 μm. The dashed line indicates the mean value of the central lobe size d0 if the Bessel beam is formed from a plane wave.

Fig. 22 and Fig. 23 show the transverse size of the central lobe of the Bessel beam versus the distance z from the axicon tip for different values of M2 parameters and the axicon apex angles of 1400 and 1600. One can see very good agreement of the experimental data with the theory.

5. Superfocusing of multimode semiconductor lasers and light-emitting diodes In this section, we will discuss the idea of ‘superfocusing’ of the multimode beams with high M2 via the ‘interference’ focusing with an axicon initially proposed in [46]. Focusing of the multimode radiation is among the most significant problems that hinder expansion of the field of application of high power semiconductor lasers and LEDs. Fig. 24a illustrates a basic problem in the focusing of multimode beams. With typical M2 values of 20–50 for high-power semiconductor lasers and reaching up to 200–500 for LEDs, a low spatial quality of the beam determines the focal spot size to be one to two orders of magnitude larger than the diffraction limit. This hinders the possibility for increasing the focal power density of the beam and the production of high optical field gradients that are necessary for many practical applications.

24

To overcome this limitation, a method of focusing multimode radiation through the generation of the quasi-Bessel beam with an axicon has been proposed by the Authors [46]. The idea of the proposed approach is based on the fact that, in the case of traditional focusing methods for multimode radiation, different curvatures of the wave fronts of various modes lead to a shift of their foci along the optical axis and, hence, to the increase of the focal spot size with increase of M2 value. The interference focusing of radiation of the semiconductor source with an axicon leads to formation of the common central lobe of a Bessel beam for all modes (Fig. 24b). It should, of course, be borne in mind that there will be a gradual increase of the transverse size of the Bessel beam formed from a collimated multimode quasi-Gaussian beam during its propagation behind the axicon because of considerable divergence of the initial beam, as already mentioned. This limits the propagation length zB of the resulting quasi-Bessel beam (Fig. 24c). However, the primary size of the central lobe in the Bessel beam may actually be a few-fold smaller than the diffraction limit of the focal spot size of a quasi-Gaussian beam having a high M2. exp(-2r2/w2)

a)

b)

J02(r)

exp(-2r2/w2)

J02(r)

zB

zB0

z0

c)

Fig. 24. Comparison of the propagation of the Gaussian and Bessel beams: (a) focusing of (solid curve) the Gaussian and (dashed curve) quasi-Gaussian beams by a lens; (b) propagation of the Bessel beam formed by an axicon; (c) propagation of the Bessel beam formed from the diverging quasi-Gaussian beam. Note a difference between the propagation length of the Bessel beam (determined by the divergence zB of the initial beam), the “geometric” propagation length zB0, and the Rayleigh range z0

Fig. 25 shows the results of our experiments on the interference focusing for overcoming the diffraction limit for radiation from high-power semiconductor lasers and LEDs. In the assessment with a semiconductor laser of 100-μm-wide stripe width (λ= 1.06 μm, M2= 22), the diameter of the focal spot is observed to be as small as ~4 μm which is less than half of the diffraction limit of focusing of such beam by an ideal optical system having a unity numerical aperture. We have also demonstrated the focusing of a LED beam (λ= 0.6 μm,

25

M2>200) into the Bessel beam with the central lobe diameter of 6 μm [46], which is an order of magnitude smaller than the diffraction limit (~40 μm)! However, despite experimental demonstration of a small focal spot size, which was to date considered absolutely unachievable for the quasi-Gaussian beams with high M2 values, there are many applications where the laser power density should also be increased. The highest power density that can be achieved by focusing of the quasi-Gaussian beam by the ideal optical system with a numerical aperture of unity can be approximately expressed as follows: πℑ (29) PG = 2 ( M 2λ ) , where ℑ is the radiation intensity. For the sake of simplicity, the average power density in the focal spot can be considered. In fact, the power density, which is maximum at the focal spot center and decays to its edge, can significantly exceed the value given by (29). However, since in most practical applications of high-power semiconductor lasers it is sufficient to take into account only the average power density in the spot, the consideration below will be restricted to the approximation (29). As can be seen from there, the highest power density of a quasi-Gaussian ray exhibits the quadratic decay with the decrease of the beam quality (increase of M2). This opens up avenue route to increase the power density by means of the interference focusing. At the same time, the power density in the central lobe of the Bessel beam (in contrast to the Gaussian and the quasi-Gaussian counterparts) is determined not only by the central lobe size d0 on a 1/e2 level but also by the total number of fringes m in the Bessel beam: PB =

4ℑ π md 02

(30)

so that the ratio of the power density in the central lobe of the Bessel beam to that in the quasi-Gaussian focal spot is: PB 1 ⎛ 2 M 2 λ ⎞ = ⎜ ⎟ PG m ⎝ π d 0 ⎠

a)

2

(31)

b)

10 um Fig. 25. Superfocusing of (a) the semiconductor laser beam (λ= 1.06 μm, M2= 22) into a spot with diameter about 4μm and (b) the LED radiation (λ= 0.6 μm, M2= 200) into the Bessel beam with the central lobe diameter of 6 μm.

Formula (31) suggests that the interference focusing presents a means for overcoming the diffraction limit of the power density of the quasi-Gaussian ray only for very large values of M2. In other words, the interference focusing of a beam with a reasonable beam quality does not lead to any gain in power density, because the term squared in formula (31) cannot exceed unity for M2≈1. Therefore, in order to assess the range of practical applicability of the interference focusing, it is necessary to consider the ratio of the wavelength to the central lobe

26

size (λ/d0) and the number of fringes m of the Bessel beam. Using (6) for the central lobe size d0 on a zero level and taking into account the roots of the zero-order Bessel function of the first kind one can with good accuracy determine m as follows: 2 α (32) m ≈ M 2 ( n − 1) cos π 2 Upon carrying out simple transformations and taking into account that in practice the fraction of power transmitted through the aperture ω0 of the forming quasi-Gaussian beam is ~86.5%, one can express the formula (31) in the convenient form: PB α (33) ≈ 1.77(n − 1) M 2 NA cos 2 PG , As can be readily seen from (33), the interference focusing of a semiconductor laser beam with M2=17 the diffraction limit of the power density to be overcome with very realistic parameters of an axicon made of BK-7 glass with α=1400 and a numerical aperture of the initial beam NA>0.2. The realization of such a method of overcoming the diffraction limit of the power density with the interference focusing of the high-M2 beams is, unfortunately, hindered by the effect of a round tip on the axicon. Nevertheless, based on the demonstration of the reduction of the size of a focal spot of high-power semiconductor lasers and LEDs to a level unachievable by means of traditional focusing, we believe that the improvement of the axicon tip quality will open the way to the proportional increase of the power density available from semiconductor light sources. 6. Optical trapping with Bessel beams generated from semiconductor lasers In this section we demonstrate, for the first time to the best of our knowledge, the utilization of the Bessel beams generated from a semiconductor laser for optical trapping and manipulation of microscopic particles including living cells. A schematic view of the experimental setup is shown in Fig. 26.

Fig. 26. Schematic representation of the experimental setup for optical trapping and manipulation of microscopic particles with Bessel beams generated from a semiconductor laser.

27

The edge-emitting semiconductor laser used for the generation of Bessel beams in these experiments was fiber-coupled with the output power from the fiber up to 600 mW and the emission spectrum centered at the wavelength of 1065 nm. The output laser beam was collimated and refocused by the optical system including a set of interchangeable microscope lenses with magnification ranging from x8 to x60 and a dichroic mirror transmitting backlight (see Fig.26). The Bessel beam was generated with the axicon of 1600 apex angle, which allowed for the 1/e2–defined transverse size of the central lobe of ~7 μm (see Fig.4). The size of the Bessel beam features was measured with a graticule (10 μm scales, 1 mm range). This graticule was placed in the focal plane of the microscope lens so as to enable detection of the Bessel beam passing through the scaled region. This ensured precise control of magnification of the microscope detection system. The optical power in the central lobe of the Bessel beam with losses in the optical scheme taken into account could be varied up to 20 mW. The setup requirements for the optical trapping experiments control were enabled by highprecision micropositioners introduced into the optical scheme and a video-recording microscope with a CCD camera. The micro-objects for the study were polystyrene beads (10 μm mean diameter) in a water solution as the high refractive index objects and water bubbles in oil as the low-index particles. The solution containing the micro-objects was placed on a glass slide covered with a cover glass and controlled with a micropositioner.

Fig. 29. Optical trapping of a living red blood cell in the central lobe of the Bessel beam generated from the semiconductor laser. In this experiment, the sample was moved relative to the Bessel beam with the object trapped in its central lobe. Arrows indicate the movement of the non-trapped object while the trapped object remains in the central lobe of the Bessel beam.

We also studied the optical trapping of living red cells of rat blood. In the assessment, rat blood was dissolved in water with the addition of heparin to prevent clotting. The mean size of the cells was approximately 5 μm (measured with graticule, similar to measurement of the size of the features of the transverse projection of the Bessel beam). Fig.29 shows a series of experimental images demonstrating the two-dimensional optical trapping and the manipulation of a living red cell of rat blood in the central lobe of the Bessel beam generated from the semiconductor laser. In our experiment, the sample was moved relative to the Bessel beam with the object trapped in its central lobe. Therefore, arrows indicate the movement of

28

the non-trapped object while the trapped object remains in the central lobe of the Bessel beam. No damage of the living cells was detected in the available optical power range. Our assessments to date have demonstrated a very good potential for the Bessel beams generated from semiconductor lasers with the likelihood that they can replace their vibronic counterparts in applications relating to optical trapping and the manipulation of micro-objects including living biological cells. Conclusion The future prospects for Bessel-beam generation from semiconductor light sources seem to be very promising. Recent progress in the generation of the non-diffracting beams from various types of laser diodes up to a multi-Watt power level for central lobe diameters of a few to tens of micrometers reported in this paper ensures their wide-ranging applications in many fields of science and technology that require compact sources of ‘needle-like’ light. Reduction of the central lobe size of the Bessel beam generated with the high-power broad-stripe semiconductor lasers and LEDs to the level unachievable by means of the traditional focusing together with the possibility of exceeding the limit of power density for focusing of radiation with the high beam propagation parameter M2 opens up the new avenues in cutting-edge applications of the semiconductor light sources. Finally, the optical trapping/tweezing with semiconductor-laser-generated Bessel beams as described in this paper paves the way to replace their vibronic-generated counterparts for a range of implementations in the optical manipulation of micro-objects and living cells towards novel lab-on-chip configurations. Acknowledgments The authors would like to thank Dr. Aram Mooradian for providing the semiconductor lasers used in this work and Dr. Mantas Butkus whose efforts have made these experiments possible. This research was partially supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) through SeNDBeams (grant agreement No.255646) and FAST-DOT (grant agreement No.224338) projects. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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