Experimental study of wavefront distortion in the dark hollow beam generated using axicon

Accepted Manuscript Title: Experimental study of wavefront distortion in the dark hollow beam generated using axicon Authors: Rajeev Dwivedi, VK Jaiswal, Parag Sharma, Ranjana Mehrotra PII: DOI: Reference:

S0030-4026(17)30462-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.04.059 IJLEO 59101

To appear in: Received date: Accepted date:

23-1-2017 16-4-2017

Please cite this article as: Rajeev Dwivedi, VK Jaiswal, Parag Sharma, Ranjana Mehrotra, Experimental study of wavefront distortion in the dark hollow beam generated using axicon, Optik - International Journal for Light and Electron Opticshttp://dx.doi.org/10.1016/j.ijleo.2017.04.059 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 proof before it is published in its final 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.

Experimental study of wavefront distortion in the dark hollow beam generated using axicon

Rajeev Dwivedi1,2, VK Jaiswal1, 2 *, Parag Sharma1, 2 and Ranjana Mehrotra1, 2

1

Academy of Scientific and Innovative Research (AcSIR) at CSIR-NPL Campus,

Dr K S Krishnan Marg, New Delhi 110012, India 2

CSIR-National Physical Laboratory, Dr K S Krishnan Marg, New Delhi 110012

E-mail: [email protected] Phone: +91-01145608228, Fax: +91-01145609310

1

Abstract Axicons are the optical components widely used for the generation of dark hollow beam in a controlled manner. In certain circumstances, distortion in the transverse intensity profile (i.e. wavefront distortion) appears in DHB produced by axicon-lens assembly. In this paper, we performed the detailed experimental study for exploring the reasons for the appearance of wavefront distortion including its control and removal in partially coherent dark hollow beam produced from axicon-lens assembly. This experimental study find its crucial importance in the applications, where axicon is used for producing dark hollow beams, e.g. in eye surgery, atom trapping, laser machining and laser resonator. We also observed that the effect of chromatic aberration near the region of induced wavefront distortion in the depth of focus changes with propagation and can be controlled by varying spatial coherence of the beam. This control on the effect of chromatic aberration will be helpful in attaining achromatic dark hollow beam from axicon.

Keywords: Polychromatic DHB; Monochromatic DHB; Wavefront distortion; Coherence.

1. Introduction In recent time, optical beams having dark region in their intensity profile have found significant importance in various applications and most of them fall in a new branch of modern physics known as singular optics [1]. Therefore the field of singular optics has evolved rapidly as an interesting and significant branch for research [2-4] finding ever increasing applications starting from non invasive atom confinement [5], Biomedical 2

applications [6, 7], optical tweezing [8], laser machining [9], laser cooling and trapping of neutral atoms [10, 11]. A lot of studies is being conducted on generation of dark hollow beam (DHB), which is a special kind of singular hollow beam having high intensity ring surrounding almost null intensity dark region and possesses topological charges [12]. One such kind of popular and well studied optical beam carrying dark central region along its axis, known as optical vortex beam, also manifests helical phase structure [13, 14]. In general dark hollow beams can be generated in controlled manner in a laboratory with different techniques which includes axicon-lens system [15], photonic crystal fiber [16], spatial light modulator [17], and optical holographic technique [18]. Among these, axicon-lens system is the most convenient way of generating dark hollow beam. In most of the studies, monochromatic laser light source is utilized to generate optical vortex having topological charge behavior [19, 20]. However, studying singularities generated in partially coherent polychromatic light fields is interesting as they show spectral modulations at singular points [21] and therefore make its own importance. Recently, evolution of generic singularities and chromatic effects at the centre of partially coherent polychromatic DHB have been reported, also featuring wavefront ‘tearing like’ distortion near the centre of the beam intensity profile [22]. This wavefront distortion [23-26] can be easily observed within the depth of focus of intensity profile of DHB, generated with axicon-lens assembly and affects its beam quality. It has been also reported that the lateral shift of few mm in the position of axicon results in the asymmetric intensity distribution of considerable amount in the beam profile [27, 28]. In specific beam delivery system, ring shaped DHB is carefully produced whose perfection plays a crucial role in performing the noncontact eye surgery [7]. A small misalignment of axicon could disturb the precision of the surgery and may lead to catastrophic damage to the eye. Laser resonators [29] for producing DHB, laser machining [9] for fine cutting applications and laser trapping 3

with DHB are few other potential applications which also suffer greatly with misalignment and need to be addressed and studied adequately. To the best of our knowledge least attention was paid in qualitative analysis of misalignment of such crucial optics in the setup resulting in the reduced beam quality. In the present work wavefront distortion featuring DHB was demonstrated with partially coherent light field and thoroughly investigated its complete characteristics, including control over its formation and removal in the region before the emergence of DHB. The partially coherent monochromatic light field was preferred over coherent light field, due to its lower sensitivity towards speckles. In addition, its behavior with propagation, lateral shift of the axicon-lens assembly, and change in coherence of the incident light field were studied. We also analyze the effect of chromatic aberration near the region of wavefront distortion. For the detailed analysis experiments were also performed with polychromatic light field. The electric field distribution of a Gaussian beam in the plane perpendicular to the direction of beam propagation is expressed as, −𝑟2

𝐸(𝑟) = 𝐸0 𝑒 𝑤2

… (1)

where, E0 is the electric field amplitude of Gaussian beam at origin, r is the radial distance in the plane perpendicular to the direction of the beam propagation, w is the beam width incident on the axicon and can be given by 𝑤 = 𝑤0 √1 + 𝑍/𝑍𝑅 . ZR is the Rayleigh range defined as 𝑍𝑅 =

𝜋𝑤0 2 𝜆

, 𝑤0 is the beam waist at focus.

A partially coherent monochromatic DHB is generated when a converging lens of focal length f is placed after axicon with condition Z0 < f, where, Z0 is the distance between axicon and converging lens. A linear phase shift of 𝜏 = 𝑒 −𝑖𝑏𝑟 is introduced while passing through axicon where, b= 2𝑛𝜋 tan(𝑎) /𝜆 , ‘λ’ is the wavelength of the incident beam, ‘n’ is 4

the refractive index of axicon, alpha ‘a’ is the angle between the conical and the flat surfaces of axicon. Electric field after axicon- lens system [30] can be expressed as, 𝐸(0, 𝑍) =

−2𝜋𝑖 𝜆𝐵

where, 𝜇 =

1 𝑤2

𝐸0 𝑒 𝑖𝑘𝑍 { −

𝑖𝑘𝐴 2𝐵

1 2𝜇

−

𝑖𝑏

𝜋

√ 𝑒 4𝜇 𝜇

2

(−𝑏 ⁄4𝜇 ) [1

− ɸ(

𝑖𝑏 2√𝜇

)]}

…. (2)

and ɸ is the Fresnel integral. A and B are the elements of the ABCD 𝑥2

matrix and defined as, 𝑨 = (

𝑥1

𝑥

)

Ѳ1 =0

and 𝑩 = ( 2 )

Ѳ1 𝑥 =0 1

. Here 𝑥1 and 𝑥2 are the distances

of the ray on input and output planes from the optical axis respectively, similarly Ѳ1 and Ѳ2 are the angles made with optical axis. In the present setup, Z0 = 50 mm and f = 120 mm. The on-axis intensity of the beam produced after axicon-lens system at a distance Z in the direction of the propagation of the beam can be written as, 𝐼(0, 𝑍) = |𝐸(0, 𝑍)|2

…. (3)

2. Experimental Setup Schematic diagram of the setup is shown in figure 1. The monochromatic light beam generated from intensity stabilized He-Ne laser (1 mW, 633 nm) passes through a rotating diffuser plate D. The desired spatial coherence can be generated in the resulting beam using a pinhole. Generated spatial coherence is calculated by the lateral coherence length 𝐿𝑐 = 0.16𝜆0 𝑍1 𝑑

, where, λ0 is the source wavelength and 𝑑 is the diameter of the pin hole. Spatial

coherence of 𝐿𝑐 ≈ 303.840 μm was built by allowing light coming out from a pinhole of diameter 0.020 mm in free space to a distance of Z1 = 600 mm between pin hole A and entrance plane of axicon 𝐴𝑋 (having diameter 25.4 mm, refractive index of 1.515 and apex angle 178°) in order to create a Bessel beam followed by a monochromatic DHB. In order to produce a focused DHB, a converging lens ‘L’ (focal length f =120 mm) is placed at Z0 = 50 mm from Ax, fulfilling the condition Z0 < f [15]. The output pattern was imaged using a

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CCD camera (Q-Imaging 5 MP with pixel size 3.4 µm×3.4 µm) placed on a 3-D mount. To generate polychromatic DHB, source S is replaced with a tungsten halogen lamp (24 V, 10 A) operated with highly stabilized AC power supply, keeping the rest of the setup intact. 3. Result and Discussion The present work was carried out in two parts; (i) experiment was conducted for understanding the conditions for appearance of wavefront distortion in partially coherent monochromatic dark hollow beam (PCMDHB), and (ii) study of observed effects in polychromatic DHB. The wavefront distortion near the centre of the beam was observed prior to the evolution of the DHB, as reported earlier for partially coherent polychromatic DHB [22]. To understand the dynamics of the generation of wavefront distortion within the depth of focus, we studied its longitudinal and transversal characters. It was observed that the region of wavefront distortion expands with increase in the propagation distance along Z-axis and converts into the central dark region of the DHB as shown in figure 2(a) – 2(f). Maximum horizontal expansion of wavefront distortion with propagation distance is measured and plotted in figure 2(g). Figure 3(a) – (o) and table 2 depicts the changes in wavefront distortion followed by the lateral shift of axicon in +X, -X, +Y and –Y directions (keeping its position fixed at Z = 143 mm). A star like bright structure at the center of the beam, as appeared in figure 3(a) and 3(g), stretches in two opposite directions with increasing lateral shift of axicon, leaving behind a diminishing partial curly ring pattern in the beam, as evidenced in figure 3(d) – 3(f) and figure 3(k) – 3(l). An initial offset in the axicon position can be easily noticed in nonidentical appearance of wavefront distortion in the beam, e.g. at X = 2 mm and X = -2 mm [as shown in figure 3(b) and figure 3(i)]. It all shows the variation of wavefront distortion as a function of lateral shift of axicon and can be made symmetric. In a process, eventually we achieved a distortion free beam in the depth of focus zone by aligning axicon to a point in the 6

lateral plane as observed in figure 4(a) – 4(d). It is observed that this distortion free beam has a symmetric intensity distribution profile throughout propagation as shown in figure 4(a) – 4(d). It infers that a distortion of known amount can be introduced into the beam and controlled by shifting axicon laterally in a measured way. Furthermore, it is also observed that a beam generated from perfectly aligned setup did not show any wavefront distortion in depth of focus while propagating; however beam still contain star like structure at its center, which appears due to the interference between conical waves, refracting out from axicon [see figure 4(a) – 4(d)]. As the peak intensity of the ring vary with radial distribution along propagation, we define a factor Ic, the central intensity in comparison with the mean intensity of the ring surrounding the dark region, and can be calculated as, 𝐼𝑐 =

2∗𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑐𝑒𝑛𝑡𝑟𝑒

…... (4)

𝐼1 +𝐼2

Where, 𝐼1 and 𝐼2 are the peak intensities of the ring along diametrically opposite points on the left and right side of the dark region as plotted in figure 5(a). The visibility of the beam is also calculated for different values of Z by using same method as in [31] and plotted in figure 5(b). The visibility of PCMDHB peaks initially and then decreases with propagation distance Z as shown in figure 5(b), which may occur due to the focusing nature of converging lens L. In addition effect of change in spatial coherence by changing the speed of rotating diffusing plate was also analyzed. It is observed that change in speed of the rotating diffuser plate does not show any considerable effect on the resulting profile of the PCMDHB as shown in figure 6(a) – 6(d). An experimental investigation for better understanding of the behavior of wavefront distortion in partially coherent polychromatic DHB was also done and a similar observation of change in the region of wavefront distortion with the lateral shifts was observed and shown 7

in figure 7(a) – 7(h), which confirms that the appearance and spread in the region of wavefront distortion is a function of lateral shifts of axicon. It is also important and interesting to look at the chromatic characteristic of such beam in this region. We demonstrated that the effect of chromatic aberration [32] (results in the separation of colors as shown in the radial intensity distribution of the beam) is found prominent near the region of wavefront distortion. It is also observed that the separation of colors increases as we move forward in the depth of focus region along Z-axis and attains maximum at Z = 170 mm, where wavefront distortion starts appearing as shown in figure 8(a) – 8(h). We compared the radial intensity distribution of beam for different aperture sizes (d = 0.10mm, d = 0.05mm and d = 0.02mm) at Z = 170 mm. It is observed that the radial separation of colors is large for smaller value of aperture size and maximum for d = 0.02 mm as shown in figure 9(a) – figure 9(c), which inferred that the effect of chromatic aberration can be controlled by changing spatial coherence of the beam. The width of the DHB of different colors and the separation between them in the center of the beam are plotted in figure 10(a) – 10(b). It is noted that the width of DHB is larger for beam having a lower wavelength, in accordance with the geometrical predictions. These observations will be helpful in attaining the experimental control on the chromatic aberration in DHB and producing achromatic beam from axicon. 4. Conclusions In the present paper, we performed the experimental study of the wavefront distortion appeared in the transverse intensity profile of DHB, generated using an axiconlens system. The geometrical factors responsible for wavefront distortion were identified and it has been shown to control by changing these geometrical parameters in the 8

experimental setup in a measured way. Experiments were performed with both partially coherent monochromatic and polychromatic light fields and concluded that this fine control on wavefront distortion can be obtained in all partially coherent dark hollow beams. This study of wavefront distortion find its potential in removal of the factors affecting the symmetric intensity distribution of the DHB and hence finds its major importance in the applications where especially DHB is used e.g. eye surgery, laser machining, atom trapping, laser scanning and laser imaging. It is also observed that the separation of colors due to chromatic aberration is maximum near the region of wavefront distortion in the depth of focus and changes with propagation. It is also inferred that the separation of colors can be controlled by varying spatial coherence of the generating beam and will be helpful in attaining the experimental control on the chromatic aberration in DHB and producing achromatic beam from axicon.

5. Acknowledgement The authors thank Director, CSIR-National Physical Laboratory for granting the permission and support for this research work.

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[24] D.L. Fried, Statistics of a geometric representation of wavefront distortion, JOSA, 55 (1965) 1427-1435. [25] S.L. Prins, A.C. Barron, W.C. Herrmann, J.R. McNeil, Effect of stress on performance of dense wavelength division multiplexing filters: optical properties, Appl. Opt., 43 (2004) 626-632. [26] J. Nye, M. Berry, Dislocations in wave trains, in: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 1974, pp. 165-190. [27] D. Lin, S. Alam, A. Malinowski, K. Chen, J. Hayes, J. Flannagan, V. Geddes, J. Nilsson, S. Ingram, S. Norman, Temporally and spatially shaped fully‐fiberized ytterbium‐doped pulsed MOPA, Laser Physics Letters, 8 (2011) 747-753. [28] M. Couture, M. Piché, Focusing properties of an axicon pair, Canadian journal of physics, 71 (1993) 70-78. [29] M. Endo, Doughnut like beam generation by a W-axicon resonator with variable geometry, Japanese Journal of Applied Physics, 46 (2007) 593. [30] J.-H. Lin, M.-D. Wei, H.-H. Liang, K.-H. Lin, W.-F. Hsieh, Generation of supercontinuum bottle beam using an axicon, Opt. Express, 15 (2007) 2940-2946. [31] V.G. Shvedov, Y.V. Izdebskaya, A.V. Rode, A. Desyatnikov, W. Krolikowski, Y.S. Kivshar, Generation of optical bottle beams by incoherent white-light vortices, Opt. Express, 16 (2008) 20902-20907. [32] D. Mas, J. Espinosa, J. Perez, C. Illueca, Three dimensional analysis of chromatic aberration in diffractive elements with extended depth of focus, Opt. Express, 15 (2007) 17842-17854.

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Figure Captions

Figure 1: Schematic diagram of experimental setup. S is the source, D is the rotating ground glass diffuser, AX is the axicon, L is the convex lens, CCD is the Charge Coupled Device camera, Z1 is the distance between the pinhole and axicon, Z0 is the distance between the lens and axicon and Z is the distance between the lens and CCD. Figure 2: CCD images of PCMDHB, showing the expansion of wavefront distortion with increasing distance of CCD on Z-axis. Z1 = 600 mm (distance between the pinhole and axicon), Z0 = 70 mm (distance between the lens and axicon) and graph shows the plot of horizontal expansion of wavefront distortion with propagation of the beam. Figure 3: Transverse images of PCMDHB showing the effect of lateral shift of axicon [in left (a) - (f), right (g) – (i) and vertical (m) – (o)] in the region of wavefront distortion, keeping Z = 143 mm, Z1 = 600 mm (distance between the pinhole and axicon), Z0 = 50 mm (distance between lens and axicon) and graph showing horizontal expansion of the beam with lateral shifts of axicon in the right direction. Figure 4: Images of tearing free PCMDHB at different position of CCD camera on Z – axis. Z1 = 600 mm (distance between pinhole and axicon), Z0 = 50 mm (distance between the lens and axicon). Figure 5: (a) Visibility of the monochromatic dark hollow beam, (b) Intensity at the beam center in comparing with the surrounded ring for different values of Z. Figure 6: Effect of speed of rotating ground glass diffuser on PCMDHB. Speed is in increasing order from (a) – (d) keeping Z constant. Figure 7: Effect of lateral shift on wavefront distortion in polychromatic dark hollow beam, keeping 𝑍 = 143 mm constant Z1 = 600 mm (distance between the pinhole and axicon), Z2 = 50 mm (distance between the lens and axicon). Figure 8: Radial intensity distribution of RGB wavelengths (a) – (d) for different values of Z keeping pinhole diameter 0.02 mm and (e) – (h) CCD image showing separation of colors near wavefront distortion in polychromatic DHB for different values of Z. Figure 9: Radial intensity distribution of RGB wavelengths, (e) – (g) for different pinhole diameters (d = 0.02 mm, 0.05 mm, 0.10 mm) keeping Z = 170 mm. Figure 10: (a) Plots of width of polychromatic dark hollow beam and (b) Separations between different wavelengths (Red – Blue) at different positions of CCD camera on Z – axis.

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Figures

Figure 1: Schematic diagram of experimental setup. S is the source, D is the rotating ground glass diffuser, AX is the axicon, L is the convex lens, CCD is the Charge Coupled Device camera, Z1 is the distance between the pinhole and axicon, Z0 is the distance between the lens and axicon and Z is the distance between the lens and CCD.

Figure 2: CCD images of PCMDHB, showing the expansion of wavefront distortion with increasing distance of CCD on Z-axis. Z1 = 600 mm (distance between the pinhole and 14

axicon), Z0 = 70 mm (distance between the lens and axicon) and graph shows the plot of horizontal expansion of wavefront distortion with propagation of the beam.

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Figure 3: Transverse images of PCMDHB showing the effect of lateral shift of axicon [in left (a) - (f), right (g) – (i) and vertical (m) – (o)] in the region of wavefront distortion, keeping Z = 143 mm, Z1 = 600 mm (distance between the pinhole and axicon), Z0 = 50 mm (distance between lens and axicon) and graph showing horizontal expansion of the beam with lateral shifts of axicon in the right direction.

Figure 4: Images of tearing free PCMDHB at different position of CCD camera on Z – axis. Z1 = 600 mm (distance between the pinhole and axicon), Z0 = 50 mm (distance between the lens and axicon).

Figure 5: (a) Visibility of the monochromatic dark hollow beam, (b) Intensity at the beam center in comparing with the surrounded ring for different values of Z. 16

Figure 6: Effect of speed of rotating ground glass diffuser on PCMDHB. Speed is in increasing order from (a) – (d) keeping Z constant.

Figure 7: Effect of lateral shift on wavefront distortion in polychromatic dark hollow beam, keeping 𝑍 = 143 mm constant Z1 = 600 mm (distance between pinhole and axicon), Z2 = 50 mm (distance between the lens and axicon).

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Figure 8: Radial intensity distribution of RGB wavelengths (a) – (d) for different values of Z keeping pinhole diameter d = 0.02 mm and (e) – (h) CCD image showing separation of colors near wavefront distortion in polychromatic DHB for different values of Z.

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Figure 9: Radial intensity distribution of RGB wavelengths, (e) – (g) for different pinhole diameters (d = 0.02 mm, 0.05 mm, 0.10 mm) keeping Z = 170 mm.

Figure 10: (a) Plots of width of polychromatic dark hollow beam and (b) Separations between different wavelengths (Red – Blue) at different positions of CCD camera on Z – axis.

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Tables

Table 1: Maximum horizontal expansion of wavefront distortion with the beam propagation. S.No.

Z (mm)

Expansion (mm)

1

140

0.0442

2

144

0.1428

3

149

0.2141

4

154

0.3400

Table 2: The values of maximum horizontal expansion of wavefront distortion in the intensity profile of the beam with lateral shifts of axicon in right direction and keeping Z = 143 mm constant. S.No. 1 2 3 4 5 6

Lateral shift (mm) 0 1 2 3 4 5

Expansion (mm) 0.0238 0.0306 0.0442 0.0578 0.068 0.0748

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