Young’s experiment with waves near zeros Paramasivam Senthilkumaran* and Monika Bahl Department of Physics, Indian Institute of Technology Delhi, New Delhi, 110016, India *[email protected]
Abstract: We report an interesting observation in the formation of Young’s fringes from a two pinhole arrangement illuminated by waves from the neighborhood of a zero of an optical phase singularity. Spacing of the Young’s fringes appears to defy the dependence of pin-hole separation. But for larger pinhole separation such an anomalous phenomenon is not discernible. The experiments show that the fringe spacing is governed by the stronger local phase gradient near the vortex core that also has a radial part. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have only azimuthal and longitudinal components. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient. ©2015 Optical Society of America OCIS codes: (260.0260) Physical optics; (010.7350) Wave-front sensing; (050.1940) Diffraction; (260.1960) Diffraction theory; (260.6042) Singular optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
T. Young, “An account of some cases of the production of colours, not hitherto described,” Philos. Trans. R. Soc. Lond. 92, 387–397 (1802). T. Young, “The Bakerian lecture: experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. Lond. 94, 1–16 (1804). R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, and H. M. Wiseman, “A double-slit ‘which-way’ experiment on the complementarity–uncertainty debate,” New J. Phys. 9(8), 287 (2007). O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66(21), 2689–2692 (1991). C. Jönsson, “Elektroneninterferenzen an mehrerenk unstlichhergestellter Feinspalten,” Z. Phys. 161, 454–474 (1961). C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. 42(1), 4–11 (1974). A. Zelinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe, “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. 60(4), 1067–1073 (1988). M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C(60) molecules,” Nature 401(6754), 680–682 (1999). Z. N. Ozer, H. Chaluvadi, M. Ulu, M. Dogan, B. Aktas, and D. Madison, “Young’s double-slit interference for quantum particles,” Phys. Rev. A 87(4), 042704 (2013). F. Shimizu, K. Shimizu, and H. Takuma, “Double-slit interference with ultracold metastable Neon atoms,” Phys. Rev. A 46(1), R17–R20 (1992). O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66(21), 2689–2692 (1991). H. J. Lugt, Vortex Flow in Nature and Technology (John Weily & Sons Inc, 1983). J.-Z. Wu, H.-Y. Ma, and M. D. Zhou, Vorticity and Vortex Dynamics (Springer Verlog, 2006). P. G. Saffman, Vortex Dynamics (Cambridge University Press, 1992). J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single slit diffraction of an optical beam with phase singularity,” Opt. Lasers Eng. 47(1), 123–126 (2009). H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31(7), 999– 1001 (2006).
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10968
20. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Light’s Orbital Angular Momentum,” Phys. Rev. Lett. 105(5), 053904 (2010). 21. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). 22. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011). 23. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001). 24. J. W. Goodman, Introduction to Fourier Optics (Roberts and Co., 2007). 25. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14(25), 11919–11924 (2006). 26. J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50(10), 501573 (2003). 27. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15(7), 073022 (2013). 28. M. J. Padgett, J. Courtial, and L. Allen, “Light's orbital angular momentum,” Phys. Today 57(5), 35–40 (2004). 29. B. K. Singh, M. Bahl, D. S. Mehta, and P. Senthilkumaran, “Study of internal energy flows in dipole vortex beams by knife edge test,” Opt. Commun. 293, 15–21 (2013). 30. M. V. Berry, Proceedings of the OSA, Washington DC, 2007. 31. M. V. Berry, “Quantum coherence and reality,” J.S. Anandan, J.L. Safko (Eds.), Celebration of the 60th Birthday of YakirAharonov, World Scientific, Singapore, 55–65 (1994). 32. Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64(25), 2965–2968 (1990). 33. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223(1-3), 31–37 (2003). 34. M. V. Berry, “Waves near zeros,” Conference on Coherence and Quantum Optics. Optical Society of America, Washington DC 2008, Eds: N.P.Bigelow, J.H.Ebery & C.R.Stroud, 37–41 (2008). 35. M. V. Berry, “Phase vortex spirals,” J. Phys. Math. Gen. 38(45), L745–L751 (2005). 36. A. E. Siegman, Lasers, Oxford Univ. Press, (1986).
1. Introduction Young’s interference experiment [1–3] is one of the fundamental experiments in Physics that proved the wave nature of light. By replacing light by atoms, electrons, neutrons and molecules [4–11] it has been shown that similar interference patterns can be obtained thereby establishing duality between particle and wave natures. In the Young’s double slit experiment, coherent states of light is wavefront split and a spatial overlap of electromagnetic fields from these slits is observed on a screen that depicts the independent evolution of these fields. The superposition results in interference fringes which can be understood only by wave theory. Interference fringes are there, only as long as, through which slit the light has passed through, is unknown. Any attempt to find which way light has passed through destroys the interference pattern. The choice of exhibiting one of the two natures namely the particle and wave natures is complementarity. Vortices are ubiquitous and are being studied in various branches of science and engineering [12–14]. Spiral galaxies, tornados, cyclones, vortices in water and in tea cups, wing-tip vortices of a moving aircraft, Abrikosov lattices, quantum Hall-effect, chaos, Bose Einstein condensates, superfluidity are some of the examples where vortices are present. The circulatory motion of matter or any other physical parameter about the vortex core is being studied for its positive and negative impacts. An optical phase singularity [15–17], also called as a vortex is a space curve of amplitude null, around which the optical currents curl and is observed as a point phase defect in any two dimensional observation plane. The core of a singularity is a zero of the field where phase is indeterminate. The wavefronts are helical ramps winding about the vortex core. The interest in optical vortices has grown leaps and bounds recently, as their presence, usefulness and their properties have evoked more curiosity in the research community. Efforts to generate, detect, examine and use or avoid them are growing day by day. Diffraction experiments involving vortices are strikingly interesting. The diffraction patterns obtained by non-singular beam illumination of apertures are completely different to their singular beam counterparts. The single slit, double slit and triangular aperture diffraction patterns [18–22] reveal the topological charge of the incident singular wave. Here we report the anomalous phenomenon that the fringe spacing does not depend on pin-hole separation in
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10969
the Young’s experiment with waves near zero of an optical phase singularity. But for larger pinhole separation such an anomalous phenomenon is not discernible and the usual Young’s fringes appear. The anomalous phenomenon on fringe spacing can be understood if one considers the presence of an additional radial component to the phase gradient field near the core. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have components that are only azimuthal and longitudinal. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient. 2. Young’s experiment Superposition of Huygen’s secondary waves from two pinholes S1 and S2 illuminated by a coherent monochromatic plane wave results in the interference pattern, in which the formation of bright fringes are governed by the phase difference condition k ( l1 − l2 ) = 2qπ where k is the propagation constant, q is an integer and l1 , l2 are respectively the distances S1P and S2P in Fig. 1. The Fraunhofer diffraction pattern [23,24] and the aperture transmittance function are connected by a Fourier transform – an operation which is shift invariant. Shift invariance means that the diffracted field amplitudes from the two individual slits overlap exactly in the Fraunhofer diffraction plane. Such an overlap of wavefunctions is partial when the pattern is observed in the near field – nevertheless the arguments presented here are applicable where interference fringes are seen. Under plane wave illumination at normal incidence, the phase distribution across the aperture plane is constant and the propagation vector points in the direction of the optical axis of the system. But when the two pinholes are illuminated by a singular beam, the propagation vector is no longer unidirectional and is now a function of spatial coordinates. At the two pinhole locations the local propagation vectors point in two different directions and are represented here by vectors k1 and k2 respectively. Consider a scalar optical field ψ ( r ) = A( r ) exp {i χ ( r )} which has a phase singularity at r = 0 . The optical field in the immediate neighbourhood of the core of the vortex is characterized by high phase gradients. Phase gradient which is normal to phase contour surface χ ( r ) = constant is the propagation vector k = ∇χ . The propagation vector (in cylindrical coordinate system) for a singular beam of charge m is given by k = kr rˆ + kφ φˆ + k z zˆ . The origin of the coordinate system is fixed at the vortex core in the aperture plane and zˆ coincides with the optical axis of the experimental setup. The azimuthal component is kφ = m r and the longitudinal component is k z = k 2 − kr2 − kφ2 . The presence of azimuthal components has been experimentally studied by many research groups [25–29]. Since the azimuthal component varies as ( m r ) , in the regions very close to the vortex core, the phase gradient will be very high. In this region waves can vary arbitrarily faster than the wavelength [30–32] and can have an evanescent wave component [33]. The phase gradient ( ∇χ ) in this superosicllatory region, with faster than Fourier behaviour, must have a radial component in addition to the pure circulating phase gradients associated with a vortex. Here we provide the first experimental evidence to the presence of a radial component.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10970
Fig. 1. Schematic of the Young’s Experiment
As the direction of the propagation vector is not constant over the wavefront, the phase distribution and the angle of incidence at the two apertures are not same. Assuming that the two pinholes are small, the phase at the two pin-hole locations are χ1 and χ 2 respectively. There is also a relative tilt between k1 and k2 vectors. These relative tilt components lead to the change in the number of Young’s fringes. The Poynting vector (time-averaged energy flow) in optical fields is given by j ( r ) = Imψ * ( r ) ∇ψ ( r ) = I ∇χ , where I is the intensity. Also known as optical current, it flows in a direction normal to the wavefronts, which is the direction of the propagation vector. Berry [34,35] has suggested that the geometry of these lines are such that they can spiral into or out of the core of a vortex, if the scalar field ψ ( r ) is expanded about its zero, up to third order in the distance r from the vortex. Berry [35] has shown that optical beams whose amplitude distribution A(r ) is a function of the form r n , where n is an integer, have radial phase gradient. In a typical example, consider a LG beam [36] given by ψ pm ( r , φ , z ) =
m r 2 m 2r 2 z r2 ikr 2 exp ( imφ ) exp i ( 2 p + m + 1) tan −1 Lp 2 exp − 2 exp − 2 2 2 2 (1 + z zR ) w ( z ) w ( z ) w ( z ) 2 ( z + zR ) zR
C
where apart from the Gaussian amplitude variation, the amplitude variation due to the Laguerre Polynomial Lmp for p > 1 , has terms that are functions of r n . In the above equation,
C is a constant, w ( z ) is the beam waist and z R is the Rayleigh length. 3. Experimental results and discussion In our experiment (Fig. 1), we have used two small circular apertures each of radius a = 0.3mm . The four pin hole separations that were tried are d = 0.9mm, 1.2mm, 1.5mm and 2.4mm. We actually tried to have the pin-hole separation to be as small as possible, but as the distance between the pin-holes is reduced, the intensity of light diffracted also becomes very low and that puts limitation on the detection. We have used Imagingsource DFK51AU02 Color camera and its gain has to be adjusted to capture low intensity levels. Thus the intensity is presented in arbitrary units. A spatially filtered collimated light from a He-Ne laser source
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10971
( λ = 632.8nm ) is infested with a phase singularity by a spiral phase plate (RPC Photonics). This singular beam is incident on the two pin-hole arrangement and the diffraction pattern is observed by using a camera. Use of two long slits was avoided, as there would be considerable phase variation across each of the slits. Hence the radius of each of the circular apertures is kept small and at the same time they allow enough light to pass through to carry out the experiment. The Young’s fringes are hence modulated by diffraction envelope which is given by the Airy function ( 2 J1 (ν ) ν ) , where J1 (ν ) is the Bessel function of first kind,
ν = ( 2π λ ) a sin θ and θ is the diffraction angle.
Fig. 2. Experimentally observed Young’s fringe patterns
In the Young’s experiment performed in our lab, we have varied m, the magnitude of the topological charge of the vortex from 1 to 4. The two pin-holes have been positioned equidistant from the core such that the pin-holes and the vortex core are collinear in the aperture plane. Results obtained by varying their separation are presented in rows whereas columns present the results when the magnitude of the topological charge is varied. The first column shows the Young’s fringes obtained under non-singular beam ( m = 0 ) illumination. From the results shown in Fig. 2, one can observe that there is an increase in the number of maxima and minima as the topological charge is increased even though the pin-hole separation distance is kept constant. A comparison with the fringes obtained with nonsingular beam illumination shows this increase in the number of fringes. This can be accounted only by the presence of a radial component of phase gradient near the core of the vortex. This result points to the fact that as the charge of vortex is increased the stronger phase gradient at the pin-hole location with concomitant increase in the radial component. Similarly the variation in the radial component of phase gradient as we go closer to the core can also be noticed. Since the experiment involves low intensity levels of the vortex core neighbourhood, we also computed the Fraunhofer diffraction pattern (Fig. 3). Even if at the aperture plane the vortex phase distribution is considered to have only azimuthal phase gradient, diffraction due to propagation plays a greater role in the propagation vector spiraling inwards. The first three rows of Figs. 2 and 3 present the interference patterns for the increasing pin-hole separation.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10972
The last row depicts the case where the pin-holes are well separated so that they are not in the regions of high vortex phase gradient [22].
Fig. 3. Simulated Young’s fringe patterns
To put things in a simpler perspective, non-singular plane wave illumination corresponds to the case where the propagation vectors at the two pinhole locations do not have a relative tilt between them where as, in the case of singular beam the two propagation vectors have a relative tilt between them and this extra tilt is responsible for the extra fringes in the diffraction pattern. As the charge of the vortex or the distance between the two pinholes is changed the tilt between the propagation vectors at the pinhole locations also changes and hence the number of fringes. 4. Conclusion An interesting observation in the formation of Young’s fringes has been seen and reported in this work. The number of fringes has shown to defy the general relation between the fringe spacing and pinhole separation. This has been attributed to the presence of the radial component of the phase gradient of singular beams. The transverse component of the phase gradient, spirals in and out of the vortex core. This behaviour is seen very near to the phase singularity where faster than Fourier phenomenon is observed. Outside this superoscillatory region, it circulates about the singularity. This radial component is responsible for the observed change in the Young’s fringe pattern and has been experimentally verified. This aspect has not been seen and reported so far. This work is the first experimental evidence of this phenomenon. Acknowledgements Research grant from Department of Science and Technology (DST), India, SR/S2/LOP22/2013 is thankfully acknowledged.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10973
Abstract: We report an interesting observation in the formation of Young’s fringes from a two pinhole arrangement illuminated by waves from the neighborhood of a zero of an optical phase singularity. Spacing of the Young’s fringes appears to defy the dependence of pin-hole separation. But for larger pinhole separation such an anomalous phenomenon is not discernible. The experiments show that the fringe spacing is governed by the stronger local phase gradient near the vortex core that also has a radial part. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have only azimuthal and longitudinal components. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient. ©2015 Optical Society of America OCIS codes: (260.0260) Physical optics; (010.7350) Wave-front sensing; (050.1940) Diffraction; (260.1960) Diffraction theory; (260.6042) Singular optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
T. Young, “An account of some cases of the production of colours, not hitherto described,” Philos. Trans. R. Soc. Lond. 92, 387–397 (1802). T. Young, “The Bakerian lecture: experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. Lond. 94, 1–16 (1804). R. Mir, J. S. Lundeen, M. W. Mitchell, A. M. Steinberg, J. L. Garretson, and H. M. Wiseman, “A double-slit ‘which-way’ experiment on the complementarity–uncertainty debate,” New J. Phys. 9(8), 287 (2007). O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66(21), 2689–2692 (1991). C. Jönsson, “Elektroneninterferenzen an mehrerenk unstlichhergestellter Feinspalten,” Z. Phys. 161, 454–474 (1961). C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys. 42(1), 4–11 (1974). A. Zelinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe, “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys. 60(4), 1067–1073 (1988). M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger, “Wave-particle duality of C(60) molecules,” Nature 401(6754), 680–682 (1999). Z. N. Ozer, H. Chaluvadi, M. Ulu, M. Dogan, B. Aktas, and D. Madison, “Young’s double-slit interference for quantum particles,” Phys. Rev. A 87(4), 042704 (2013). F. Shimizu, K. Shimizu, and H. Takuma, “Double-slit interference with ultracold metastable Neon atoms,” Phys. Rev. A 46(1), R17–R20 (1992). O. Carnal and J. Mlynek, “Young’s double-slit experiment with atoms: A simple atom interferometer,” Phys. Rev. Lett. 66(21), 2689–2692 (1991). H. J. Lugt, Vortex Flow in Nature and Technology (John Weily & Sons Inc, 1983). J.-Z. Wu, H.-Y. Ma, and M. D. Zhou, Vorticity and Vortex Dynamics (Springer Verlog, 2006). P. G. Saffman, Vortex Dynamics (Cambridge University Press, 1992). J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single slit diffraction of an optical beam with phase singularity,” Opt. Lasers Eng. 47(1), 123–126 (2009). H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31(7), 999– 1001 (2006).
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10968
20. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Light’s Orbital Angular Momentum,” Phys. Rev. Lett. 105(5), 053904 (2010). 21. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003). 22. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011). 23. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001). 24. J. W. Goodman, Introduction to Fourier Optics (Roberts and Co., 2007). 25. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express 14(25), 11919–11924 (2006). 26. J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50(10), 501573 (2003). 27. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15(7), 073022 (2013). 28. M. J. Padgett, J. Courtial, and L. Allen, “Light's orbital angular momentum,” Phys. Today 57(5), 35–40 (2004). 29. B. K. Singh, M. Bahl, D. S. Mehta, and P. Senthilkumaran, “Study of internal energy flows in dipole vortex beams by knife edge test,” Opt. Commun. 293, 15–21 (2013). 30. M. V. Berry, Proceedings of the OSA, Washington DC, 2007. 31. M. V. Berry, “Quantum coherence and reality,” J.S. Anandan, J.L. Safko (Eds.), Celebration of the 60th Birthday of YakirAharonov, World Scientific, Singapore, 55–65 (1994). 32. Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, “Superpositions of time evolutions of a quantum system and a quantum time-translation machine,” Phys. Rev. Lett. 64(25), 2965–2968 (1990). 33. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223(1-3), 31–37 (2003). 34. M. V. Berry, “Waves near zeros,” Conference on Coherence and Quantum Optics. Optical Society of America, Washington DC 2008, Eds: N.P.Bigelow, J.H.Ebery & C.R.Stroud, 37–41 (2008). 35. M. V. Berry, “Phase vortex spirals,” J. Phys. Math. Gen. 38(45), L745–L751 (2005). 36. A. E. Siegman, Lasers, Oxford Univ. Press, (1986).
1. Introduction Young’s interference experiment [1–3] is one of the fundamental experiments in Physics that proved the wave nature of light. By replacing light by atoms, electrons, neutrons and molecules [4–11] it has been shown that similar interference patterns can be obtained thereby establishing duality between particle and wave natures. In the Young’s double slit experiment, coherent states of light is wavefront split and a spatial overlap of electromagnetic fields from these slits is observed on a screen that depicts the independent evolution of these fields. The superposition results in interference fringes which can be understood only by wave theory. Interference fringes are there, only as long as, through which slit the light has passed through, is unknown. Any attempt to find which way light has passed through destroys the interference pattern. The choice of exhibiting one of the two natures namely the particle and wave natures is complementarity. Vortices are ubiquitous and are being studied in various branches of science and engineering [12–14]. Spiral galaxies, tornados, cyclones, vortices in water and in tea cups, wing-tip vortices of a moving aircraft, Abrikosov lattices, quantum Hall-effect, chaos, Bose Einstein condensates, superfluidity are some of the examples where vortices are present. The circulatory motion of matter or any other physical parameter about the vortex core is being studied for its positive and negative impacts. An optical phase singularity [15–17], also called as a vortex is a space curve of amplitude null, around which the optical currents curl and is observed as a point phase defect in any two dimensional observation plane. The core of a singularity is a zero of the field where phase is indeterminate. The wavefronts are helical ramps winding about the vortex core. The interest in optical vortices has grown leaps and bounds recently, as their presence, usefulness and their properties have evoked more curiosity in the research community. Efforts to generate, detect, examine and use or avoid them are growing day by day. Diffraction experiments involving vortices are strikingly interesting. The diffraction patterns obtained by non-singular beam illumination of apertures are completely different to their singular beam counterparts. The single slit, double slit and triangular aperture diffraction patterns [18–22] reveal the topological charge of the incident singular wave. Here we report the anomalous phenomenon that the fringe spacing does not depend on pin-hole separation in
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10969
the Young’s experiment with waves near zero of an optical phase singularity. But for larger pinhole separation such an anomalous phenomenon is not discernible and the usual Young’s fringes appear. The anomalous phenomenon on fringe spacing can be understood if one considers the presence of an additional radial component to the phase gradient field near the core. Many diffraction experiments reported so far have missed this aspect as the phase gradient in a vortex beam is normally considered to have components that are only azimuthal and longitudinal. This work reveals the vortex core structure and is the first experimental evidence to the existence of a radial component of this phase gradient. 2. Young’s experiment Superposition of Huygen’s secondary waves from two pinholes S1 and S2 illuminated by a coherent monochromatic plane wave results in the interference pattern, in which the formation of bright fringes are governed by the phase difference condition k ( l1 − l2 ) = 2qπ where k is the propagation constant, q is an integer and l1 , l2 are respectively the distances S1P and S2P in Fig. 1. The Fraunhofer diffraction pattern [23,24] and the aperture transmittance function are connected by a Fourier transform – an operation which is shift invariant. Shift invariance means that the diffracted field amplitudes from the two individual slits overlap exactly in the Fraunhofer diffraction plane. Such an overlap of wavefunctions is partial when the pattern is observed in the near field – nevertheless the arguments presented here are applicable where interference fringes are seen. Under plane wave illumination at normal incidence, the phase distribution across the aperture plane is constant and the propagation vector points in the direction of the optical axis of the system. But when the two pinholes are illuminated by a singular beam, the propagation vector is no longer unidirectional and is now a function of spatial coordinates. At the two pinhole locations the local propagation vectors point in two different directions and are represented here by vectors k1 and k2 respectively. Consider a scalar optical field ψ ( r ) = A( r ) exp {i χ ( r )} which has a phase singularity at r = 0 . The optical field in the immediate neighbourhood of the core of the vortex is characterized by high phase gradients. Phase gradient which is normal to phase contour surface χ ( r ) = constant is the propagation vector k = ∇χ . The propagation vector (in cylindrical coordinate system) for a singular beam of charge m is given by k = kr rˆ + kφ φˆ + k z zˆ . The origin of the coordinate system is fixed at the vortex core in the aperture plane and zˆ coincides with the optical axis of the experimental setup. The azimuthal component is kφ = m r and the longitudinal component is k z = k 2 − kr2 − kφ2 . The presence of azimuthal components has been experimentally studied by many research groups [25–29]. Since the azimuthal component varies as ( m r ) , in the regions very close to the vortex core, the phase gradient will be very high. In this region waves can vary arbitrarily faster than the wavelength [30–32] and can have an evanescent wave component [33]. The phase gradient ( ∇χ ) in this superosicllatory region, with faster than Fourier behaviour, must have a radial component in addition to the pure circulating phase gradients associated with a vortex. Here we provide the first experimental evidence to the presence of a radial component.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10970
Fig. 1. Schematic of the Young’s Experiment
As the direction of the propagation vector is not constant over the wavefront, the phase distribution and the angle of incidence at the two apertures are not same. Assuming that the two pinholes are small, the phase at the two pin-hole locations are χ1 and χ 2 respectively. There is also a relative tilt between k1 and k2 vectors. These relative tilt components lead to the change in the number of Young’s fringes. The Poynting vector (time-averaged energy flow) in optical fields is given by j ( r ) = Imψ * ( r ) ∇ψ ( r ) = I ∇χ , where I is the intensity. Also known as optical current, it flows in a direction normal to the wavefronts, which is the direction of the propagation vector. Berry [34,35] has suggested that the geometry of these lines are such that they can spiral into or out of the core of a vortex, if the scalar field ψ ( r ) is expanded about its zero, up to third order in the distance r from the vortex. Berry [35] has shown that optical beams whose amplitude distribution A(r ) is a function of the form r n , where n is an integer, have radial phase gradient. In a typical example, consider a LG beam [36] given by ψ pm ( r , φ , z ) =
m r 2 m 2r 2 z r2 ikr 2 exp ( imφ ) exp i ( 2 p + m + 1) tan −1 Lp 2 exp − 2 exp − 2 2 2 2 (1 + z zR ) w ( z ) w ( z ) w ( z ) 2 ( z + zR ) zR
C
where apart from the Gaussian amplitude variation, the amplitude variation due to the Laguerre Polynomial Lmp for p > 1 , has terms that are functions of r n . In the above equation,
C is a constant, w ( z ) is the beam waist and z R is the Rayleigh length. 3. Experimental results and discussion In our experiment (Fig. 1), we have used two small circular apertures each of radius a = 0.3mm . The four pin hole separations that were tried are d = 0.9mm, 1.2mm, 1.5mm and 2.4mm. We actually tried to have the pin-hole separation to be as small as possible, but as the distance between the pin-holes is reduced, the intensity of light diffracted also becomes very low and that puts limitation on the detection. We have used Imagingsource DFK51AU02 Color camera and its gain has to be adjusted to capture low intensity levels. Thus the intensity is presented in arbitrary units. A spatially filtered collimated light from a He-Ne laser source
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10971
( λ = 632.8nm ) is infested with a phase singularity by a spiral phase plate (RPC Photonics). This singular beam is incident on the two pin-hole arrangement and the diffraction pattern is observed by using a camera. Use of two long slits was avoided, as there would be considerable phase variation across each of the slits. Hence the radius of each of the circular apertures is kept small and at the same time they allow enough light to pass through to carry out the experiment. The Young’s fringes are hence modulated by diffraction envelope which is given by the Airy function ( 2 J1 (ν ) ν ) , where J1 (ν ) is the Bessel function of first kind,
ν = ( 2π λ ) a sin θ and θ is the diffraction angle.
Fig. 2. Experimentally observed Young’s fringe patterns
In the Young’s experiment performed in our lab, we have varied m, the magnitude of the topological charge of the vortex from 1 to 4. The two pin-holes have been positioned equidistant from the core such that the pin-holes and the vortex core are collinear in the aperture plane. Results obtained by varying their separation are presented in rows whereas columns present the results when the magnitude of the topological charge is varied. The first column shows the Young’s fringes obtained under non-singular beam ( m = 0 ) illumination. From the results shown in Fig. 2, one can observe that there is an increase in the number of maxima and minima as the topological charge is increased even though the pin-hole separation distance is kept constant. A comparison with the fringes obtained with nonsingular beam illumination shows this increase in the number of fringes. This can be accounted only by the presence of a radial component of phase gradient near the core of the vortex. This result points to the fact that as the charge of vortex is increased the stronger phase gradient at the pin-hole location with concomitant increase in the radial component. Similarly the variation in the radial component of phase gradient as we go closer to the core can also be noticed. Since the experiment involves low intensity levels of the vortex core neighbourhood, we also computed the Fraunhofer diffraction pattern (Fig. 3). Even if at the aperture plane the vortex phase distribution is considered to have only azimuthal phase gradient, diffraction due to propagation plays a greater role in the propagation vector spiraling inwards. The first three rows of Figs. 2 and 3 present the interference patterns for the increasing pin-hole separation.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10972
The last row depicts the case where the pin-holes are well separated so that they are not in the regions of high vortex phase gradient [22].
Fig. 3. Simulated Young’s fringe patterns
To put things in a simpler perspective, non-singular plane wave illumination corresponds to the case where the propagation vectors at the two pinhole locations do not have a relative tilt between them where as, in the case of singular beam the two propagation vectors have a relative tilt between them and this extra tilt is responsible for the extra fringes in the diffraction pattern. As the charge of the vortex or the distance between the two pinholes is changed the tilt between the propagation vectors at the pinhole locations also changes and hence the number of fringes. 4. Conclusion An interesting observation in the formation of Young’s fringes has been seen and reported in this work. The number of fringes has shown to defy the general relation between the fringe spacing and pinhole separation. This has been attributed to the presence of the radial component of the phase gradient of singular beams. The transverse component of the phase gradient, spirals in and out of the vortex core. This behaviour is seen very near to the phase singularity where faster than Fourier phenomenon is observed. Outside this superoscillatory region, it circulates about the singularity. This radial component is responsible for the observed change in the Young’s fringe pattern and has been experimentally verified. This aspect has not been seen and reported so far. This work is the first experimental evidence of this phenomenon. Acknowledgements Research grant from Department of Science and Technology (DST), India, SR/S2/LOP22/2013 is thankfully acknowledged.
#233196 - $15.00 USD (C) 2015 OSA
Received 13 Feb 2015; revised 11 Apr 2015; accepted 13 Apr 2015; published 20 Apr 2015 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.010968 | OPTICS EXPRESS 10973
