Abstract

We have previously proposed and demonstrated the targeted-light delivery capability of wave-guided optical waveguides (WOWs). As the WOWs are maneuvered in 3D space, it is important to maintain efficient light coupling through the waveguides within their operating volume. We propose the use of dynamic diffractive techniques to create diffraction-limited spots that will track and couple to the WOWs during operation. This is done by using a spatial light modulator to encode the necessary diffractive phase patterns to generate the multiple and dynamic coupling spots. The method is initially tested for a single WOW and we have experimentally demonstrated dynamic tracking and coupling for both lateral and axial displacements.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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2014 (1)

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

2013 (3)

D. Palima and J. Glückstad, “Gearing up for optical microrobotics: micromanipulation and actuation of synthetic microstructures by optical forces,” Laser Photon. Rev. 7(4), 478–494 (2013).
[CrossRef]

S. H. Simpson, D. B. Phillips, D. M. Carberry, and S. Hanna, “Bespoke optical springs and passive force clamps from shaped dielectric particles,” J. Quant. Spectrosc. Radiat. Transf. 126, 91–98 (2013).
[CrossRef]

D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, T. Aabo, P. Ormos, and J. Glückstad, “Optical forces through guided light deflections,” Opt. Express 21(1), 581–593 (2013).
[CrossRef] [PubMed]

2012 (3)

2011 (2)

J. Glückstad, “Optical manipulation: Sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
[CrossRef]

G. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011).
[CrossRef]

2010 (2)

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

2008 (3)

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

D. Palima and J. Glückstad, “Comparison of generalized phase contrast and computer generated holography for laser image projection,” Opt. Express 16(8), 5338–5349 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

2003 (1)

2002 (1)

2000 (1)

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185(1-3), 77–82 (2000).
[CrossRef]

1996 (1)

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130(4-6), 225–230 (1996).
[CrossRef]

1991 (1)

Aabo, T.

Andilla, J.

Artusio-Glimpse, A. B.

G. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011).
[CrossRef]

Bañas, A. R.

Bastmeyer, M.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Bøggild, P.

Bowman, R.

Carberry, D. M.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

S. H. Simpson, D. B. Phillips, D. M. Carberry, and S. Hanna, “Bespoke optical springs and passive force clamps from shaped dielectric particles,” J. Quant. Spectrosc. Radiat. Transf. 126, 91–98 (2013).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Chamorovskiy, Y.

Churchman, L. S.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Costa, R.

Dam, J. S.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Eriksen, R. L.

Filippov, V.

Fischer, J.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Flyvbjerg, H.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Franz, C. M.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Gammelgaard, L.

Gibson, G. M.

Glückstad, J.

D. Palima and J. Glückstad, “Gearing up for optical microrobotics: micromanipulation and actuation of synthetic microstructures by optical forces,” Laser Photon. Rev. 7(4), 478–494 (2013).
[CrossRef]

D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, T. Aabo, P. Ormos, and J. Glückstad, “Optical forces through guided light deflections,” Opt. Express 21(1), 581–593 (2013).
[CrossRef] [PubMed]

D. Palima, A. R. Bañas, G. Vizsnyiczai, L. Kelemen, P. Ormos, and J. Glückstad, “Wave-guided optical waveguides,” Opt. Express 20(3), 2004–2014 (2012).
[CrossRef] [PubMed]

J. Glückstad, “Optical manipulation: Sculpting the object,” Nat. Photonics 5(1), 7–8 (2011).
[CrossRef]

D. Palima and J. Glückstad, “Comparison of generalized phase contrast and computer generated holography for laser image projection,” Opt. Express 16(8), 5338–5349 (2008).
[CrossRef] [PubMed]

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. Perch-Nielsen, and J. Glückstad, “Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps,” Opt. Express 13(18), 6899–6904 (2005).
[CrossRef] [PubMed]

R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Multiple-beam optical tweezers generated by the generalized phase-contrast method,” Opt. Lett. 27(4), 267–269 (2002).
[CrossRef] [PubMed]

J. Glückstad, “Phase contrast image synthesis,” Opt. Commun. 130(4-6), 225–230 (1996).
[CrossRef]

Haist, T.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185(1-3), 77–82 (2000).
[CrossRef]

Hanna, S.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

S. H. Simpson, D. B. Phillips, D. M. Carberry, and S. Hanna, “Bespoke optical springs and passive force clamps from shaped dielectric particles,” J. Quant. Spectrosc. Radiat. Transf. 126, 91–98 (2013).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Ho, Y.-L. D.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

Jiang, Z.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Keiding, S.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Kelemen, L.

Kerttula, J.

Kitamura, N.

Klein, F.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Koshioka, M.

Liesener, J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185(1-3), 77–82 (2000).
[CrossRef]

Linden, S.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Martín-Badosa, E.

Martinelli, M.

Masuhara, H.

Melloni, A.

Miles, M. J.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Misawa, H.

Mogensen, P. C.

Monguzzi, P.

Montes-Usategui, M.

Mortensen, K. I.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Okhotnikov, O. G.

Ormos, P.

Padgett, M. J.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Palima, D.

Palima, D. Z.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Perch-Nielsen, I.

Perch-Nielsen, I. R.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Peterson, T. J.

G. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011).
[CrossRef]

Phillips, D. B.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

S. H. Simpson, D. B. Phillips, D. M. Carberry, and S. Hanna, “Bespoke optical springs and passive force clamps from shaped dielectric particles,” J. Quant. Spectrosc. Radiat. Transf. 126, 91–98 (2013).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Pleguezuelos, E.

Plet, C.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Raisanen, A. D.

G. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011).
[CrossRef]

Reicherter, M.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185(1-3), 77–82 (2000).
[CrossRef]

Rill, M. S.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Rodrigo, P. J.

Sasaki, K.

Simpson, S. H.

D. B. Phillips, M. J. Padgett, S. Hanna, Y.-L. D. Ho, D. M. Carberry, M. J. Miles, and S. H. Simpson, “Shape-induced force fields in optical trapping,” Nat. Photonics 8(5), 400–405 (2014).
[CrossRef]

S. H. Simpson, D. B. Phillips, D. M. Carberry, and S. Hanna, “Bespoke optical springs and passive force clamps from shaped dielectric particles,” J. Quant. Spectrosc. Radiat. Transf. 126, 91–98 (2013).
[CrossRef]

D. B. Phillips, G. M. Gibson, R. Bowman, M. J. Padgett, S. Hanna, D. M. Carberry, M. J. Miles, and S. H. Simpson, “An optically actuated surface scanning probe,” Opt. Express 20(28), 29679–29693 (2012).
[CrossRef] [PubMed]

Spudich, J. A.

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods 7(5), 377–381 (2010).
[CrossRef] [PubMed]

Stapelfeldt, H.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Staude, I.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Striebel, T.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

Swartzlander, G.

G. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, “Stable optical lift,” Nat. Photonics 5(1), 48–51 (2011).
[CrossRef]

Thiel, M.

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Thøgersen, J.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Tiziani, H. J.

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computer-generated holograms,” Opt. Commun. 185(1-3), 77–82 (2000).
[CrossRef]

Ulriksen, H.-U.

H.-U. Ulriksen, J. Thøgersen, S. Keiding, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Stapelfeldt, and J. Glückstad, “Independent trapping, manipulation and characterization by an all-optical biophotonics workstation,” J. Eur. Opt. Soc. Rapid Publ. 3, 08034 (2008).
[CrossRef]

Ustimchik, V.

Vizsnyiczai, G.

von Freymann, G.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

M. S. Rill, C. Plet, M. Thiel, I. Staude, G. von Freymann, S. Linden, and M. Wegener, “Photonic metamaterials by direct laser writing and silver chemical vapour deposition,” Nat. Mater. 7(7), 543–546 (2008).
[CrossRef] [PubMed]

Wegener, M.

F. Klein, T. Striebel, J. Fischer, Z. Jiang, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Elastic fully three-dimensional microstructure scaffolds for cell force measurements,” Adv. Mater. 22(8), 868–871 (2010).
[CrossRef] [PubMed]

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Supplementary Material (5)

» Media 1: MOV (723 KB)     
» Media 2: MOV (181 KB)     
» Media 3: MOV (163 KB)     
» Media 4: MOV (94 KB)     
» Media 5: MOV (108 KB)     

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Figures (7)

Fig. 1
Fig. 1

Side-view microscope image showing experimental visualization of the focused light coupling to a free-standing WOW. Here the trapped WOW is brought to the focus of a static beam for optimal light coupling. In subsequent results, this green coupling light will be diffractively generated on an SLM to enable dynamic addressing and full 3D targeted light delivery.

Fig. 2
Fig. 2

Array of wave-guided optical waveguides fabricated by two-photon polymerization on a glass substrate. Each WOW consists of a bent waveguide held by sphere handles to aid in optical manipulation. Scale bar: 40 µm. Insets show snapshots from the side-view microscope showing a free-floating, optically manipulated WOW (see Media 1).

Fig. 3
Fig. 3

Schematic diagram of the Biophotonics Workstation equipped with a diffractive SLM setup. The workstation generates counter-propagating beams for the optical manipulation of the WOW-handles while the SLM creates dynamic beams that dynamically track and couple to the waveguide-parts of each WOW.

Fig. 4
Fig. 4

Diffractive SLM-addressing workflow. For proper calibration, imaging plane displacements follow a series of conversions and scalings before calculating the necessary input phase pattern on the SLM. Focal lengths given here also correspond to the lenses given in Fig. 3.

Fig. 5
Fig. 5

Axial propagation profiles of multiple focal spots at different axial and lateral positions visualized by side-view microscopy. This 3D addressing capability offered by the diffractive SLM-setup enables tracking and coupling to individual WOWs as they are displaced at multiple positions within their operating volume.

Fig. 6
Fig. 6

Holographic coupling of a single WOW translated axially. (a) Without diffractive addressing, coupling only occurs at certain axial positions (see Media 2). (b) The green coupling-beam emerges through the sharp tip when diffractive addressing is performed (see Media 3). The first frame in both set of figures shows a static beam for reference. We stack together individual images and create an animation to aid visually the axial movement and coupling and provided them as multimedia files.

Fig. 7
Fig. 7

An optically manipulated WOW translated laterally. (a) For a static coupling-beam, the bright spot at the tip occurs only when the WOW passes the beam (see Media 4). (b) Holographically-addressed WOW demonstrating coupling through the sharp tip at different lateral positions (see Media 5). An animation of stacked images is provided as visual aid showing with and without diffractive addressing.

Equations (5)

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NA= n waveguide 2 n background 2 =0.69
V= Dπ λ NA=4.075
ϕ lateral ( x,y )= 2π λ f 1 ( xΔ x +yΔ y )
ϕ axial ( x,y )= πΔ z λ f 1 2 ( x 2 + y 2 )
ϕ eff ( x,y )=mod( ϕ offset + ϕ lateral + ϕ axial ,2π )

Metrics