Abstract

We demonstrate two-dimensional optical lattice generation at 1064nm wavelength using vertically embedded multimode-interference (MMI) square-core polymer waveguides on a silicon chip. We demonstrate tuning of the effective waveguide length by longitudinally offsetting the waveguide input end-face from the input beam waist. Our measurement results of the waveguides with different cross-sectional dimensions at different effective waveguide lengths exhibit lattice patterns spanning from 4 × 4 to 10 × 10 arrays at the waveguide output end-face. Our theoretical analysis reveals that the offset causes additional mode-dependent phase changes. Our numerical modeling results using the three-dimensional beam-propagation method are consistent with our experimental results and theory.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. F. M. Dickey and T. E. Lizotte, eds., Laser Beam Shaping Applications (CRC, 2017).
  2. M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  5. D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
    [Crossref] [PubMed]
  6. B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
    [Crossref] [PubMed]
  7. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
    [Crossref]
  8. D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11(2), 158–166 (2003).
    [Crossref] [PubMed]
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    [Crossref]
  10. S. He, X. Ao, and V. Romanov, “General properties of N x M self-images in a strongly confined rectangular waveguide,” Appl. Opt. 42(24), 4855–4859 (2003).
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    [Crossref]
  13. Z. Yao and A. W. Poon, “Vertically embedded multimode-interference waveguide-based optical stretchers for mechanical characterization of cells,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2017) (Optical Society of America, 2017), paper SM4C.2.
    [Crossref]
  14. Z. Yao and A. W. Poon, “Optical Lattice-Based Cell Guiding and Stretching Using Integrated Vertical Multimode-Interference Waveguides,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2018) (Optical Society of America, 2018), Accepted.
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    [Crossref]
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2015 (1)

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

2014 (2)

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

L. Meng and J. Yan, “Mechanism study of sidewall damage in deep silicon etch,” Appl. Phys., A Mater. Sci. Process. 117(4), 1771–1776 (2014).
[Crossref]

2013 (2)

2004 (1)

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting by periodic potential energy landscapes: Optical fractionation,” Phys. Rev. E 70(1), 010901 (2004).
[Crossref] [PubMed]

2003 (3)

2002 (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

2001 (1)

I. Zubel and M. Kramkowska, “The effect of isopropyl alcohol on etching rate and roughness of (1 0 0) Si surface etched in KOH and TMAH solutions,” Sens. Actuators A Phys. 93(2), 138–147 (2001).
[Crossref]

1995 (1)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

1994 (1)

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

1973 (1)

Alpmann, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Ao, X.

Bachmann, M.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Baird, M. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Beach, J. R.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Besse, P. A.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Betzig, E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Bishof, M.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Bloom, B. J.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Bromley, S. L.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Campbell, S. L.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Chen, B.-C.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

Davidson, M. W.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Denz, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

D. McGloin, G. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11(2), 158–166 (2003).
[Crossref] [PubMed]

Esseling, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Grier, D. G.

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting by periodic potential energy landscapes: Optical fractionation,” Phys. Rev. E 70(1), 010901 (2004).
[Crossref] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

Hale, G. M.

Hammer, J. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

He, S.

Kasza, K.

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting by periodic potential energy landscapes: Optical fractionation,” Phys. Rev. E 70(1), 010901 (2004).
[Crossref] [PubMed]

Kirchhausen, T.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

Kramkowska, M.

I. Zubel and M. Kramkowska, “The effect of isopropyl alcohol on etching rate and roughness of (1 0 0) Si surface etched in KOH and TMAH solutions,” Sens. Actuators A Phys. 93(2), 138–147 (2001).
[Crossref]

Ladavac, K.

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting by periodic potential energy landscapes: Optical fractionation,” Phys. Rev. E 70(1), 010901 (2004).
[Crossref] [PubMed]

Lei, T.

Li, D.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

McGloin, D.

Melchior, H.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Melville, H.

Meng, L.

L. Meng and J. Yan, “Mechanism study of sidewall damage in deep silicon etch,” Appl. Phys., A Mater. Sci. Process. 117(4), 1771–1776 (2014).
[Crossref]

Milkie, D. E.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Moses, B.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Nicholson, T. L.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Pasham, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

Poon, A. W.

Querry, M. R.

Romanov, V.

Shao, L.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Sibbett, W.

Smit, M. K.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Spalding, G.

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

Williams, J. R.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Woerdemann, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Xu, P.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Yan, J.

L. Meng and J. Yan, “Mechanism study of sidewall damage in deep silicon etch,” Appl. Phys., A Mater. Sci. Process. 117(4), 1771–1776 (2014).
[Crossref]

Ye, J.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Zhang, M.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Zhang, W.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Zhang, X.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Zubel, I.

I. Zubel and M. Kramkowska, “The effect of isopropyl alcohol on etching rate and roughness of (1 0 0) Si surface etched in KOH and TMAH solutions,” Sens. Actuators A Phys. 93(2), 138–147 (2001).
[Crossref]

Appl. Opt. (2)

Appl. Phys., A Mater. Sci. Process. (1)

L. Meng and J. Yan, “Mechanism study of sidewall damage in deep silicon etch,” Appl. Phys., A Mater. Sci. Process. 117(4), 1771–1776 (2014).
[Crossref]

J. Lightwave Technol. (2)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[Crossref]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[Crossref]

Laser Photonics Rev. (1)

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013).
[Crossref]

Nature (2)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003).
[Crossref] [PubMed]

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10-18 level,” Nature 506(7486), 71–75 (2014).
[Crossref] [PubMed]

Opt. Commun. (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1), 169–175 (2002).
[Crossref]

Opt. Express (2)

Phys. Rev. E (1)

K. Ladavac, K. Kasza, and D. G. Grier, “Sorting by periodic potential energy landscapes: Optical fractionation,” Phys. Rev. E 70(1), 010901 (2004).
[Crossref] [PubMed]

Science (1)

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349(6251), 010901 (2015).
[Crossref] [PubMed]

Sens. Actuators A Phys. (1)

I. Zubel and M. Kramkowska, “The effect of isopropyl alcohol on etching rate and roughness of (1 0 0) Si surface etched in KOH and TMAH solutions,” Sens. Actuators A Phys. 93(2), 138–147 (2001).
[Crossref]

Other (4)

F. M. Dickey and T. E. Lizotte, eds., Laser Beam Shaping Applications (CRC, 2017).

Z. Yao and A. W. Poon, “Integrated optofluidic cell stretchers using optical lattices generated from vertically embedded multimode-interference waveguides,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2016) (Optical Society of America, 2016), paper SW4G.2.
[Crossref]

Z. Yao and A. W. Poon, “Vertically embedded multimode-interference waveguide-based optical stretchers for mechanical characterization of cells,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2017) (Optical Society of America, 2017), paper SM4C.2.
[Crossref]

Z. Yao and A. W. Poon, “Optical Lattice-Based Cell Guiding and Stretching Using Integrated Vertical Multimode-Interference Waveguides,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (2018) (Optical Society of America, 2018), Accepted.
[Crossref]

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Figures (14)

Fig. 1
Fig. 1 (a) Schematic of optical lattice generation using a SU-8-filled vertically embedded square-core MMI waveguide in a silicon substrate. (b)-(c) Illustrations of (b) a positive and (c) a negative longitudinal offset of the waveguide bottom end-face from the input beam waist (O).
Fig. 2
Fig. 2 (a) Side-view of the 3-D BPM-simulated field-amplitude distribution inside the MMI waveguide upon a Gaussian beam input at the center of the waveguide bottom end-face (d = 0 μm). (b) Zoom-in view of the top region including a range of ΔH = 20 μm in air outside the waveguide output end-face. (c) Top-view of the simulated field-amplitude distribution at the waveguide output end-face. (d) Simulated field-amplitude profile along the dashed line indicated in (c). (e) Simulated input field-amplitude profile along the center of the beam at the waveguide bottom end-face. (f) Simulated Lwg as a function of W for optical lattice generation with N = 4 ~10.
Fig. 3
Fig. 3 (a) Side-view of the 3-D BPM-simulated field-amplitude distribution inside the MMI waveguide upon d = 10 μm. (b) Top-view of the simulated field-amplitude distribution at the waveguide output end-face. (c) Simulated field-amplitude profile along the dashed line indicated in (b). (d) Simulated Lwg as a function of d to generate an optical lattice with N = 5 from waveguides with W = 31.5 μm.
Fig. 4
Fig. 4 (a)-(h) Schematic fabrication process flow. (i)-(j) Optical transmission patterns from (i) a waveguide with a bubble inside the core and (j) a bubble-free waveguide.
Fig. 5
Fig. 5 (a) SEM image of the side-view profile of a silica-coated through-wafer square hole. (b)-(d) Zoom-in-view SEM images of the (b) top, (c) middle and (d) bottom regions of the hole as indicated in (a). (e) Extracted local widths of four representative through-wafer square holes as a function of depth.
Fig. 6
Fig. 6 (a)-(b) Optical microscope images of (a) top- and (b) bottom-end-faces of a fabricated square-core waveguide with W ≈30 μm. (c)-(d) Optical microscope images of (c) top- and (d) bottom-end-faces of a fabricated square-core waveguide with W ≈43 μm. (e)-(f) Measured (e) top- and (f) bottom-widths in the x and y directions from 31 square-core waveguides. (g) Average bottom-widths as a function of top-widths.
Fig. 7
Fig. 7 Experimental setup for the optical lattice characterization. MO: microscope objective. Inset (i): calibration of d along the z direction of the chip bottom surface from the input beam waist. Inset (ii): end-firing of the beam into the waveguide with d > 0.
Fig. 8
Fig. 8 Optical lattice patterns generated from fabricated waveguides with self-image numbers of (a) 4 × 4 (W ≈28 μm and d ≈-6 μm), (b) 5 × 5 (W ≈30 μm and d ≈-17 μm), (c) 6 × 6 (W ≈34 μm and d ≈-8 μm, (d) 7 × 7 (W ≈36 μm and d ≈-12 μm), (e) 8 × 8 (W ≈41 μm and d ≈-1 μm), (f) 9 × 9 (W ≈43 μm and d ≈-8 μm) and (g) 10 × 10 (W ≈46 μm and d ≈6 μm). The intensity is normalized to the maximum intensity of each pattern.
Fig. 9
Fig. 9 Fourier-transformed k-space diagrams of optical lattice patterns in Fig. 8 with self-image numbers of (a) 4 × 4, (b) 5 × 5, (c) 6 × 6, (d) 7 × 7, (e) 8 × 8, (f) 9 × 9 and (g) 10 × 10.
Fig. 10
Fig. 10 Statistics of the normalized power distributions from representative optical lattice patterns with self-image numbers of (a) 4 × 4, (b) 5 × 5, (c) 6 × 6, (d) 7 × 7, (e) 8 × 8, (f) 9 × 9 and (g) 10 × 10. The Gaussian function-based fitting curve and the extracted RSD value are shown in each figure. The power of each bright spot is normalized to the total power of the pattern.
Fig. 11
Fig. 11 Measured longitudinal waveguide offsets for various squared measured waveguide widths to generate optical lattices with N = 4 to N = 10. Blue dotted lines indicate the waveguides that generate two optical lattices. Red lines are linear fittings assuming Lwg = 296 μm.
Fig. 12
Fig. 12 (a)-(d) Measured 7 × 7 optical lattices from a waveguide with W ≈36 μm and d ≈-12 μm, upon Δx of (a) ∼0 μm, (b) ∼1 μm, (c) ∼2 μm and (d) ∼3 μm. (e)-(h) Simulated 7 × 7 optical lattice patterns from a waveguide with W = 36 μm, Lwg = 270 μm and d = 8 μm, upon Δx of (e) 0 μm, (f) 1 μm, (g) 2 μm and (h) 3 μm. (i)-(l) Measured 7 × 7 optical lattice patterns from the same waveguide as in (a)-(d) with Δx ≈0 μm, upon Δd of (i) ∼0 μm, (j) ∼3 μm, (k) ∼6 μm and (l) ∼9 μm. (m)-(p) Simulated 7 × 7 optical lattices from a waveguide with the same parameters as in (e)-(h) with Δx = 0 μm, upon Δd of (m) 1 μm, (n) 3 μm, (o) 6 μm and (p) 9 μm.
Fig. 13
Fig. 13 (a)-(b) Measured 4 × 4 optical lattice patterns from a waveguide with W ≈28 μm and d ≈-6 μm, upon (a) NA ≈0.2 and (b) NA ≈0.1. (c)-(d) Measured 10 × 10 optical lattice patterns from a waveguide with W ≈46 μm and d ≈6 μm, upon (c) NA ≈0.2 and (d) NA ≈0.1.
Fig. 14
Fig. 14 (a)-(b) Schematics of the waveguide mode wavevectors at the waveguide input-end-air interface upon (a) d > 0 and (b) d < 0. The beam waists are treated as point sources. (c) Zoom-in view of the wavevectors at the waveguide-air interface region.

Tables (2)

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Table 1 Extracted 1st-order kx and ky wavenumbers and extracted average lattice pitches

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Table 2 Linear fitting results and the extracted a values from Fig. 11

Equations (18)

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N= 1 L wg n( λ ) λ W eff 2
N 1 L eff n( λ ) λ W eff 2
L eff n(λ)d+ L wg .
L eff ad+ L wg
d n Nλa W 2 L wg a
sinθ=nsinθ'
dtanθ=d'tanθ'
d'nd
Ψ(x,0)= ν m1 c ν ψ ν (x)
Ψ(x, L wg )= ν m1 c ν ψ ν (x)exp(j β ν L wg )
β ν n k 0 (ν+1) 2 π λ 0 4n W eff 2
Ψ(x, L wg )exp[ j k 0 ( n L wg ) ] ν m1 c ν ψ ν (x)exp[ j (ν+1) 2 π λ 0 4n W eff 2 L wg ]
Ψ(x,d+ L wg )= ν m1 c ν ψ ν (x)exp[ j( β in,ν d+ β ν L wg ) ]
β in,ν = k 0 2 (1 n 2 )+ β ν 2
β in,ν k 0 1 (ν+1) 2 λ 0 2 4 W eff 2 + [ (ν+1) 2 λ 0 2 8n W eff 2 ] 2
β in,ν k 0 1 (ν+1) 2 λ 0 2 4 W eff 2 k 0 [ 1 (ν+1) 2 λ 0 2 8 W eff 2 ]= k 0 (ν+1) 2 π λ 0 4 W eff 2
Ψ(x,d+ L wg )exp[ j k 0 ( d+n L wg ) ] ν m1 c ν ψ ν (x)exp[ j (ν+1) 2 π λ 0 4n W eff 2 ( nd+ L wg ) ]
L eff nd+ L wg

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