Abstract

We propose a novel design to produce a free space diffractionless beam by adiabatically reducing the difference of the refractive index between the core and the cladding regions of a single mode tapered slab waveguide. To ensure only one propagating eigenmode in the adiabatic transition, the correlation of the waveguide core width and the refractive index is investigated. Under the adiabatic condition, we demonstrate that our waveguide can emit a diffractionless beam in free space up to 500 micrometers maintaining 72% of its original peak intensity. The proposed waveguide could find excellent applications for imaging purposes where an extended depth of field is required.

© 2009 Optical Society of America

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  1. P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2008

2007

G. Mikuła, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc and M. Sypek, "Imaging with extended focal depth by means of lenses with radial and angular modulation," Opt. Express 15, 9184-9193 (2007).
[CrossRef] [PubMed]

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

2006

2002

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

2001

2000

1998

S. J. Al-Bader and H. A. Jamid, "Perfectly matched layer absorbing boundary conditions for the method of lines modelling scheme," IEEE Microw. Guided Wave Lett. 8, 357-359 (1998)
[CrossRef]

1995

1994

J. J. Gerdes, "Bidirectional eigenmode propagation analysis of optical waveguides based on method of lines," Electron. Lett. 30, 550-551 (1994).
[CrossRef]

1992

1991

1990

1989

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

1988

1987

1954

Ackerman, D. A.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Al-Bader, S. J.

S. J. Al-Bader and H. A. Jamid, "Perfectly matched layer absorbing boundary conditions for the method of lines modelling scheme," IEEE Microw. Guided Wave Lett. 8, 357-359 (1998)
[CrossRef]

M. Imtaar and S. J. Al-Bader, "Analysis of diffraction from abruptly -terminated optical fibers by the method of lines," J. Lightwave Technol. 13, 137-141 (1995).
[CrossRef]

Arimoto, R.

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Bará, S

Benish, D.

J. Gerdes, B. Lunitz, D. Benish and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett. 28, 1013-1015 (1992).
[CrossRef]

Cathey, W. T.

Chang, C.-K.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Chen, C.-H.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Chen, Z.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Cheng, T.-D.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Cox, A. J.

Dholakia, K.

J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Diaz, A.

Dibble, D. C.

Ding, Z.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Dong, B.-Z.

Dowski, E. R.

Durnin, J.

Friberg, Ari T.

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

García, J. A.

García, M. G

Gaylord, T. K.

Gerdes, J.

J. Gerdes, B. Lunitz, D. Benish and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett. 28, 1013-1015 (1992).
[CrossRef]

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Am. B 8, 389-394 (1991).
[CrossRef]

Gerdes, J. J.

J. J. Gerdes, "Bidirectional eigenmode propagation analysis of optical waveguides based on method of lines," Electron. Lett. 30, 550-551 (1994).
[CrossRef]

Glebov, L. B.

Grann, E. B.

Greger, K.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Gu, B.-Y.

Henry, C. H.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Imtaar, M.

M. Imtaar and S. J. Al-Bader, "Analysis of diffraction from abruptly -terminated optical fibers by the method of lines," J. Lightwave Technol. 13, 137-141 (1995).
[CrossRef]

Jamid, H. A.

S. J. Al-Bader and H. A. Jamid, "Perfectly matched layer absorbing boundary conditions for the method of lines modelling scheme," IEEE Microw. Guided Wave Lett. 8, 357-359 (1998)
[CrossRef]

Jaroszewicz, Z

Jaroszewicz, Z.

Kawata, S.

Kistler, R. C.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Kolodziejczyk, A

Kolodziejczyk, A.

Kubitscheck, U.

Lee, C.-K.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Lee, R. K.

Lin, D.-Z.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Liu, J.

Lunitz, B.

J. Gerdes, B. Lunitz, D. Benish and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett. 28, 1013-1015 (1992).
[CrossRef]

Marcello, M.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Mcleod, J. H.

Mikula, G.

Moharam, M. G.

Nelson, J. S.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Ojeda-Castañeda, J.

Orlowsky, K. J.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Pampaloni, F.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Petelczyc, K

Petelczyc, K.

Pommet, D. A.

Pregla, R.

J. Gerdes, B. Lunitz, D. Benish and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett. 28, 1013-1015 (1992).
[CrossRef]

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines," J. Opt. Soc. Am. B 8, 389-394 (1991).
[CrossRef]

Ren, H.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Ritter, J. G.

Saloma, C.

Shani, Y.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Siebrasse, J.-P.

Stelzer, E. H. K.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Swoger, J.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Sypek, M.

Tanaka, T.

Tepichin, E.

Tsai, C.-C.

Turunen, Jari

Vasara, Antti

Veith, R.

Verveer, P. J.

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Xu, Y.

Yang, G.-Z.

Yariv, A.

Yeh, C.-S.

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Zeldovich, B. Ya.

Zhao, Y.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett

D.-Z. Lin, C.-H. Chen, C.-K. Chang, T.-D. Cheng, C.-S. Yeh, and C.-K. Lee, "Subwavelength nondiffraction beam generated by a plasmonic lens," Appl. Phys. Lett.  92, 233106-1-3 (2008).
[CrossRef]

Appl. Phys. Lett.

Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, "Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon," Appl. Phys. Lett. 55, 2389-2391 (1989).
[CrossRef]

Electron. Lett.

J. Gerdes, B. Lunitz, D. Benish and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett. 28, 1013-1015 (1992).
[CrossRef]

J. J. Gerdes, "Bidirectional eigenmode propagation analysis of optical waveguides based on method of lines," Electron. Lett. 30, 550-551 (1994).
[CrossRef]

IEEE Microw. Guided Wave Lett.

S. J. Al-Bader and H. A. Jamid, "Perfectly matched layer absorbing boundary conditions for the method of lines modelling scheme," IEEE Microw. Guided Wave Lett. 8, 357-359 (1998)
[CrossRef]

J. Lightwave Technol.

M. Imtaar and S. J. Al-Bader, "Analysis of diffraction from abruptly -terminated optical fibers by the method of lines," J. Lightwave Technol. 13, 137-141 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Methods

P. J. Verveer, J. Swoger, F. Pampaloni, K. Greger, M. Marcello and E. H. K. Stelzer, "High-resolution three dimensional imaging of large specimens with light sheet-based microscopy," Nat. Methods 4, 311-313 (2007).
[PubMed]

Opt Lett.

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, "High-resolution optical coherence tomography over a large depth range with an axicon lens," Opt Lett. 27, 243-245 (2002).
[CrossRef]

Opt. Commun.

J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Other

25. J. Hecht, Understanding Fiber Optics, (Prentice Hall, 2006).

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Figures (6)

Fig. 1.
Fig. 1.

The application of non-diffracted beam emitted from adiabatic tapered-waveguide to single plane illumination microscopy (SPIM).

Fig. 2.
Fig. 2.

(a). Divergence of beam emitting from a single-mode symmetric slab waveguide into air with incident wavelength λ=0.405 µm, core width w=0.8 µm, core refractive index nc =1.5 and substrate ns =1.48. The waveguide length is 200 µm and the propagation distance in air is also 200 µm. (b). Diffractionless beam coming out from the grazing waveguide with the same configuration as in (a) except the core refractive index nc =1.4801. (c) Transverse beam profiles of (a) in the air at z=0, 10, 20, 50, 100 µm respectively with intensity I(z)/I(0) normalized to 1 at z=0. (d) Transverse beam profiles of (b) I(z)/I(0) up to propagating distance 100 µm in air with almost constant FWHM=18.81 µm.

Fig. 3.
Fig. 3.

(a). Diffractionless beam emitting from the grazing waveguide with the same condition as in Fig. 1 (b) but with an increased core width w=5.4 µm. (b) Transverse beam profiles of (a) I(z)/I(0) at z=0, 10, 20, 50, 100 µm in air with almost constant FWHM=7.38 µm.

Fig. 4.
Fig. 4.

(a). The adiabatic condition for transforming the lowest eigenmode with λ=0.405 µm in a single-mode waveguide with w=0.8 µm, n c=1.5, n s=1.48 into the diffractionless mode in the grazing waveguide with w=5.4 µm, n c=1.4801, n s=1.48. The core refractive index n c(z) and width w (z) are kept at constant value within every segment i with ΔL=2000 µm in the stair-case approximation of 46000 µm in 23 stairs. (b) Curve of effective refractive index neff and |dneff |/(neff k 0)dz obtained from Eq.(2) and (3), according to the parameters in (a).

Fig. 5.
Fig. 5.

(a) Side view of the adiabatic-grazing slab waveguide, with the configuration according to Fig. 3(a). (b) Intensities of TE wave inside the waveguide before emitting out. (c) Intensities of beam propagating in air after emitting out. (d) The transverse profiles (cross sections) of the beam in (c) at z=0, 10, 20, 50, 100, 150, 200, and 250 µm respectively, with intensities I(z)/I(0) normalized at z=0.

Fig. 6.
Fig. 6.

(a). The adiabatic condition for transforming a eigenmode with λ=0.405 µm in a single-mode waveguide with w=0.8 µm, n c=1.5, n s=1.48 into the diffractionless mode in the grazing waveguide with w=12 µm, n c=1.48001, n s=1.48. The core refractive index n c(z) and width w (z) are kept at constant value within every segment i with ΔL=200 µm in the stair-case approximation of 11200 µm in 56 partitions. (b) Intensities of TE wave inside the waveguide before emitting out. (c) Intensities of beam propagating in the air after emitting out. (d) Transverse profiles (cross sections) of the beam in (c) I (z)/I(0) at z=0, 100, 200, 300, 400, and 500 µm respectively.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

d 2 E d z 2 + ( n 2 ( z ) n eff 2 k 0 2 ) E = 0
n eff = ( γ c , s k 0 ) 2 + n c , s 2 ( z ) , γ c = 1 w ( z ) { 2 tan 1 ( γ c γ s ) + π }
1 n eff k 0 d n eff d z n eff ( n , w )

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