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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 21723–21731
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Diffractionless beam in free space with adiabatic changing refractive index in a single mode tapered slab waveguide

Chang-Ching Tsai, Claudio Vinegoni, and Ralph Weissleder  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 21723-21731 (2009)
http://dx.doi.org/10.1364/OE.17.021723


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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

1. Introduction

Diffractionless beams with large depth of focus (DOF) have significant applications in three dimensional biomedical imaging, such as optical coherence tomography (OCT) and single plane illumination microscopy (SPIM) [1

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]

2

2. 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]

]. For two dimensional imaging, the image resolution in the transverse plane (x-y) is determined by the beam size r(x,y) on the focal plane at the distance z. For three dimensional imaging, to keep the same image resolution in a light propagation interval Δz inside the sample, an invariant beam size along the z-axis is required. In other words, the longitudinal image resolution is limited by the DOF of the beam.

Many efforts have been made so far to increase the DOF in different optical systems. Most of the techniques focus on modulating the phase transmission function over the lens [3

3. 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]

5

5. B.-Z. Dong, J. Liu, B.-Y. Gu, and G.-Z. Yang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18, 1465–1470 ( 2001). [CrossRef]

], or the wavefronts coding [6

6. E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 ( 1995). [CrossRef] [PubMed]

]. The use of cylindrical or plasmonic lenses can provide high image resolution, but their DOFs are not large enough for deep bio-medical imaging [7

7. J. G. Ritter, R. Veith, J.-P. Siebrasse, and U. Kubitscheck, “High-contrast single-particle tracking by selective focal plane illumination microscopy,” Opt. Express 16, 7142–7152 ( 2008). [CrossRef] [PubMed]

8

8. 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]

]. Optical power-absorbing apodizers can offer larger DOF, but the drawback of losing the incident energy alternately reduces the image resolution and can be only applied to strong illumination sources [9

9. J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 ( 1990). [CrossRef] [PubMed]

]. Typically, the diffractionless Bessel beams produced by axicons or Fabry-Perot interferometers or holograms [10

10. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 ( 1992). [CrossRef] [PubMed]

13

13. Jari Turunen, Antti Vasara, and Ari T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 ( 1988). [CrossRef] [PubMed]

] are often employed to complete the task.

Fig. 1. The application of non-diffracted beam emitted from adiabatic tapered-waveguide to single plane illumination microscopy (SPIM).

2. Theoretical background and principle of design

Limited aperture sizes and finite wavelengths are the causes of diffraction as described by the Huygens-Fresnel principle. The Airy pattern of light diffracted by a circular aperture of diameter D in the Fraunhofer region gives diffraction a quantitatively description in terms of the wavelength and aperture size. By inspecting its first zero radius J 1(x)≈1.22 f·λ/D, it indicates that the beam spreads out severely if the aperture is sufficiently small compared with the wavelength λ. However as pointed out by Mcleod [14

14. J. H. Mcleod, “The Axicon: A New Type of Optical Element,” J. Opt. Soc. Am. 44, 592–597 ( 1954) [CrossRef]

], a ring aperture can form a line of image without severe diffraction. This is later labeled as Bessel beam, which is indeed a non-diffracted solution of the Helmholtz wave equation [15

15. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 ( 1987). [CrossRef]

]. Basically, Bessel beams are light of ring-patterns originally due to either the Fourier spectrum of an annular slit or the conical lens of virtual ring structure in the far field [14

14. J. H. Mcleod, “The Axicon: A New Type of Optical Element,” J. Opt. Soc. Am. 44, 592–597 ( 1954) [CrossRef]

] and it can be considered as the earliest scheme of modulating the phase transmission function on the aperture to produce diffractionless beam. Most of the other schemes today basically follow the manipulation of the phase transmission function on different optical elements. In this paper, we turn our attention out of the Fourier diffraction and seek for alternative approaches.

Let us first consider the beam inside a waveguide, which is the optical mode that satisfies the boundary condition of Maxwell equations in bounded space. The wave propagating inside the waveguide is confined in the small core region, so the lateral beam size is about the same order of the core width. In a natural way, the waveguide generates a continuous small size beam without the concept of focusing. However, this property is only supported inside the waveguide. Due to the small core width and the discontinuity of the refractive index in the facets of waveguides or optical fibers, the beams suffer severe divergence when emitted out in free space. The challenge then is to find a way to let the beam pass smoothly through the facet with the minimum of diffraction. For simplicity, we are particularly interested in investigating the single-mode waveguide, since the multi-modes will create more complicated beam profile in the transverse plane. Furthermore, we will discuss the symmetric waveguide with examples of TE polarized wave only.

Disregarding the nature of the mode in the waveguide, the core sandwiched in the substrate and cladding at the end of a waveguide is analog to the phase aperture in the Fresnel-Kirchhoff diffraction theory. Hence, if we let the core refractive index n c infinitesimally approach that of the substrate n s (the same refractive index in cladding), the phase aperture essentially diminishes. In this way, we would expect the output beam to be diffractionless. In this extreme condition (n cn s), we should first explore the possibility of the existence of a single propagating mode. The propagating mode is decided by the wave incident angle on the boundary of the waveguide, which should be larger than the critical angle θc=sin-1(n s/n c). In our diminishing case, if n cn s, then θc→90°. For this situation, the propagating mode exists as long as n c > n s and its existence can be proved from the analytical solution of the slab waveguide. We will label this type of waveguide as “grazing waveguide,” accounting for the fact that the incident angle of the ray bouncing inside the waveguide is nearly 90°, i.e. almost tangent to the boundary.

To demonstrate that the diminishing of the refractive index difference can lead to a diffractionless beam, in Figs. 2(a) and 2(b) we compare the numerical results of the output intensities from a single-mode waveguide and the “grazing waveguide” respectively. These two waveguides have the same configurations and parameters except for the refractive index value nc within the core region. In Fig. 2(a) we choose n s=1.48, and n c=1.5 while we let n c=1.4801 in Fig. 2(b). Comparing the two sets of the transverse beam intensities at z=0, 10, 20, 50, 100 µm in Figs. 2(c) and 2(d), it is clear that the grazing waveguide generates a less diffractive beam, which keeps above 92% of the original peak power. However, for such low contrast of refractive index, the beam size broadens a lot as the FWHM inside the waveguide increases from 0.70 µm in Fig. 2(c) to 18.81 µm in Fig. 2(d). On the other side increasing the core width w in the grazing waveguide can help reducing the beam size as clearly shown in Fig. 3(b). In this case, the FWHM is reduced to 7.38 µm with w=5.4 µm while keeping the same parameters as in Fig. 2(b).

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. (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.

It is known in many physical systems with operators of different eigen-problems, that if the operator changes slowly (with time or distance or both) the corresponding eigenmode can transform into the other without loosing the characteristics of mode. This is known as the principle of adiabatic mode transition. Generally, the condition of adiabatic condition can be written as | f/ x|≪f, where f is the vector of mode-characterized parameters and x is a vector of dimensionless variables either in time- or space-domain or both. In optics the adiabatic transitions of optical modes are typically made by variation of the material properties or the geometrical structure. This concept has been applied to waveguide-fiber couplers, slab-photonic crystal waveguides coupling, and volume holograms [16

16. 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]

18

18. C.-C. Tsai, L. B. Glebov, and B. Ya. Zeldovich, “Adiabatic three-wave volume hologram: large efficiency independent of grating strength and polarization,” Opt. Lett. 31, 718–720 ( 2006) [CrossRef] [PubMed]

].

We utilized the adiabatic mode-evolution to change the propagating mode in a high divergent waveguide to a diffractionless mode in a grazing waveguide. Here the Helmholtz wave equation inside the waveguide can be written as,

d2Edz2+(n2(z)neff2k02)E=0
(1)

neff=(γc,sk0)2+nc,s2(z),γc=1w(z){2tan1(γcγs)+π}
(2)

where γ c,s are the transverse propagation constants in the core and substrate region respectively. The adiabatic transition criterion can then be here expressed as

1neffk0dneffdzneff(n,w)
(3)

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).

3. Numerical results of the diffractionless grazing waveguide

For our simulations we use the Method of Line-Beam Propagation Method (MoL-BPM) [19

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

20

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

] with perfect matched layer (PML) boundary condition [21

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

]. This method has been demonstrated to give accurate results for beams propagating inside a multi-junctions waveguide and into free space from abrupt-terminated waveguide facet including radiation and absorption effects [22

22. 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]

23

23. 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]

]. We modeled the continuously changing waveguide profile by stair-case approximation. To avoid the numerical overflow during the eigen-matrices iterating and inverting processes we used Enhanced Transmittance Matrix Approach (ETMA) to do the computation [24

24. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 ( 1995). [CrossRef]

]. Theoretically, in a smoothly adiabatic process there is no reflection during the eigenmode transformation. In the stair-case approximation, there is a negligible amount of reflection at each interface of segment. Numerical experiments show that the total reflectivity is almost the same as the light emitted from a normal single waveguide into the air. Following the parameter curve of Fig. 4(a) we design the appropriate waveguide structure for the diffractionless beam in Fig. 5(a). The simulation condition is that we started from the lowest eigenmode in a slab waveguide with λ=0.405 µm, w=0.8 µm, n c=1.5, n s=1.48 and transform it into the diffractionless mode in the grazing waveguide with w=5.4 µm, n c=1.4801, n s=1.48. In stair-case approximation, the core refractive index n c(z) and width w (z) are kept at constant value within each stair i of ΔL=2000 µm in a total length L=46000 µm with 23 partitions. The beam intensities inside the waveguide as well as in the air are calculated and shown in Figs. 5(b) and 5(c) respectively. Fig. 5(d) gives the transverse beam profile at different propagation distance z=0, 10, 20, 50, 100, 150, 200, 250 µm in the air. We see that Fig. 5(d) depicts similar non-diffraction feature at z=0, 10, 20, 50, 100 µm as in Fig. 3(b) after the adiabatic transition. The non-diffracted quality of the beam can be determined by examining the peak intensities and the FWHMs of the above beam profiles. By setting the peak value equal to 1 at z=0 µm (calculated FWHM=7.41 µm), we obtain at z=100 µm a peak value of 0.901 with a FWHM=8.95 µm. After propagating 100 µm in free space, the peak still keeps 90% of its value and the FWHM broadens only 20%. The grazing waveguide indeed can emit a high diffractionless beam without any external lens attached. At large distance z=250 µm, the peak value is 0.728, which keeps 72% of the initial intensity at z=0. This demonstrates the feasibility of efficient light coupling into the diffractionless grazing waveguide by stair-case transition.

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.

Since there is no cut-off condition for the lowest mode in a symmetric slab waveguide, we further check the transition of propagating mode for the effective refractive index neff very close to the refractive index of cladding n c in a shorter partition length ΔL in the following.

Based on the adiabatic condition Eq. (3) and the waveguide structure of Fig. 5(a), we investigate the diffractionless property of the grazing waveguide by reducing the core refractive index n c another numerical order to 1.48001. This time we divide the waveguide into 56 slices with each partition ΔL=200 µm. The parametric curve of the waveguide configuration n cL) and wL) is plotted in Fig. 6(a). We show the results of the beam propagation in the waveguide and air in Fig. 6(b) and 6(c) respectively. Fig. 6(d) gives the cross sections of the intensity distribution of the beam of Fig. 6(c) at distance z=0, 100, 200, 300, 400, and 500 µm. Data clearly indicate that the peak of the beam keeps the value above 0.983 in the first 100 µm with FWHM=14 µm. Then it reduces to 0.88 at z=200 µm with FWHM=15.95 µm. At z=500 µm, the peak intensity is 0.723, i.e. 72% of its value after emitting with a beam size with FWHM=20.8 µm.

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.

We emphasize that the performance of the grazing waveguide is up to the precision of the refractive index difference between the core and cladding. Practically, the change of refractive index can reach an accuracy of 10-5 in a germania-doped silica-glass [25

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

].

4. Conclusion

In conclusion, we have introduced a new way of producing a high diffractionless beam based on diminishing the optical phase contrast at the end of a slab waveguide. We were able to generate such grazing mode by virtue of the concept of the adiabatic transition. Therefore, a light sheet produced by the slab waveguide can be beneficial for many optical systems. Furthermore, for beams coming out from the slab waveguide, a lens system is usually employed to collimate light from diffraction. The lens system is either bulky or complicated, which limits the application to micro-systems. On the contrary, the tiny and simple design of the grazing waveguide provides a unique solution that can be directly integrated with other optical instruments. The same concept can apply to an optical fiber. The generation of a diffractionless beam by only one single optical element would be very useful in integrated optics. For example, in bio-imaging application, the light sheet can provide uniform deeper micro-illumination for SPIM.

Acknowledgements

This work was supported in part by NIH grants 1-RO1 EB006432 and U24 CA 092782.

*The author contributed in conceptual development, designing the device and numerical modeling of this work.

2,3These authors contributed equally in supervising this work.

References and links

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]

2.

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]

3.

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]

4.

J. A. García1, S. Bará, M. G. García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, “Imaging with extended focal depth by means of the refractive light sword optical element,” Opt. Express 16, 18371–18378 ( 2008). [CrossRef]

5.

B.-Z. Dong, J. Liu, B.-Y. Gu, and G.-Z. Yang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18, 1465–1470 ( 2001). [CrossRef]

6.

E. R. Dowski Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 ( 1995). [CrossRef] [PubMed]

7.

J. G. Ritter, R. Veith, J.-P. Siebrasse, and U. Kubitscheck, “High-contrast single-particle tracking by selective focal plane illumination microscopy,” Opt. Express 16, 7142–7152 ( 2008). [CrossRef] [PubMed]

8.

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]

9.

J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 ( 1990). [CrossRef] [PubMed]

10.

R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 ( 1992). [CrossRef] [PubMed]

11.

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

12.

A. J. Cox and D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry-Perot resonator,” J. Opt. Soc. Am. A 9, 282–286 ( 1992). [CrossRef]

13.

Jari Turunen, Antti Vasara, and Ari T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 ( 1988). [CrossRef] [PubMed]

14.

J. H. Mcleod, “The Axicon: A New Type of Optical Element,” J. Opt. Soc. Am. 44, 592–597 ( 1954) [CrossRef]

15.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 ( 1987). [CrossRef]

16.

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]

17.

Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757 ( 2000). [CrossRef]

18.

C.-C. Tsai, L. B. Glebov, and B. Ya. Zeldovich, “Adiabatic three-wave volume hologram: large efficiency independent of grating strength and polarization,” Opt. Lett. 31, 718–720 ( 2006) [CrossRef] [PubMed]

19.

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

20.

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

21.

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

22.

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]

23.

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]

24.

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 ( 1995). [CrossRef]

25.

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

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(130.0130) Integrated optics : Integrated optics
(230.7400) Optical devices : Waveguides, slab

ToC Category:
Optical Devices

History
Original Manuscript: July 22, 2009
Revised Manuscript: October 7, 2009
Manuscript Accepted: October 12, 2009
Published: November 12, 2009

Citation
Chang-Ching Tsai, Claudio Vinegoni, and Ralph Weissleder, "Diffractionless beam in free space with adiabatic changing refractive index in a single mode tapered slab waveguide," Opt. Express 17, 21723-21731 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21723


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References

  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]
  2. 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]
  3. 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]
  4. J. A. García, S . Bará, M. G . García, Z . Jaroszewicz, A . Kolodziejczyk, and K . Petelczyc, "Imaging with extended focal depth by means of the refractive light sword optical element," Opt. Express 16, 18371-18378 (2008). [CrossRef]
  5. B.-Z. Dong, J. Liu, B.-Y. Gu, and G.-Z. Yang, "Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution," J. Opt. Soc. Am. A 18, 1465-1470 (2001). [CrossRef]
  6. E. R. Dowski, Jr. and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
  7. J. G. Ritter, R. Veith, J.-P. Siebrasse and U. Kubitscheck, "High-contrast single-particle tracking by selective focal plane illumination microscopy," Opt. Express 16, 7142-7152 (2008). [CrossRef] [PubMed]
  8. 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]
  9. J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, "Zone plate for arbitrarily high focal depth," Appl. Opt. 29, 994-997 (1990). [CrossRef] [PubMed]
  10. R. Arimoto, C. Saloma, T. Tanaka, and S. Kawata, "Imaging properties of axicon in a scanning optical system," Appl. Opt. 31, 6653-6657 (1992). [CrossRef] [PubMed]
  11. J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001). [CrossRef]
  12. A. J. Cox and D. C. Dibble, "Nondiffracting beam from a spatially filtered Fabry-Perot resonator," J. Opt. Soc. Am. A 9, 282-286 (1992). [CrossRef]
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