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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 4 — Feb. 16, 2009
  • pp: 2797–2804
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Large longitudinal electric fields (Ez ) in silicon nanowire waveguides

Jeffrey B. Driscoll, Xiaoping Liu, Saam Yasseri, Iwei Hsieh, Jerry I. Dadap, and Richard M. Osgood, Jr.  »View Author Affiliations


Optics Express, Vol. 17, Issue 4, pp. 2797-2804 (2009)
http://dx.doi.org/10.1364/OE.17.002797


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Abstract

We demonstrate the presence of strong longitudinal electric fields (Ez ) in silicon nanowire waveguides through numerical computation. These waveguide fields can be engineered through choice of waveguide geometry to exhibit amplitudes as high as 97% that of the dominant transverse field component. We show even larger longitudinal fields created in free space by a terminated waveguide can become the dominant electric field component, and demonstrate Ez has a large effect on waveguide nonlinearity. We discuss the possibility of controlling the strength and symmetry of Ez using a dual waveguide design, and show that the resulting longitudinal field is sharply peaked beyond the diffraction limit.

© 2009 Optical Society of America

1. Introduction

The longitudinal electric field component (Ez) of a propagating electromagnetic wave has recently been the subject of increasing interest largely as a result of two unique attributes. First, it can be shown to focus tighter than the diffraction limit [1–4

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 1–4 (2003). [CrossRef]

], which lends itself to applications such as lithography [5

5. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

], near-field microscopy [6

6. J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Imaging and Time-Resolved Spectroscopy of Single Molecules at an Interface” Science 272, 255–258 (1996). [CrossRef]

, 7

7. L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998). [CrossRef]

], and optical data storage [8

8. A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006). [CrossRef]

]. Second, because of its directionality, applications have been proposed to take advantage of this unique polarization, such as particle acceleration [9

9. M. O. Scully, “A Simple Laser Linac,” Appl. Phys. B. 51, 238–241 (1990). [CrossRef]

], absorption dipole moment probing [10

10. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]

], multiple quantum well heterostructure excitation [11

11. G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, Ch. Dotzler, A. Winkler, G. Leuchs, G. H. Dohler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” J. Appl. Phys. 100, 023112 1–6 (2006). [CrossRef]

], optical tweezing [12

12. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004) [CrossRef] [PubMed]

], grating couplers in high-index-contrast slab waveguides [13

13. N. Destouches, B. Sider, A. V. Tishchenko, and O. Parriaux, “Optimization of a Waveguide Grating for Normal TM Mode Coupling,” Opt. Quantum Electron. 38, 123–131 (2006). [CrossRef]

], as well as nonlinear optical response probing [14

14. Y. Kozawa and S. Sato, “Observation of the longitudinal field of a focused laser beam by second-harmonic generation,” J. Opt. Soc. Am. B 25, 175–179 (2008). [CrossRef]

]. The longitudinal field originates from the spatial derivative of the transverse fields [15

15. A. Boivin and E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. 138, B1561–B1565 (1965). [CrossRef]

]; hence by means of strong optical confinement a large Ez component can be generated. For a Gaussian beam, the generation of a large Ez is nontrivial as it can be shown to be small for the paraxial case [16

16. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]

]. A common approach to creating large longitudinal fields, as used in many of the above applications, is through radially or azimuthally polarized beams [17

17. V. G. Niziev and A. V. Nesterov, “Longitudinal fields in cylindrical and spherical modes,” J. Opt. A: Pure Appl. Opt 10, 085005 1–7 (2008). [CrossRef]

, 18

18. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

].

We propose an alternative, integrated approach to the generation of large longitudinal electric fields, especially desirable due to the recent trend of miniaturizing photonic devices and the growing interest in lab-on-chip micro-systems. Since Ez originates from the spatial gradient of the transverse fields, it is reasonable to expect strong optical confinement associated with silicon (Si) photonic wire waveguides (Si nanowires) to yield a large Ez. Si nanowires based on the silicon-on-insulator (SOI) platform have been heavily investigated recently due to their high potential for nanoscale photonic devices and photonic integrated circuits (PICs). The ultratight optical confinement of Si nanowires has led to the realization of unique optical properties. As a recent example, it was shown that the dispersion in these guides is dominated by the cross-section geometry as opposed to material properties, allowing waveguide dispersive properties to be engineered [19

19. X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006). [CrossRef]

, 20

20. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006). [CrossRef] [PubMed]

]. The high electric field intensity has also led to the exploration of nonlinear optical effects in silicon [21

21. J. I. Dadap, N. C. Panoiu, X. Chen, I. Hsieh, X. Liu, C. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

]. In fact, a recent work proposed an optical isolator based on nonreciprocal Raman gain resulting from the longitudinal electric field in Si waveguides [22

22. M. Krause, H. Renner, and E. Brinkmeyer, “Optical isolation in silicon waveguides based on nonreciprocal Raman amplification,” Elect. Lett. 44, 691–693 (2008). [CrossRef]

].

2. Basic theoretical considerations

The origin of the longitudinal electric field within a waveguide can be understood through a consideration of Maxwell’s equations. In a waveguide, the electric field vector may be expressed as E(r) =E 0(r)exp(- jβz) where β is the propagation constant and z the direction of propagation [15

15. A. Boivin and E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. 138, B1561–B1565 (1965). [CrossRef]

]. Because of waveguide translational symmetry, E 0(r) =E 0(r T), where r T is the transverse position vector. Assuming an invariant dielectric profile in the z direction, through Gauss’s law, the longitudinal electric field can be expressed as ∂Ez/dz = - (1/n 2) T · (n 2 E T)[23

23. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994). [CrossRef]

]. In piecewise homogeneous regions, one can show that Ez = T · E T/, where E T denotes the transverse electric field complex amplitudes (Ex,Ey), and T the transverse gradient. Equiva-lently,

Ez=λ0j2πNeffT·ET
(1)

where N eff is the effective index experienced by the mode and λ 0 the free space wavelength. For quasi-TE and -TM modes, T · E T∂Ex/dxand T · E T∂Ey/dy respectively. For the following discussion, we consider quasi-TE mode propagation. Examination of Eq. (1) shows that the complex amplitude of the longitudinal field is purely imaginary (by convention that the transverse fields are purely real) meaning that it is in quadrature (π/2) phase with respect to the transverse fields. Similarly, Ez is in quadrature phase with the transverse magnetic fields and examination of the complex Poynting vector, S=12E×H*,, reveals no real contribution via the longitudinal components. Therefore Ez does not contribute to net energy transport and carries no net momentum; however it does contribute to the total energy density, meaning that the longitudinal component acts locally as an energy reservoir, which can be tapped.

Equation (1) is useful within the usual limits of the effective index approximation. We do note exact solutions for the electric field have been discussed for the high-index-contrast cylindrical waveguide [24

24. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

] including the effect of varying radial extent, while no exact solutions exist for the channel case leaving numerical analysis as the best approach for investigating these wire waveguides; the difference between the cylindrical and channel cross-sectional waveguides will be discussed in more detail below. Nevertheless, Eq. (1) does serve as a useful approximation for conveying the impact of tight confinement on increasing the longitudinal field.

3. Calculation and optimization of the longitudinal electric field

Fig. 1. The Ex(Ey) and Ez components of the fundamental quasi-TE(-TM) mode supported by a 260 × 4002 nm Si nanowire waveguide surrounded by SiO2 cladding. Graded yellow and blue colors indicate a π-phase difference.

As an example, a 330 × 320 nm2 Si nanowire on an SiO2 substrate with air cladding yields ∣E z(max)/E T(max)∣= 97%(89%) for the quasi-TE(-TM) mode. This result shows that it is possible to design a single waveguide such that its Ez component is comparable to that of the transverse field for both quasi-TE and -TM modes. This is in stark contrast to standard fiber optic cable where Ez is only a few percent the transverse field and is often ignored in analysis [25

25. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001).

]. Comparing Figs. 2(a) and 2(b) shows that a slightly smaller Ez exists for the quasi-TM mode compared with the quasi-TE mode for the air/SiO2 cladding system and arises from a larger SiO2 area being sampled by the field for the quasi-TM mode. Similarly, Ez is smaller for the full SiO2 cladding system [Figs. 2(c)–2(d)] compared to the air/SiO2 system [Figs. 2(a–2(b)].

To provide a comparison with our results for a channel waveguide, we investigated computationally the longitudinal field of the fundamental HE11 mode for cylindrical-waveguides. Note that these computations agree with analytic results of Tong et al [24

24. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

], thus providing a validating case for the computations. In addition, the results show that the increased confinement factor associated with a cylindrical Si nanowire allows for a larger Ez normalized to the transverse field [Fig. 3]. In general, this comparison shows clearly that for comparable size the longitudinal field in a channel guide is less than that in a cylindrical waveguide. For example, we find that a 400 nm diameter cylindrical silicon waveguide with SiO2 cladding supports an Ez component ~ 0.69 times that of the transverse field [Fig. 3], while a square Si nanowire waveguide (400 × 400 nm2) with SiO2 cladding supports an Ez component ~ 0.58 times that of the transverse field for both the quasi-TE and -TM modes [Figs. 2(c)–2(d)]. This difference makes it important to provide specific calculations for the channel guide. Numerical analysis of the channel structure also allows for the study of unsymmetric cladding and rectangular cross-sections which are not well approximated by circular cross-sections. Finally, we note that rectangular channel guides have the practical geometry now being used in virtually all SOI PICs today.

Fig. 2. Ez(max)/E T(max)∣ for the quasi-TE and -TM modes of Si nanowires with variable dimensions, SiO2 substrate, and air (a,b) or SiO2 (c,d) cladding.
Fig. 3. Ez(max)/E T(max)∣ for the HE11 mode in a cylindrical Si nanowire of variable diameter, and air (top blue line) or SiO2 (bottom red line) cladding.

4. Harnessing and controlling the longitudinal electric field

Figures 4(a) and 4(b) show the Ex and Ez components respectively for a single wire, where both have been normalized to the maximum Ex component. Note that a jump in the Ez amplitude occurs due to the continuity of the normal electric displacement D [Fig. 4(b)]. The transverse components of the electric field, on the other hand, are continuous across the interface by virtue of tangential electric field continuity [Fig. 4(a)], and numerical calculations are in accord with the boundary conditions. An examination of the fields beyond the waveguide’s endface clearly show a substantial presence of Ez, and Fig. 4(c) shows a contour plot of ∣Ez2 40 nm from the endface normalized to ∣E x(max)2 in the wire.

Fig. 4. The (a) Ex and (b) Ez electric field components of a mode propagating in a 260×500 nm2 Si nanowire with SiO2 cladding. (c) ∣Ez/E T(max)2 40 nm from the endface.

Fig. 5. The (a) transverse (Ex) and (b) longitudinal (Ez) field components of the fundamental antisymmetric system mode supported by (c) dual waveguides where each waveguide has cross-section dimensions 260 × 500 nm2 separated by a 50 nm gap.
Fig. 6. The(a)Ez and(b)Ex electric field components at the endface of dual 260 × 500 nm2 Si nanowire waveguides with SiO2 cladding, 50 nm gap, terminated into air, and excited by the fundamental antisymmetric system mode. (c) ∣Ez2 near the edge of the dual waveguide structure. Contour plot (d) and line scan (e) of ∣Ez2 40 nm from the edge of the waveguide.

For this antisymmetric waveguide design, the mode propagation of the Ex and Ez components into air was found through FDTD [Figs. 6(a)–6(b)]. At the output, the longitudinal field dominates that of the transverse field and is further enhanced in free space due to constructive interference [Fig. 6(c)]. A plot of the spatial contours of ∣Ez2 40 nm from the endface [Figs. 6(d)–6(e)] reveals a very narrow peak; in fact it is beyond the diffraction limit and has a 320 nm (~ λ/5) full-width half-maximum (FWHM) along the x-direction. Furthermore, at this distance, ∣Ez2 is ~ 1.4 times that of ∣Ex2 in the waveguide. It is important to reiterate the significant difference between the slot waveguide structures [26

26. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

, 27

27. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

] and the design discussed here; the slot waveguide is optimized to enhance the transverse fields in the gap, while here we are interested in enhancing the longitudinal electric fields at the terminated output; thus different geometry and phase relations must be considered.

5. The effect of the longitudinal field on waveguide nonlinearity

The presence of a large longitudinal electric field can also be shown to be important in devices based on optical nonlinearities. Our preliminary calculations show that the longitudinal electric field may have an adverse effect on the effective nonlinear parameter (γ). These preliminary calculations clearly show the importance of considering Ez in predicting γ As an example, following the method presented in Ref. [19

19. X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006). [CrossRef]

] to calculate γ, we find the effective nonlinear parameter for a 260 × 500 nm2 SiO2 clad Si nanowire to be ~ 245 W-1 m-1 when the longitudinal field is included in the calculation and ~ 345 W-1m-1 when the longitudinal field is excluded. As the waveguide’s confinement factor increases via size reduction, the total transverse electric field intensity increases however so does the longitudinal electric field; these results have an opposite effect on γ and is in clear contrast to similar calculations in fiber optics where the longitudinal field is ignored under the assumption its magnitude is negligible [25

25. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001).

]. Such an approximation is valid for low-index-contrast fiber optic cables, however the longitudinal field plays a critical role in the case of high-index-contrast Si nanowires. Currently a more extensive investigation of this phenomenon is underway in our laboratory and will be reported elsewhere.

6. Conclusions

In this paper we have shown numerically that the longitudinal electric-field component may become comparable to the amplitude of the transverse field in high-index-contrast Si nanowire waveguides. Ez may be engineered via waveguide geometry and optimized to be ~ 97% that of the transverse field, which allows Si nanowire supported Ez fields to have potential uses in novel integrated waveguide applications such as optical isolation. Using the antisymmetric system mode of a dual waveguide design, a purely longitudinal electric field exists in the gap and the longitudinal field becomes enhanced and sharply peaked (~ λ/5) at the output, which serves as an important new route to ultrahigh-resolution applications such as subwavelength optical microscopy. We further show that Ez has a large effect when calculating the nonlinear parameter of a Si nanowire and, in contrast to low-index-contrast platforms such as standard fiber optic cables, cannot be ignored in analysis.

Acknowledgments

This research was supported by the Air Force Office of Scientific Research (AFOSR) Grant FA9550-05-1-0428 and in part by National Science Foundation (NSF) Grant DMR-0806682.

References and links

1.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 1–4 (2003). [CrossRef]

2.

H. P. Urbach and S. F. Pereira, “Field in Focus with a Maximum Longitudinal Electric Component,” Phys. Rev. Lett. 100, 123904 1–4 (2008). [CrossRef]

3.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

4.

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2000).

5.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001). [CrossRef]

6.

J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Imaging and Time-Resolved Spectroscopy of Single Molecules at an Interface” Science 272, 255–258 (1996). [CrossRef]

7.

L. Novotny, E. J. Sanchez, and X. S. Xie, “Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams,” Ultramicroscopy 71, 21–29 (1998). [CrossRef]

8.

A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, “High-density optical data storage,” Rep. Prog. Phys. 69, 2323–2363 (2006). [CrossRef]

9.

M. O. Scully, “A Simple Laser Linac,” Appl. Phys. B. 51, 238–241 (1990). [CrossRef]

10.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]

11.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, Ch. Dotzler, A. Winkler, G. Leuchs, G. H. Dohler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” J. Appl. Phys. 100, 023112 1–6 (2006). [CrossRef]

12.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004) [CrossRef] [PubMed]

13.

N. Destouches, B. Sider, A. V. Tishchenko, and O. Parriaux, “Optimization of a Waveguide Grating for Normal TM Mode Coupling,” Opt. Quantum Electron. 38, 123–131 (2006). [CrossRef]

14.

Y. Kozawa and S. Sato, “Observation of the longitudinal field of a focused laser beam by second-harmonic generation,” J. Opt. Soc. Am. B 25, 175–179 (2008). [CrossRef]

15.

A. Boivin and E. Wolf, “Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam,” Phys. Rev. 138, B1561–B1565 (1965). [CrossRef]

16.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]

17.

V. G. Niziev and A. V. Nesterov, “Longitudinal fields in cylindrical and spherical modes,” J. Opt. A: Pure Appl. Opt 10, 085005 1–7 (2008). [CrossRef]

18.

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

19.

X. Chen, N. C. Panoiu, and R. M. Osgood Jr., “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42, 160–170 (2006). [CrossRef]

20.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006). [CrossRef] [PubMed]

21.

J. I. Dadap, N. C. Panoiu, X. Chen, I. Hsieh, X. Liu, C. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

22.

M. Krause, H. Renner, and E. Brinkmeyer, “Optical isolation in silicon waveguides based on nonreciprocal Raman amplification,” Elect. Lett. 44, 691–693 (2008). [CrossRef]

23.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994). [CrossRef]

24.

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef] [PubMed]

25.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 2001).

26.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29, 1209–1211 (2004). [CrossRef] [PubMed]

27.

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

28.

M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(130.4310) Integrated optics : Nonlinear
(180.4243) Microscopy : Near-field microscopy
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Integrated Optics

History
Original Manuscript: November 14, 2008
Revised Manuscript: January 11, 2009
Manuscript Accepted: January 29, 2009
Published: February 11, 2009

Citation
Jeffrey B. Driscoll, Xiaoping Liu, Saam Yasseri, Iwei Hsieh, Jerry I. Dadap, and Richard M. Osgood, "Large longitudinal electric fields (Ez) in silicon nanowire waveguides," Opt. Express 17, 2797-2804 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2797


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References

  1. R. Dorn, S. Quabis, and G. Leuchs, "Sharper Focus for a Radially Polarized Light Beam," Phys. Rev. Lett. 91, 233901 1-4 (2003). [CrossRef]
  2. H. P. Urbach and S. F. Pereira, "Field in Focus with a Maximum Longitudinal Electric Component," Phys. Rev. Lett. 100, 123904 1-4 (2008). [CrossRef]
  3. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
  4. Q. Zhan and J. Leger, "Focus shaping using cylindrical vector beams," Opt. Express 10, 324-331 (2000).
  5. L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens systems," Opt. Commun. 191, 161-172 (2001). [CrossRef]
  6. J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, "Imaging and Time-Resolved Spectroscopy of Single Molecules at an Interface" Science 272, 255-258 (1996). [CrossRef]
  7. L. Novotny, E. J. Sanchez, and X. S. Xie, "Near-field optical imaging using metal tips illuminated by higher-order Hermite-Gaussian beams," Ultramicroscopy 71, 21-29 (1998). [CrossRef]
  8. A. S. van de Nes, J. J. M. Braat, and S. F. Pereira, "High-density optical data storage," Rep. Prog. Phys. 69, 2323-2363 (2006). [CrossRef]
  9. M. O. Scully, "A Simple Laser Linac," Appl. Phys. B. 51, 238-241 (1990). [CrossRef]
  10. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal Field Modes Probed by Single Molecules," Phys. Rev. Lett. 86, 5251-5254 (2001). [CrossRef] [PubMed]
  11. G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, Ch. Dotzler,A. Winkler, G. Leuchs, G. H. Dohler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, "Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam," J. Appl. Phys. 100, 023112 1-6 (2006). [CrossRef]
  12. Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004) [CrossRef] [PubMed]
  13. N. Destouches, B. Sider, A. V. Tishchenko, and O. Parriaux, "Optimization of a Waveguide Grating for Normal TM Mode Coupling," Opt. Quantum Electron. 38,123-131 (2006). [CrossRef]
  14. Y. Kozawa and S. Sato, "Observation of the longitudinal field of a focused laser beam by second-harmonic generation," J. Opt. Soc. Am. B 25, 175-179 (2008). [CrossRef]
  15. A. Boivin and E. Wolf, "Electromagnetic Field in the Neighborhood of the Focus of a Coherent Beam," Phys. Rev. 138, B1561-B1565 (1965). [CrossRef]
  16. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975). [CrossRef]
  17. V. G. Niziev and A. V. Nesterov, "Longitudinal fields in cylindrical and spherical modes," J. Opt. A: Pure Appl. Opt. 10, 085005 1-7 (2008). [CrossRef]
  18. K. Youngworth and T. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7, 77-87 (2000). [CrossRef] [PubMed]
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