In recent years, a lot of research has been focusedon developing various micro- and nanophotonics de-vices designed to couple two waveguides of differenttype, for example a conventional single-mode fiberwith a wire or planar waveguide, or a planar wave-guide with a photonic crystal (PhC) waveguide.Presently, known nanophotonics devices for couplingtwo waveguide structures include the following:(i) adiabatically-tapered ridge waveguides intendedfor coupling with PhC waveguides [1–7]. Note thatnot only can the waveguide structures be joinedoutput-to-input, but they can be overlapped parallelto each other as well [8]; (ii) Bragg diffraction grat-ings in a waveguide [9–12] to output light. In thiscase, the Bragg-grating-aided fiber can be locatedon the surface of a planar fiber [13]; (iii) a parabolicmicromirror put at an angle to couple light into aplanar waveguide [14]; (iv) conventional refractivelenses or microlenses [15–18]; (v) Veselago super-lenses with negative refraction, which may be planar
[19–26] or concave [27–29]; (vi) microwave couplingdevices and superlenses [30,31]; (vii) PhC lenses[32]. The nanophotonic devices also work in couplingtwo different PhC waveguides[33].
The tapered waveguides can have high coupling ef-ficiency if the modes in the ridge waveguide and inthe PhC waveguide are comparable in width. In thiscase, the coupling efficiency—defined as the ratio ofthe output energy to the input energy—can be aslarge as 80%–90% [1,2,4,6]. If the width of a ridgewaveguide (1:6 μm) is several times greater than thatof a PhC waveguide (200nm), the coupling efficiencyis decreased to 60% [3]. For greater differences inwidth of the waveguides under coupling, the size ofthe adiabatically-tapered waveguide portion be-comes relatively large: when matching the mode ofa single-mode fiber of core diameter 4:9 μm to themode of a planar waveguide of width 120nm, the ta-per’s length is 40 μm, [5], whereas the diameter of the0:3 × 0:5 μm waveguide can be narrowed to a dia-meter of 75nm at a distance of 150 μm [7]. It is worthkeeping in mind that the optical mode size is largerthan the waveguide’s physical dimension.
Coupling devices to couple light from a single-mode fiber into a planar or PhC waveguide by means
of on-waveguide diffraction gratings also have ta-pered portions. For instance, a Gaussian beam witha 14 μmwaist can be narrowed to match a waveguideof width 1 μm by a taper as short as 14 μm [9,10].Note that the experimental coupling efficiency hasbeen reported to be 35% [10] without a mirror layerand 57% [9] with a mirror on the waveguide’sunderside. A Gaussian beam of wavelength 1:3 μmhas been coupled into a waveguide using an on-waveguide diffraction grating [10]. A similar cou-pling device comprising an on-Si-waveguide gratingof period 630nm with a 20–40 μm taper for the wave-length of 1:55 μm has been reported to have an ex-perimental coupling efficiency of 33% (54% withthe mirror) [11]. A higher-quality device comprisingan in-Si diffraction grating of period 610nm andwidth 10 μm designed to couple output light from asingle-mode fiber into a wire waveguide of width3 μm has been shown to have an experimental cou-pling efficiency of 69% [12]. A calculated coupling ef-ficiency of over 90% has been reported for a J coupler,designed to couple a wide waveguide (10 μm) with aPhC waveguide (420nm) through a 15 × 20 μm para-bolic mirror for a wavelength of 1:3 μm [14].Conventional refractive lenses and microlenses
have also been successfully employed when solvingcoupling problems. For example, a Si waveguide ofwidth 1–2 μm provided with an on-end lens can becoupled with a Si PhC waveguide with an estimatedefficiency of 90% [15]. The modeling [17] has shown
that a single-mode fiber of diameter 10:3 μm (λ ¼1:55 μm wavelength) can be coupled with a PhCwaveguide propagating a 0:19 × 0:27 μm mode usinga Si focusing microlens of aperture radius 123 μm,achieving a 80% efficiency.
Of special notice are coupling devices based on 2Dsuperlenses, which utilize the negative refractionphenomenon. A superlens with an effective refractiveindex of −1 can be obtained on the basis of photoniccrystals. The superlens is used for imaging a pointsource. Notice that the original image is formed with-in the lens and the second image—behind the lens ata distance of 2B-A, where B is the thickness of aplane-parallel lens and A is the lens-to-source dis-tance [19,23]. It has been shown [21] that describinga 2D point source by the Hankel function H0ðkrÞ,where k is the wave number and r is the distancefrom the source to the observation point, yields theresulting image proportional to the Bessel functionJ0ðkrÞ. Thus, the superlens produces a spot of sizeFWHM ¼ 0:35λ. A simulation of the 2D superlensperformance [24] has shown that a lens comprisingtwo layers of dielectric rods (permittivity ε ¼ 12:96),operating at wavelength λ ¼ 1:55 μm, has a refractiveindex of n ¼ −1, imaging a point source put at dis-tance A ¼ 0:26λ at about the same distance on theother side of the lens and producing a spot size ofFWHM ¼ 0:36λ. In some articles, a Veselago lensversion with a concave surface instead of a plane-parallel PhC layer was considered [27,28].
Fig. 1. (a) 2D PhC–ML comprising 12 × 17 holes in silicon of size 3 × 4 μm; (b) diffraction field from a plane TE wave or a 2D intensitydistribution jExj2 (y is the vertical axis and z is the horizontal axis); (c) axial intensity distribution; (d) intensity distribution in the focalplane.
Another type of the PhC lens was proposed in [34–36]. Such a 2D lens has an array of holes with a con-stant period but the hole sizes are varying as a cer-tain function. The familiar gradient Mikaelian lens[37] operates by focusing all the rays parallel to theoptical axis and incident perpendicularly to its planesurface in a point on the opposite plane surface. Therefractive index of such an axisymmetric gradientlens is related to the radial coordinate as
nðrÞ ¼ n0
�cosh
�πr2L
��−1; ð1Þ
where n0 is the refractive index on the optical axisand L is the lens thickness along the optical axis. In[34], a 2DMikaelian lens of aperture 12 μmwasmod-eled. The reported efficiency was 55% [34]. In [35,36],a similar PhC Mikaelian lens, but having differentparameters, was modeled. The focal spot size wasFWHM ¼ 0:42λ.In this work, we have modeled, fabricated, and
characterized a new ultracompact nanophotonic de-vice that allows 2D waveguides of different width tobe efficiently coupled by means of a PhC Mikaelianlens (PhC–ML). The device is fabricated on silicon-on-insulator (SOI), the input waveguide width is 4:5 μm,the outputwaveguidewidth is 1 μm, and thePhC–MLsize is 3 × 4 μm. The lens comprises a 12 × 17 hole ar-ray, in which the array period is 250nm and the holediameter ranges from center to periphery from 160 to200nm. The device operates in a wavelength range of1:5–1:6 μm.Depending on the inputwaveguidewidth,
the calculated coupling efficiency is found to be in arangefrom40%to80%.ThePhC–MLfocuseslight intoa small focal spot in the air directly behind the lens,characterized by FWHM ¼ 0:32λ, which is smallerthan in [21,24,28,36].
2. Simulation
A. Simulation of a 2D PhC–ML
The PhC–ML under simulation comprises a 12 × 17hole array in silicon (the effective refractive index ofthe silicon-on-insulator for TE wave is n ¼ 2:83), thehole array period is 250nm, the minimal hole dia-meter on the optical axis is 186nm, and the maximalhole diameter on the lens edge is 250nm. The thick-ness along the optical axis is 3 μm, and the lens width(aperture) is 5 μm. The wavelength is λ ¼ 1:55 μm.
The simulation was conducted using a finite-differ-ence time-domain (FDTD) scheme for solving Max-well’s equations implemented in C++. Figure 1(a)shows the above-described 2D PhC–ML in SOI,and Fig. 1(b) shows a half-tone time-averaged diffrac-tion pattern of the plane TE wave of intensity I ¼jExðy; zÞj2 (the x axis is perpendicular to the planeof Fig. 1). Figures 1(c) and 1(d) show the intensity dis-tributions I on optical axis z and on the line parallelto the y axis and passing through the focus. FromFigs. 1(c) and 1(d), it is seen that the focal spot sizeat half intensity is FWHM ¼ 0:36λ, whereas the axialfocus size is FWHM ¼ 0:52λ.
Fig. 2. (a) 2D PhC lens in waveguide output; (b) half-tone diffraction pattern of the plane TEwave of amplitude ex incident on the input ofthe 5 μm waveguide with a 3 μm lens at the output, and the intensity distribution (c) on the axis and (d) in the lens focus.
Here we model a PhC lens with the same parametersas in Fig. 1(a), which is located in the output of a Siwaveguide of width 5 μm and length 5 μm [Fig. 2(a)].The time-averaged diffraction field of intensity I ¼
jExðy; zÞj2 calculated by the FDTD method is shownin Fig. 2(b) (wavelength 1:45 μm). Figure 2(c) showsthe intensity distribution I on the optical axis.Figures 1(c), 1(d), 2(c), and 2(d) show the ratio I=I0along the ordinate axis (where I0 is the intensityof the beam lighting the in-waveguide PhC lens).In comparison with Fig. 1(c), the intensity in focus inFig. 2(c) is seen to have increased (6.5 and 7, respec-tively) and the intensity modulation amplitude with-in the lens is seen to have decreased. This is becausethe refractive index contrast between the lens andthe waveguide [Fig. 2(c)] is much less than that be-
tween the lens and the air [Fig. 1(c)], resulting in asmaller amplitude of waves scattered from the medi-um interface. In Fig. 2(d), the intensity distributionI ¼ jExðy; zÞj2 in the lens focus in air on a line parallelto the y axis shows that the focal spot size at half-intensity is FWHM ¼ 0:32λ. From the comparisonof Figs. 2(d) and 1(d), it can be seen that alongsidethe focal spot decrease due to the in-waveguidePhC lens, the side lobes of the diffraction patternin focus have also been decreased.
Note that in the 2D case, the scalar theory de-scribes the diffraction-limited focus by the sinc func-tion: Exðy; zÞ ¼ sincð2πyNA=λÞ, whence follows thatat a maximal NA ¼ 1, the diffraction limit of the focalspot size at half-intensity is FWHM ¼ 0:44λ. For asuperlens [21], the limiting value of the focal spot isdefined by the Bessel function J0ðkrÞ, producing the
Fig. 3. (Color online) (a) Schematic diagram of coupling two planar waveguides with a PhC lens; (b) instantaneous diffraction pattern of aTE wave, calculated by the FDTD method using the FullWAVE 6.0; (c) magnified fragment of the pattern at the output of the 0:5 μmwaveguide; (d) output intensity distribution.
focal spot size at half maximum of FWHM ¼ 0:35λ.Hence, the lens in Fig. 2(a) focuses light into a spotsmaller than the diffraction limit. It can be explainedbecause the focal spot is generated in the vicinity ofthe lens surface and the spot is partially created bysurface (evanescent) waves with a wavelength smal-ler than λ.Here and in Subsections 2.C and 2.D, simulation
by the FDTD method is implemented with samplingsteps λ=100 in space and λ=ð50cÞ in time, where c isthe speed of light in vacuum.
C. Simulation of Coupling Two Waveguides Using a PhCLens
Figure 3(a) depicts a device for coupling two 2Dwaveguides by means of a PhC–ML. The input wave-guide has a width of 5 μm, the output waveguide is
0:5 μm wide, and both waveguides are 6 μm long.The PhC lens in Si (the effective index n ¼ 2:83) com-prises a 12 × 19 hole array of period 0:25 μm, the holediameters being the same as in Subsections 2.A and2.B. The wavelength is 1:55 μm.
The simulation was conducted by FDTD usingFullWAVE 6.0 (by RSoft, USA). Note that theFullWAVE simulation engine calculates only in-stantaneous 2D fields of amplitude [Figs. 3(b) and4(b)] while our C++ program calculates the 2D fieldof time-averaged intensity [Figs. 1(b) and 2(b)].Figure 3(b) depicts the instantaneous amplitude dis-tribution Ex of the TE wave. The coupling efficiencywas found to be 45%. The light was partially (about20%) backscattered from the lens inside the inputwaveguide, with some light passing through thelens but missing the narrow waveguide. Shown in
Fig. 4. (Color online) (a) 2D arrangement for coupling two waveguides by means of a PhC lens for a Δz ¼ 1 μm interwaveguide spacing(white—material, black—air); (b) instantaneous intensity distribution of the TEwave, calculated in FullWAVE; and the coupling efficiencyas a function of (c) the interwaveguide spacing Δz and (d) the output waveguide width W2.
Fig. 3(c) is a magnified fragment of the outputamplitude distribution of Fig. 3(b) of the narrowwaveguide. For the narrow waveguide, the output in-tensity distribution on the transverse y axis,jExðy; zÞj2, is shown in Fig. 3(d), fromwhich themodalspot size in air at half intensity is equal toFWHM ¼ 0:32λ. Note that, in the output waveguideof width 1 μm—the other conditions being the same—the focal spot size in waveguide is FWHM ¼ 0:21λ,where λ is the wavelength in vacuum. This is smallerthan the earlier reported value of FWHM ¼0:24λ [17].The numerical simulation of the optical scheme
stability [Fig. 3(a)] has shown that, by changingthe medium index by 5%, the coupling efficiency canchange by 20%; and changing the hole diameters by5% leads to a 17% change in the coupling efficiency.
D. Modeling the Interwaveguide-Space Effect
Figure 4 shows a 2D arrangement for coupling twocoaxial waveguides separated by a space. The inputwaveguide comprising the PhC lens is W1 ¼ 4:6 μmwide and the output waveguide is W2 ¼ 1 μm wide,the interwaveguide spacing being Δz ¼ 1 μm. Theother parameters are λ ¼ 1:55 μm and n ¼ 1:46. ThePhC lens consists of a 12 × 17 hole array of period a ¼0:25 μm with the hole diameters ranging from 186 to250nm. Shown in Fig. 4(a) in gray is the waveguide,and white is the air. Figure 4(b) depicts the instan-taneous amplitude distribution Ex of the TE wave,calculated in FullWAVE 6.0 for the arrangementin Fig. 4(a). Figure 4(c) depicts the coupling efficiencyη (the ratio of the output power of the narrow wave-guide to the input power of the wide waveguide) as a
function of interwaveguide spacingΔz. It can be seenfrom Fig. 4(c) that the maximal coupling intensity of73% is attained when the interwaveguide spacing is0:6 μm. Note that the interwaveguide spacing is filledwith the waveguide material rather than air.
Figure 4(d) depicts the coupling efficiency for thearrangement in Fig. 4(a) as a function of the outputwaveguide width W2 when the interwaveguide spa-cing is Δz ¼ 1 μm. From Fig. 4(d), the couplingefficiency is seen to increase nearly linearly with in-creasing width W2 of the output waveguide.
3. Fabrication of Two 2D Waveguides Coupled with aPhC Lens
The devices were fabricated in SOI (220 μm thick sil-icon on 2 μm of buried oxide). The pattern was ex-posed in ZEP520A resist (at a voltage of 30kV) usinga hybrid ZEISS GEMINI 1530/RAITH ELPHY elec-tron beam lithography system and developed usingxylene with ultrasonic agitation. The pattern wastransferred into the silicon using reactive ion etching(RIE) in a mixture of gases CHF3 and SF6. Theprocess is based on that of [38] The holes in thePhC lens had diameters ranging from 160 to200nm and were completely etched through the sili-con layer. The total length of the sample (the lengthof two waveguides) was 5mm. Several similar struc-tures with different values of interwaveguide spacing(Δz ¼ 0, 1, and 3 μm) and different mutual location ofthe waveguide axes (Δx ¼ 0, �0:5, and �1 μm) werefabricated simultaneously on the same substrate.Figure 5 shows a magnified (x7000) SEM (top-view)image of two fabricated waveguides located Δz ¼1 μm and combined with the PhC lens. The sample
Fig. 5. SEM (x7000) photograph of two waveguides fabricated in the Si film and coupled with the PhC lens.
in Fig. 5 has the following parameters. The designedwidth of the waveguides is W1 ¼ 4:5 μm and W2 ¼1 μm, and the PhC lens comprises a 12 × 17 hole ar-ray of period 250nm. The sensitivity to fabricationerrors of the device is similar to that reported in[39] and is on the order 0:3GHz=nm.
4. Characterization of Two Waveguides Combinedwith a PhC Lens
Figure 6 depicts an optical setup for studying thetransmission spectrum of two waveguides combinedwith a PhC lens. A broadband light emitting diode(LED) light source (1450–1700nm) is connected toan optical fiber. At the fiber output the light is colli-mated and TE polarized. Then, the light is focusedwith a micro-objective onto the surface of the inputwaveguide.A second microlens put at the narrow waveguide
output focuses the light onto the end of a multimodeoptical fiber connected with an optical spectrumanalyzer (OSA). The total coupling efficiency isapproximately 1%. Figure 7 shows the normalizedemission spectrum I=Imax (atomic units) of a sourcethat has a maximum at 1:55 μm. The intensity unitsare arbitrary. We give here the emission spectrum ofthe source to make it easier to understand the form ofthe experimental curves.Figure 8 depicts the normalized spectra of the
transmitted light I with respect to the intensity ofthe light source Imax of the samples under study inthe range 1:5–1:6 μm for the following values of inter-waveguide spacingΔz: (a) 0 μm (curve 1), 1 μm (curve2), and 3 μm (curve 3), where off-axis displacements
Δx are with respect to the output waveguide opticalaxis; (b) 0 μm (curve 1), −0:5 μm (curve 2), þ0:5 μm(curve 3), −1 μm (curve 4), and þ1 μm (curve 5).Figure 8(a) (curve 1) suggests that the transmissionspectrum has four local maxima approximately at1535, 1550, 1565, and 1590nm. Note that two ofthe maxima (at wavelengths of 1550 and 1565nm)show the intensity three times larger than for theother two. This is most likely because, at these wave-lengths, the intensity of the source (Fig. 7) is severaltimes smaller.
We find it difficult to account for the local maximain the normalized transmission spectrum (Fig. 8). Itdoes not have any connection with the Fabry–Perotcavity inside the device in Fig. 5, because a 15–20nmperiod can arise at a cavity length of about 30 μm,while in our case the PhC-lens thickness is just3 μm. Though, it is possible that some fabrication er-rors lead to the appearance somewhere of a 30 μmFabry–Perot cavity. Likewise, there is no connectionbetween the local maxima and the longitudinal shiftof the focal spot because of changing incident wave-length (longitudinal chromatic aberration) [40], sincein [40] the maximal focal intensity was shown to bereplaced by the minimal intensity when the lightwavelength changed by 125nm. This value is 10times larger than in our case. It has been shown [39]
Fig. 6. (Color online) Optical setup used for studying the nano-photonic devices (comprising a pair of waveguides combined witha PhC lens).
Fig. 7. Emission spectrum of the light source.
Fig. 8. (Color online) Normalized spectrum of the transmitted light with respect to the intensity of the light source measured using anoptical setup of Fig. 6 for the samples under study shown in Fig. 5, for the following values of the interwaveguide spacing: (a) Δz ¼ 0 μm(curve 1),Δz ¼ 1 μm (curve 2), andΔz ¼ 3 μm (curve 3), with (b) the following off-axis displacements with respect to the output waveguideoptical axis: Δx ¼ 0 μm (curve 3), Δx ¼ −0:5 μm (curve 2), Δx ¼ þ0:5 μm (curve 4), Δx ¼ −1 μm (curve 5), and Δx ¼ þ1 μm (curve 1).
that changing the hole diameters in the PhC wave-guide by 20nm leads to a 50nm shift of the centralwavelength (1550nm). In our case, the PhC lens(Fig. 5) consists of holes of three different diameters(160, 180, and 200nm). Therefore, according to [39],the local intensity maxima (Fig. 8) are expected toshift by 50–100nm in our experiment.When the interwaveguide spacing is increased to
Δz ¼ 1 μm [Fig. 8(a), curve 2), the transmission spec-trum preserves its structure on average, but the localpeaks are reduced and shifted towards the “red”spectrum region. For still larger interwaveguidespacing of Δz ¼ 3 μm [Fig. 8(a), curve 3), not onlydo the local peaks become lower but they acquire the“blue” shift as well. The “red” shift is about 10nm,and the “blue” shift is also about −10nm (for the peaknear the central wavelength of 1:55 μm). As seenfrom Fig. 8(b), when the output waveguide is dis-placed by 1 μm from the optical axis (curves 1 and5) the output intensity is 8 times decreased (wave-length 1:55 μm). This means that the focal spot sizeproduced by the PhC lens is less than 1 μm.
5. Conclusions
We have obtained the following results:
• We have fabricated a SOI 2D photonic crystallens of size 3 × 4 μm at the output of a planar wave-guide of width 4:5 μm for coupling with a planarwaveguide of width 1 μm, with its input near the fo-cus of the lens.• A 1 μm off-axis displacement of the narrow
waveguide resulted in an 8-fold decrease in the out-put light intensity, implying that the focal spot size atthe output of the in-Si lens was less than 1 μm (λ ¼1:55 μm).• The simulation has shown that the photonics
device proposed offers maximal transmittance at awavelength of 1:55 μm, providing the coupling effi-ciency between the waveguides of 73%.• The measured transmission spectrum has been
shown to have four local peaks in the range1:50–1:60 μm: 1535, 1550, 1565, and 1590nm; thesepeaks become lower, being displaced towards the“red” or “blue” region with increasing interwaveguidespacing.• The focal spot size of the PhC lens in the air
calculated at half-intensity level is FWHM ¼ 0:32λ,where λ is the wavelength, which is less than in[21,24,28,36,39].
This work was financially supported by the U.S.Civilian Research and Development Foundation(CRDF), the Russian–American program “Basic Re-search and Higher Education” (grant PG08-014-1),the Russian Foundation for Basic Research (grant08-07-99007), and the Russian Federation (RF)Presidential Grant for the leading scientific school(NSh-3086.2008.9). The devices were fabricated inthe framework of the ePIXnet (EU-FP6 NoE) Nano-structuring Platform for Photonic Integration (http://www.nanophotonics.eu)
References
1. Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling betweenconventional dielectric waveguides with discrete translationalsymmetry,” Opt. Lett. 25, 755–757 (2000).
2. A. Mekis and J. D. Joannopoulos, “Tapered couplers forefficient interfacing between dielectric and photoniccrystal waveguides,” J. Lightwave Technol. 19, 861–865(2001).
3. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapersfor ultracompact mode conversion,” Opt. Lett. 26, 1102–1104(2001).
4. A. Talneau, P. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic crystal taper for efficient coupling betweenguide sections of arbitrary widths,” Opt. Lett. 27, 1522–1524(2002).
5. V. R. Almeida, R. R. Panepucci, and M. Lipson, “Nanotaper forcompact mode conversion,” Opt. Lett. 28, 1302–1304(2003).
6. P. Bienstman, S. Assefa, S. G. Johson, J. D. Joannopoulos,G. S. Petrich, and L. A. Koloziejski, “Taper structures forcoupling into photonic crystal slab waveguides,” J. Opt. Soc.Am. B 20, 1817–1821 (2003).
7. S. J. McNab, N. Moll, and Y. A. Vlasov, “Ultra-low loss photonicintegrated circuit with membrane-type photonic crystal wave-guide,” Opt. Express 11, 2927–2939 (2003).
8. P. E. Barclay, K. Srinivasan, and O. Painter, “Design of photo-nic crystal waveguide for evanescent coupling to optical fibertapers and integration with high-Q cavities,” J. Opt. Soc. Am.B 20, 2274–2284 (2003).
9. R. Orobtchouk, A. Layadi, H. Gualous, D. Pascal, A. Koster,and S. Laval, “High-efficiency light coupling in a submicro-metric silicon-on-insulator waveguide,” Appl. Opt. 39, 5773–5777 (2000).
10. S. Lardenois, D. Pascal, L. Vivien, E. Cassan, S. Laval,R. Orobtchouk, M. Heitzmann, N. Bonzaida, and L. Mollard,“Low-loss submicrometer silicon-on-insulator rib waveguidesand corner mirrors,” Opt. Lett. 28, 1150–1153 (2003).
11. D. Taillaert, F. Van Laere, M. Ayre, W. Bogaerts, D. VanThourhout, P. Bienstman, and R. Baets, “Grating couplersfor coupling between optical fiber and nanophotonic wave-guides,” Jpn. J. Appl. Phys. 45, 6071–6077 (2006).
12. F. Van Laere, G. Roelkens, M. Ayre, D. Taillaert, D. VanThourhout, T. F. Krauss, and R. Baets, “Compact and high ef-ficient grating couplers between optical fiber and nanopho-tonic waveguides,” J. Lightwave Technol. 25, 151–156 (2007).
13. B. L. Bachim, O. O. Ogunsola, and T. K. Gaylord, “Optical fi-ber-to-waveguide coupling using carbon-dioxide-laser-inducedlong-period fiber gratings,” Opt. Lett. 30, 2080–2082 (2005).
14. D. W. Prather, J. Murakowski, S. Shi, S. Venkataraman,A. Sharkawy, C. Chen, and D. Pustai, “High-efficiencycoupling structure for a single-line-defect photonic crystalwaveguide,” Opt. Lett. 27, 1601–1603 (2002).
15. H. Kim, S. Lee, B. O. S. Park, and E. Lee, “High efficiencycoupling technique for photonic crystal waveguides using awaveguide lens,” in Frontiers in Optics, 2003 OSA TechnicalDigest Series (Optical Society of America, 2003), paper MT68.
16. J. C. W. Corbett and J. R. Allington-Smith, “Coupling starlightinto single-mode photonic crystal fiber using a field lens,”Opt.Express 13, 6527–6540 (2005).
17. D. Michaelis, C. Wachter, S. Burger, L. Zschiedrich, andA. Brauer, “Micro-optical assisted high-index waveguidecoupling,” Appl. Opt. 45, 1831–1838 (2006).
18. G. Kong, J. Kim, H. Choi, J. E. Im, B. Park, V. Paek, andB. H. Lee, “Lensed photonic crystal fiber obtained by use ofan arc discharge,” Opt. Lett. 31, 894–896 (2006).
19. A. L. Pokrovsky and A. L. Efros, “Lens based on the use of left-handed materials,” Appl. Opt. 42, 5701–5705 (2003).
20. N. Fabre, S. Fasquel, C. Legrand, X. Melique, M. Muller,M. Francois, O. Vanbesien, and D. Lippens, “Toward focusingusing photonic crystal flat lens,” Opto-Electron. Rev. 14,225–232 (2006).
21. C. Li, M. Holt, and A. L. Efros, “Far-field imaging by theVeselago lens made of a photonic crystal,” J. Opt. Soc. Am.B 23, 490–497 (2006).
22. T. Matsumoto, K. Eom, and T. Baba, “Focusing of light bynegative refraction in a photonic crystal slab superlens onsilicon-on-insulator substrate,” Opt. Lett. 31, 2786–2788(2006).
23. C. Y. Li, J. M. Holt, and A. L. Efros, “Imaging by the Veselagolens based upon a two-dimensional photonic crystal with atriangular lattice,” J. Opt. Soc. Am. B 23, 963–968(2006).
24. T. Geng, T. Lin, and S. Zhuang, “All angle negative refractionwith the effective phase index of −1,” Chin. Opt. Lett. 5,361–363 (2007).
25. T. Asatsume and T. Baba, “Abberation reduction and uniquelight focusing in a photonic crystal negative refractive lens,”Opt. Express 16, 8711–8718 (2008).
26. N. Fabre, L. Lalonat, B. Cluzel, X. Melique, D. Lippens,F. deFornel, and O. Vanbesien, “Measurement of a flat lens fo-cusing in a 2D photonic crystal at optical wavelength,” in Con-ference on Lasers and Electro-Optics/Quantum Electronicsand Laser Science Conference and Photonic Applications Sys-tems Technologies, OSA Technical Digest (CD) (OpticalSociety of America, 2008), paper CTuDD6.
27. S. Yang, C. Hong, and H. Yang, “Focusing concave lens pho-tonic crystals with magnetic materials,” J. Opt. Soc. Am. A23, 956–959 (2006).
28. P. Luan and K. Chang, “Photonic crystal lens coupler usingnegative refraction,” PIERS Online 3, 91–95 (2007).
29. S. Haxha and F. AbdelMalek, “A novel design of photonic crys-tal lens based on negative refractive index,” PIERS Online 4,296–300 (2008).
30. Z. Lu, S. Shi, C. A. Schuetz, J. A. Murakowski, and D. Prather,“Three-dimensional photonic crystal flat lens by full 3D nega-tive refraction,” Opt. Express 13, 5592–5599 (2005).
31. Z. Lu, S. Shi, C. A. Schuetz, and D. W. Prather, “Experimentaldemonstration of negative refraction imaging in both ampli-tude and phase,” Opt. Express 13, 2007–2012 (2005).
32. I. V. Minin, O. V. Minin, Y. R. Triandafilov, and V. V. Kotlyar,“Subwavelength diffractive photonic crystal lens,” Prog. Elec-tromagn. Res. B 7, 257–264 (2008).
33. J. P. Hugonin, P. Lalanne, T. P. White, and T. F. Krauss, “Coupling into a low-mode photonic crystal waveguide,” Opt.Lett. 32, 2638–2640 (2007).
34. E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin,A. Chipoline, and A. Tunnermann, “Photonic crystal lensfor photonic crystal waveguide coupling,” in Lasers andElectro-Optics and 2006 Quantum Electronics and LaserScience Conference, Technical Digest (CD) (Optical Societyof America, 2006), CThK3.
35. Ya. R. Triandafilov and V. V. Kotlyar, “A photonic crystalMikaelian lens,” Computer Opt. 31, 27–31 (2007) (in Russian).
36. Y. R. Triandafilov and V. V. Kotlyar, “Photonic crystal Mikae-lian lens,” Opt. Mem. Neural Netw. 17, 1–7 (2008).
37. A. L. Mikaelian, “Harnessing medium properties for wave fo-cusing,” Dokl. Akad. Nauk SSSR 81, 2406–2415 (1951) (inRussian).
38. L. O’Faolain, X. Yuan, D. McIntyre, S. Thoms, H. Chong,R. M. De La Rue, and T. F. Krauss, “Low-loss propagationin photonic crystal waveguides,” Electron. Lett. 42,1454–1455 (2006).
39. D. M. Beggs, L. O’ Faolain, and T. F. Krauss “Accurate deter-mination of the functional hole size in photonic crystal slabsusing optical methods”, Photon. Nanostr. Fundam. Appl. 6,213–218 (2008).
40. Q. Wu, J. M. Gibbons, and W. Park “Graded negative indexlens by photonic crystal”, Opt. Express 16, 16941–16949(2008).
