Abstract

We report on the fluid tunable transition from trapping to discrete diffraction in planar polymer waveguide arrays. A novel optofluidic polymer waveguide array platform was engineered to allow a wavelength dependent transition from a localised state where light is trapped in a defect mode to delocalised state where light is spreading through discrete diffraction. The spectral location of this transition can be controlled through a variation of the fluid’s refractive index. The platform is compatible with aqueous solutions, making it an interesting candidate for an integrated refractive index sensor to perform label-free biosensing.

© 2013 osa

1. Introduction

The combination of microfluidics and optics is receiving increasing attention and has become established as a research area called optofluidics [1]. The interaction of integrated optics with liquids enables the realisation of highly tunable micro-optic devices [2] as well as microfluidic sensors with extraordinary sensitivities [3]. Optical sensing of fluids can enable the quick and accurate analysis of a fluid’s properties and has found application in monitoring of the environment and food production, chemical and biological threat detection as well as medical diagnostics [4].

Coupled waveguide arrays, or photonic lattices, have become an area of active research interest due to their extraordinary capabilities to manipulate light propagation. The discrete character of coupled waveguide arrays enables the control of light propagation in peculiar ways [5, 6]. About four decades ago, it was shown by Somekh et al. [7] that light coupled into a homogenous waveguide array diffracts in the form of two major side lobes radiating away from the input waveguide. This phenomenon, called discrete diffraction, can be used to control the lateral spreading of light via the coupling between the individual waveguides in the array [8]. The periodicity of the array can be broken to control the guiding behaviour within the array [9]. A structural defect waveguide can be introduced into the array to allow the formation of guided defect modes [10]. Light can be trapped in the defect waveguide, even if the surrounding array exhibits a higher effective index [10]. This extra-ordinary guiding behaviour offers interesting prospects in conjunction with tunable and reconfigurable waveguides.

The use of fluids to tune infiltrated waveguide arrays has been shown mainly to form discrete solitons in the non-linear regime [11]. Tunable diffraction and self-trapping via slow nonlinearities was first shown in two-dimensional fluid-infiltrated photonics crystal fibres [11], and then extended to a one-dimensional platform through selective infiltration [12]. Recently, tunable discrete diffraction was demonstrated for fast nonlinearities [13]. A demonstration of a temperature induced refractive index change to alter discrete diffraction was shown in linear planar fluid-infiltrated waveguide arrays [14] and within two-dimensional infiltrated fibres [15]. A change in discrete diffraction based on a voltage induced fluid index change has been demonstrated in planar liquid crystal infiltrated waveguide arrays [16].

In this paper, we propose the use of fluid-interfaced polymer waveguide arrays to enable the tunable control of light propagation. We demonstrate theoretically and experimentally the fluid index and wavelength dependent transition from trapping in a defect mode to discrete diffraction through the surrounding waveguide array. Careful engineering of a shielded defect waveguide in the fluid-interfaced waveguide array allows the tunable confinement of light in a defect mode even though it exhibits lower effective index than its surrounding medium. The optofluidic platform can be used to detect small changes in the refractive index of the fluid cladding. The platform is compatible with low index aqueous fluids and thus is interesting candidate for label-fee biosensing. In addition, the device could be employed as tunable wavelength selective switch.

2. Transition of trapping to discrete diffraction in coupled waveguide arrays

In a homogenous waveguide array, light couples from waveguide to waveguide and spreads in form of two distinct side lobes as it propagates along the array [7]. This behaviour is called discrete diffraction and can be controlled through the interaction of the individual waveguides. The development of modal amplitude An of the nth waveguide can be described via coupled mode theory:

(iddz+β0)An+C(An1+An+1)=0
where β0 is the propagation constant of one isolated array waveguide and C is the coupling coefficient between these waveguides in the array. The allowed propagation states in an infinite waveguide array form distinct propagation bands. Light is only allowed to propagate through the array if its propagation constant lies within one of these bands. If the centre position of first band is at the propagation constant β0 then the upper (βU) and lower (βL) band edge can be described according to [8]:
βU=β0+2CβL=β02C

For light to be guided in a single waveguide, without lateral spreading, it must be trapped in a defect mode. The propagation constant of this defect mode must be situated outside of the propagation band of the array. The propagation constant of the defect mode either lies above βU or below βL. If the propagation constant lies above βU, then the defect mode is guided via total internal reflection since the adjacent areas exhibit lower propagation constants and the defect mode is called unstaggered [10]. If the propagation constant of the defect mode is below βL, it is guided through band gap effects and is called a staggered defect mode [10]. In practice, a defect waveguide can be realised through variation of the dimensions of one of the cores of the waveguide array.

Consider an array as illustrated in Fig. 1(a). The top cladding of the array is a fluid with varying index. A defect waveguide is located in the centre of the array. At a particular wavelength, the waveguide array is engineered such that for low fluid indices (nfl), the isolated defect index is within the allowed propagation band of the array. Hence no defect mode exists and any optical power launched into the defect waveguide will radiate via discrete diffraction. The array waveguides interact strongly with this fluid, while the defect waveguide is isolated from the fluid. Thus, if the fluid index is increased, the propagation band of the array will rise, while the defect index remains constant. At a particular fluid index, the propagation band will move above with the defect state, a guided staggered defect mode will start to exist and light will be trapped in the defect waveguide. This behaviour is illustrated in Fig. 1(a).

 

Fig. 1 Operation principle: a) Increase in defect waveguide’s output power due to a change from discrete diffraction to defect trapping upon increase of fluid index, b) drop in defect output power upon a change in wavelength, c) shift of wavelength dependent transition due to change in fluid index.

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A similar transition from staggered guiding in the defect to radiation should be observed with changing wavelength as illustrated in Fig. 1(b). The dispersion of the defect and array waveguides will be significantly different due to the low index of the fluid which is strongly coupled to the array but is only weakly coupled to the defect. As the wavelength is increased, the array waveguides will interact more strongly with the fluid and thus the index of the propagation band will drop significantly, while the index of the waveguide will drop to a much lesser degree. Hence, as the wavelength is increased, at a particular wavelength, the propagation band of the array and the index of the defect will coincide and the staggered mode of the defect will transition from guiding to radiating. This behaviour is illustrated in Fig. 1(b).

If these two effects are combined, then it would be expected that the wavelength at which the transition from guiding to radiation is observed would depend strongly on the index of the fluid as illustrated in Fig. 1(c). This concept seems a promising configuration for a refractive index sensor and is investigated in more detail in the upcoming sections.

3. Theoretical simulation of the waveguide array

Having described the general concept, a practical waveguide structure implementing this concept is proposed and simulated numerically. The simulation is used to test if a transition between trapping and discrete diffraction can be predicted and how this transition depends on both the index of the fluid cladding and the wavelength of light passing through the defect.

3.1. Proposed waveguide array structure

The proposed waveguide array consists of 49 waveguides with a defect waveguide at its center. The array is covered with fluid. The cladding has openings to allow the fluid to be in direct contact with all of the waveguide cores with the exception of the defect core. Figures 2(a) and 2(b) illustrate top and cross-section of waveguide array, respectively. The array is 21 mm long and has 15 mm long openings of the top cladding. This allows an overlap of the evanescent field with the fluid. The defect waveguide does not have a cladding opening and its evanescent field is shielded from the fluid. The cores are made using photo-curable epoxy polymers SU-8 (ncore = 1.572 at 1.55 μm wavelength) and are enclosed in KMPR (nclad = 1.547 at 1.55 μm wavelength). Seven refractive index liquids from Cargille’s Series AAA are available for use as the fluid cladding. The liquids have refractive indices of nfl = 1.325, 1.335, 1.344, 1.354, 1.363, 1.373 and 1.383 at a wavelength of 1.55 μm. Figure 2(b) shows a schematic of the cross-section of the waveguide array and the dimensions used. Modal and beam propagation analysis were carried out to model the described transition between defect mode trapping and discrete diffraction.

 

Fig. 2 a) Top view schematic of the waveguide array. The blue sections indicate 15 mm long fluid inlets to the waveguide cores. At the 3 mm long endings the cores are covered with a KMPR cladding. b) Cross-section of the shielded defect waveguide in the centre of the array and the array waveguide cores with fluid openings.

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The propagation constant of the isolated defect and the propagation band of the array were simulated using a mode solver based on finite element method (FEM) [17]. All waveguides were designed to be single mode and TE polarised (with the electric field horizontally oriented). The propagation band of the array was determined using Eq. (2). Since all waveguides are identical in a homogenous waveguide array, the coupling coefficient can be determined by analysing the field overlap between two individual waveguides, arbitrarily named Waveguide 1 and Waveguide 2, according to [18]:

C=ωε04EwglΔεEwg2dA
where ω is the angular frequency, ε0 is the vacuum permittivity, Ewg1 and Ewg2 are the normalised field distributions of the two array waveguides, and Δε is the difference in relative permittivity between the case where both Waveguide 1 and Waveguide 2 are present and the case where only Waveguide 1 is present.

3.2. Fluid-dependent tuning of guiding behaviour

To predict the fluid dependent behaviour of the structure, the effective indices (neff) of the propagation band of the array and isolated defect were simulated with fluid index (nfl) varying from nfl = 1.30 to 1.40 with the wavelength fixed at 1.55 μm. The results are presented in Fig. 3(a). The blue shaded region shows the propagation band of the waveguide array as a function of the fluid’s refractive index. The black line shows the calculated propagation constant of an isolated defect waveguide whose evanescent field does not interact with the fluid. The position of the propagation band increases with the fluid index while the propagation constant of the defect waveguide remains constant. At low fluid indices, the propagation constant of the isolated defect waveguide lies within the propagation band and hence no defect mode should exist. At a fluid refractive index of nfl = 1.34 the propagation constant of the defect passes the lower band edge. A staggered defect mode is predicted to exist, enabling light trapping in the defect waveguide.

 

Fig. 3 Results of the fluid index dependent simulations: a) Effective index (neff) and propagation constant of isolated defect (black line), and propagation band of waveguide array at varying fluid index. b) Normalised output power of centre defect waveguide after propagation through array. The triangles illustrate the fluid indices of the Cargille liquids. Insets: Plan view of array as light propagates along for fluid indices of (I) 1.30, (II) 1.33, (III) 1.40.

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In order to better understand and visualise the transition between discrete diffraction and trapping in the defect mode, light propagation along the waveguide array was simulated using beam propagation method (BPM). Beamprop from Rsoft was used as simulation tool and the dimensions were equal to those in Fig. 2. A TE fundamental mode of an isolated waveguide at 1.55 μm was launched into the centre defect waveguide. The output power of the defect waveguide was monitored. Figure 3(b) shows the normalised output power of the defect waveguide after propagation through the array for fluid indices ranging from nfl = 1.30 to 1.40. The output power starts low with low fluid index and gradually raises as the fluid index increases. This index region were the change in output power occurs overlaps nicely with the fluid index at which the defect waveguide’s propagation constant intersects with the lower band edge, as shown in Fig. 3(a). The increase in output power at intersection with the lower band egde indicates that low fluid indices light spreads across the array due to discrete diffraction, while at larger indices, light is trapped in a staggered defect mode and remains confined in the defect waveguide. The insets in Fig. 3(b) show the plan view of the BPM as light is propagating along the array at a fluid index of nfl = 1.30 (I), 1.33 (II) and 1.40 (III), respectively. At a fluid index of 1.30, lateral spreading of light in the form of two main side lobes can be observed. This is a characteristic of discrete diffraction. At a fluid index of 1.33 the spreading is reduced and a larger amount of light is confined in the defect waveguide. At fluid index of 1.40 the light is completely confined in the defect waveguide, indicating the trapping in a staggered defect mode. The simulation results confirmed that for a fixed wavelength, a fluid tunable transition between trapped and discrete diffraction is possible.

3.3. Wavelength-dependent tuning of guiding behaviour

Now that a fluid index dependent change in guiding behaviour has been predicted, the influence on change in wavelength is analysed. As described in Section 2, due to the difference in geometry of defect and array waveguide cores, it is likely the propagation band of the array and the isolated defect mode exhibit a different dispersion. Thus a wavelength dependent transition between the light trapping in a staggered defect mode and spreading via discrete diffraction should be possible. In order to verify this wavelength dependent transition, the effective index of the defect core and the propagation band was calculated for varying wavelengths using the FEM mode solver, as described in Section 3.1.

Figure 4(a) shows the effective index of the isolated core (black solid curve) and propagation band (orange shaded region) covered with a fluid of nfl = 1.363 as function of wavelength. At low wavelengths, the defect mode exhibits a smaller effective index than the lower band edge. When the wavelength is increased, the effective index of the defect mode decreases along with the lower band edge of the array. However, the defect waveguide effective index decreases with wavelength at a much slower rate than the lower band edge of the array. At a wavelength of 1.576 μm the effective index of the isolated defect waveguide coincides with the lower edge of the band. Hence an increase in wavelength will move the array from a state where light is trapped to a state where light spreads via discrete diffraction.

 

Fig. 4 Results of wavelength dependent simulations: a) Effective index (neff) of isolated defect waveguide (black), and propagation band of array covered with Cargille Series AAA 1.37 liquid (nfl = 1.363) at varying wavelength. Inset: Wavelength at which the isolated defect waveguide index intersects the lower band edge for the Cargille fluid indices. b) Normalised output power of defect waveguide for the array covered with Cargille fluids.

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As reasoned in Section 3.1, the location of the transition from trapping to discrete diffraction should depend on the index of the fluid cladding of the array. To test this hypothesis, the simulation of Fig. 4(a) was repeated for all seven Cargille liquids. The points in the insert of Fig. 4(a) show the wavelength where the propagation constant of the isolated defect waveguide intersects with the lower band edge. The intersection wavelength increases with an increase fluid index as expected.

To further understand and illustrate the wavelength dependent transition and its change with fluid index, BPM simulations were carried out for each of the seven Cargille liquids. In these BPM simulations, for each fluid index, the wavelength was swept from 1.51 μm to 1.64 μm and the power of the defect waveguide was monitored. Figure 4(b) shows the output power from the defect as function of wavelength for the different fluid indices. At short wavelengths, a large portion of power is observed at the output of the defect waveguide. With increasing wavelength the power output on the defect gradually drops until it is completely diminished. With an increase in fluid index this behaviour shifts to longer wavelengths.

From Fig. 4(b) it can be seen that when the fluid index shift from nfl = 1.325 to 1.335, the transition wavelength shifts from 1.541 to 1.551 μm. This corresponds to a sensitivity of 991 nm per refractive index unit (RIU). However, with a shift from nfl = 1.373 to 1.383, the transition wavelength shifts from 1.594 to 1.608 μm indicating a sensitivity of 1353 nm/RIU. This predicted increase in sensitivity can be explained due to the stronger overlap between the light in the array waveguides and the fluid cladding as the as the fluid index approaches the core index.

Based on the simulations, we can conclude that our assumptions for the wavelength and fluid dependent transition between lateral spreading of light via discrete diffraction and light confinement in a staggered defect mode are valid for the chosen dimensions.

4. Realisation and experimental demonstration

The modelling carried out in Section 3 predicted that proposed geometry should support a wavelength and fluid tunable transition between discrete diffraction and and trapping of light in a defect mode. Now these predictions are investigated experimentally.

4.1. Fabrication and physical characterisation

Based on the parameters used in Section 3.1, the waveguide array was fabricated and tested. The fabrication was carried out using three step standard ultraviolet(UV) photolithography of KMPR [19] and SU-8 [20] epoxy polymers on a 3 inch silicon wafer. Firstly, a 35 μm-thick KMPR buffer layer was spin-coated, UV flood exposed and thermally cured. A 2.3 μm-thick SU-8-2002 layer was then spin-coated and patterned by UV exposure through a high resolution chrome mask, cured and developed to define the waveguide cores. A 7 μm-thick KMPR layer was spin-coated, patterned by UV exposure, cured and developed to reveal the openings to the SU-8 waveguide cores. After patterning, the wafer was separated into individual chips.

Figure 5 shows a SEM micrograph of the cross-section of the shielded defect waveguide core in the centre of the array and some of the 48 array cores with openings to the fluid. Owing to the low contrast between the SU8 and KMPR materials in the SEM image of Fig. 5(a), it is difficult to distinguish the cores of the array and the defect. The two dashed rectangles indicate the locations containing a defect and an array waveguide respectively and magnified and enhanced contrast images of the defect and array waveguides are presented in Fig. 5(b) and 5(c) respectively. The width of the openings to the waveguide cores are slightly narrower than the waveguide core width, but lies within the fabrication tolerances.

 

Fig. 5 a) SEM cross-section micrograph of openings to the waveguide array cores and shielded defect in the centre. Rectangles indicate location of SU-8 waveguide cores b) close up of the defect core and c) array core.

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4.2. Characterisation of optical transmission through defect

In order to experimentally test the guiding properties of the tunable waveguide array, TE polarised light from a tunable laser source (Agilent 81642A) was launched into the defect waveguide through a polarisation maintaining lensed fibre. The lensed fiber was brought into close proximity to the chip endface and aligned to the defect waveguide using a micro-positioning stage. The output endface of the defect waveguide was imaged onto a photodiode using 60x microscope objective. A drop of Cargille refractive index liquid was placed on the chip, where it filled the openings to waveguide cores. The laser wavelength was set to 1.51 μm and the location of the input lensed fibre was adjusted to maximize the light received at the output. After alignment, the laser wavelength was tuned from 1.51 μm to 1.64 μm in 1 nm steps and the output power was recorded. After the measurement of each fluid the sample was removed from the stage, cleaned in an isopropanol bath and placed back on the stage and re-aligned for the next measurement.

Figure 6 shows the measured output power of the defect waveguide as function of wavelength for Cargille Series AAA liquids with refractive indices of nfl = 1.325, 1.335, 1.344, 1.354, 1.363, 1.373 and 1.383. The output power was normalised to the laser output power. For each fluid index, the output power drops with an increase in wavelength. As the fluid index increases, the location of the transition shifts to longer wavelengths. If we select a power threshold of 0.5 of the maximum intensity, it is possible to quantify the sensitivity by dividing the wavelength shift by the change in refractive index that produces this output power. For a fluid index shift of nfl = 1.354 to 1.363, the wavelength for threshold power shifts from 1.5325 μm to 1.5414 μm giving a sensitivity of 989 nm/RIU. For nfl = 1.363 to 1.373 this sensitivity increases to 1022 nm/RIU and for nfl = 1.373 to 1.383 the sensitivity increases further to 1104 nm/RIU. This is in good agreement with the simulations of Section 3 which predicted a maximum sensitivity of 1353 nm/RIU.

 

Fig. 6 Measured output power of waveguide array covered with Cargille refractive index liquids, normalised to the laser output spectrum.

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The power levels recorded for liquids with nfl = 1.325, 1.335, 1.344 appear lower than the other liquids. This could be attributed to discrepancies in input fibre insertion loss during realignment as the fluids were changed and the sample was cleaned, however the trend of each response is clearly consistent. In comparison to the simulation, all of the measured curves are shifted to shorter wavelengths by about 40 nm. This offset can be explained by fabrication tolerances in dimensions and material refractive indices. Overall it can be concluded that the tested device performed similar to the predictions of the modal and BPM simulations.

The measured sensitivity of the device is comparable to advanced integrated biosensors [21], such as one-dimensional photonic crystals (23 nm/RIU) [22], slot waveguides (212 nm/RIU) [23], tunable grating couplers (142 nm/RIU) [24], Mach-Zehnder interferometers (1500 nm/RIU) [25] or integrated surface plasmon resonance sensors (2000 nm/RIU) [26]. The sensitivity is mainly limited by the evanescent field overlap with the fluid as well as the interaction length. Thus the sensitivity could be improved by utilising lower index polymers and by increasing the length of the sensor, respectively. It is expected that the sensor should be very robust to absorption in the fluid since light is transmitted to the output through a waveguide that is insulated from the fluid with any power coupled to the array considered as lost. Further, the coupling to the array is based on a fluid index tunable shift of the array propagation band, which will be dominated by the real part of the fluid refractive index.

4.3. Characterisation of optical distribution through array

In order to verify that the reduction in output power is related to a transition from defect trapping to discrete diffraction, the optical distribution at output endface was analysed. The photodiode in the setup was replaced with an infrared camera (Hamamatsu C2741). The output objective was changed to 5x magnification to observe the complete endface of the array.

Figure 7(a) displays a sequence of images obtained from the output endface of a waveguide array covered with Cargille Series AAA 1.37 refractive index liquid (nfl = 1.363) with the wavelength scanned from 1.51 to 1.64 μm. At short wavelengths most of the light is confined in the centre defect waveguide. A smaller portion of light is visible in the array waveguides. With increasing wavelength, the ratio of light in the defect waveguide to light in the other waveguides becomes smaller. The light spreads far more broadly to the edges of the waveguide array. The spreading occurs symmetrically.

 

Fig. 7 a) Infrared camera image of output cross-section of waveguide array covered with Cargille Series AAA 1.37 refractive index liquid (nfl = 1.363) at varying wavelengths and b) corresponding BPM simulations.

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The reduction of the ratio of light in the defect waveguide to light in the remaining array confirms the wavelength dependent transition from light being trapped in a defect mode to light being spread across the array. The spreading occurs in two side lobes which is characteristic for discrete diffraction. The overall reduction in intensity can be attributed to the decline of the laser output power at longer wavelengths.

Figure 7(b) shows the corresponding BPM simulation of the optical power expected at the end-point of the array. Experiment and simulation follow the same trend of light confinement at short wavelengths to lateral spreading long wavelengths. The transition occurs at about 40 nm shorter wavelength in the experiment than in simulation. This discrepancy is most likely related to variations in material constants and dimensions between those simulated and those practically realised. A characteristic sign of discrete diffraction is the formation two distinct side lobes with distinct dark nulls present in both experimental and simulated distributions.

5. Conclusion

We have presented a planar polymer fluid infiltrated waveguide array with an included defect that can exhibit optical trapping or discrete diffraction behaviour. The trapping or radiation behaviour of the defect is sensitive to both the optical wavelength and the index of the fluid suggesting the utility of the platform as a sensor. In particular, this introduced platform is compatible with aqueous fluids making it an attractive candidate for label free biosensing applications. Further investigation is currently focussing on non-uniform waveguide arrays and multiple defect waveguides to enhance sensitivity and enable more flexible operation.

Acknowledgments

We acknowledge discussions on the original concept with D. N. Neshev and F. Bennet at the Nonlinear Physics Centre, The Australian National University, Canberra, Australia. The support of the Australian Research Council through its Centers of Excellence program is gratefully acknowledged.

References and links

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2. H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics 5, 598–604 (2011) [CrossRef]  .

3. C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1, 106–114 (2007) [CrossRef]  .

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5. I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012) [CrossRef]  .

6. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003) [CrossRef]   [PubMed]  .

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9. J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett. 16, 2462–2464 (2009) [CrossRef]  .

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13. M. Vieweg, T. Gissibl, Y. V. Kartashov, L. Torner, and H. Giessen, “Spatial solitons in optofluidic waveguide arrays with focusing ultrafast Kerr nonlinearity,” Opt. Lett. 37, 2454–2456 (2012) [CrossRef]   [PubMed]  .

14. E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett. 23, 1564–1566 (2011) [CrossRef]  .

15. F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A 35, 3213–3215 (2010).

16. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-trapping in liquid crystals,” Opt. Express 13, 1808–1815 (2005) [CrossRef]   [PubMed]  .

17. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed (Wiley-IEEE, 2002).

18. A. Yariv, Optical Electronics in Mordern Communications, 5th ed (Oxford University, 1997).

19. A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid. 11, 93–104 (2011) [CrossRef]  .

20. H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng. 7, 121–124 (1997) [CrossRef]  .

21. M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev. 6, 463–487 (2012) [CrossRef]  .

22. W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron. 11, 11–16 (2005) [CrossRef]  .

23. C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett. 32, 3080 (2007) [CrossRef]   [PubMed]  .

24. S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem. 161, 721–727 (2012) [CrossRef]  .

25. A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip 10, 1167–1173 (2010) [CrossRef]   [PubMed]  .

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References

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  1. D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
    [CrossRef] [PubMed]
  2. H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics5, 598–604 (2011).
    [CrossRef]
  3. C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics1, 106–114 (2007).
    [CrossRef]
  4. X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics5, 591–597 (2011).
    [CrossRef] [PubMed]
  5. I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
    [CrossRef]
  6. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
    [CrossRef] [PubMed]
  7. S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
    [CrossRef]
  8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
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  9. J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
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  10. H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express11, 3404–3411 (2003).
    [CrossRef] [PubMed]
  11. C. R. Rosberg, F. Bennet, D. N. Neshev, and P. Rasmussen, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express15, 12145–12150 (2007).
    [CrossRef] [PubMed]
  12. F. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun.283, 4069–4073 (2010).
    [CrossRef]
  13. M. Vieweg, T. Gissibl, Y. V. Kartashov, L. Torner, and H. Giessen, “Spatial solitons in optofluidic waveguide arrays with focusing ultrafast Kerr nonlinearity,” Opt. Lett.37, 2454–2456 (2012).
    [CrossRef] [PubMed]
  14. E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
    [CrossRef]
  15. F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).
  16. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-trapping in liquid crystals,” Opt. Express13, 1808–1815 (2005).
    [CrossRef] [PubMed]
  17. J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed (Wiley-IEEE, 2002).
  18. A. Yariv, Optical Electronics in Mordern Communications, 5th ed (Oxford University, 1997).
  19. A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
    [CrossRef]
  20. H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
    [CrossRef]
  21. M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
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  22. W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
    [CrossRef]
  23. C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett.32, 3080 (2007).
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  24. S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
    [CrossRef]
  25. A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
    [CrossRef] [PubMed]
  26. J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
    [CrossRef]

2012 (4)

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
[CrossRef]

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

M. Vieweg, T. Gissibl, Y. V. Kartashov, L. Torner, and H. Giessen, “Spatial solitons in optofluidic waveguide arrays with focusing ultrafast Kerr nonlinearity,” Opt. Lett.37, 2454–2456 (2012).
[CrossRef] [PubMed]

2011 (4)

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics5, 598–604 (2011).
[CrossRef]

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics5, 591–597 (2011).
[CrossRef] [PubMed]

2010 (3)

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

F. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun.283, 4069–4073 (2010).
[CrossRef]

2009 (1)

J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
[CrossRef]

2007 (3)

2006 (1)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
[CrossRef] [PubMed]

2005 (2)

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-trapping in liquid crystals,” Opt. Express13, 1808–1815 (2005).
[CrossRef] [PubMed]

2003 (2)

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express11, 3404–3411 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

2001 (1)

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

1997 (1)

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

Alvarez, M.

M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
[CrossRef]

Amuli, I. A.

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

Assanto, G.

Barrios, C. A.

Belabas, N.

J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
[CrossRef]

Bennet, F.

F. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun.283, 4069–4073 (2010).
[CrossRef]

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

C. R. Rosberg, F. Bennet, D. N. Neshev, and P. Rasmussen, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express15, 12145–12150 (2007).
[CrossRef] [PubMed]

Bennet, F. H.

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

Bräuer, A.

Brynda, E.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Brzdakiewicz, K. A.

Carlson, J.

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

Casquel, R.

Cerullo, G.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Chrimes, A. F.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

Crespi, A.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Ctyroky, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

De La Rue, R. M.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

de Ridder, R. M.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Despont, M.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Domachuk, P.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics1, 106–114 (2007).
[CrossRef]

Dongre, C.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Dostálek, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Driessen, A.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Eggleton, B. J.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics1, 106–114 (2007).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

Estevez, M. C.

M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
[CrossRef]

Fahrni, N.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Fan, X.

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics5, 591–597 (2011).
[CrossRef] [PubMed]

Farnell, J.

F. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun.283, 4069–4073 (2010).
[CrossRef]

Fratalocchi, A.

Garanovich, I.

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

Garvin, H.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

Giessen, H.

Gilchrist, K.

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

Gissibl, T.

Grego, S.

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

Griol, A.

Gu, Y.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Gylfason, K. B.

Hawkins, A. R.

H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics5, 598–604 (2011).
[CrossRef]

Hoekstra, H. J.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Hoekstra, H. J. W. M.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Holgado, M.

Homola, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Hopman, W. C.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Hunsperger, R.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

Jin, J. M.

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed (Wiley-IEEE, 2002).

Kalantar-Zadeh, K.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Karpierz, M. A.

Kartashov, Y. V.

Kayani, A. A.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Khoshmanesh, K.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Kivshar, Y.

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

Krolikowski, W.

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

LaBianca, N.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Lambeck, P. V.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Lechuga, L. M.

M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
[CrossRef]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Bräuer, “Tailoring guided modes in waveguide arrays,” Opt. Express11, 3404–3411 (2003).
[CrossRef] [PubMed]

Levenson, J. A.

J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
[CrossRef]

Longhi, S.

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

Lorenz, H.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Michaelis, D.

Minot, C.

J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
[CrossRef]

Mitchell, A.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

Moison, J. M.

J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett.16, 2462–2464 (2009).
[CrossRef]

Monat, C.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics1, 106–114 (2007).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

Nekvindová, P.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Neshev, D.

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

Neshev, D. N.

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

C. R. Rosberg, F. Bennet, D. N. Neshev, and P. Rasmussen, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express15, 12145–12150 (2007).
[CrossRef] [PubMed]

Ngamsom, B.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Osellame, R.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Pertsch, T.

Peschel, U.

Pollnau, M.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Pottier, P.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Psaltis, D.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
[CrossRef] [PubMed]

Quake, S. R.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
[CrossRef] [PubMed]

Ramponi, R.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Rasmussen, P.

Renaud, P.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Rosberg, C. R.

Sánchez, B.

Schmidt, H.

H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics5, 598–604 (2011).
[CrossRef]

Schröfel, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

Sivan, V.

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Skalsky, M.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Škvor, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Sohlström, H.

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

Špirková, J.

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

Stoner, B.

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

Streppel, U.

Sukhorukov, A.

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

Torner, L.

Trompeter, H.

van den Vlekkert, H. H.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

van Lith, J.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Vettiger, P.

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

Vieweg, M.

Watts, P.

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

White, I. M.

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics5, 591–597 (2011).
[CrossRef] [PubMed]

Yang, C.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
[CrossRef] [PubMed]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

A. Yariv, Optical Electronics in Mordern Communications, 5th ed (Oxford University, 1997).

Yudistira, D.

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

Zeller, E.

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Appl. Phys. Lett. (1)

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett.22, 46–47 (1973).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

W. C. Hopman, P. Pottier, D. Yudistira, J. van Lith, P. V. Lambeck, R. M. De La Rue, A. Driessen, H. J. Hoekstra, and R. M. de Ridder, “Quasi-one-dimensional photonic crystal as a compact building-block for refractometric optical sensors,” IEEE J. Sel. Top. Quantum Electron.11, 11–16 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. Zeller, F. H. Bennet, D. N. Neshev, and A. Mitchell, “Laminated air structured and fluid infiltrated polymer waveguides,” IEEE Photon. Technol. Lett.23, 1564–1566 (2011).
[CrossRef]

J. Micromech. Microeng. (1)

H. Lorenz, M. Despont, N. Fahrni, N. LaBianca, P. Renaud, and P. Vettiger, “SU-8: A low-cost negative resist for MEMS,” J. Micromech. Microeng.7, 121–124 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

F. Bennet, I. A. Amuli, A. Sukhorukov, W. Krolikowski, D. Neshev, and Y. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” J. Opt. Soc. Am. A35, 3213–3215 (2010).

Lab Chip (1)

A. Crespi, Y. Gu, B. Ngamsom, H. J. W. M. Hoekstra, C. Dongre, M. Pollnau, R. Ramponi, H. H. van den Vlekkert, P. Watts, G. Cerullo, and R. Osellame, “Three-dimensional Mach-Zehnder interferometer in a microfluidic chip for spatially-resolved label-free detection,” Lab Chip10, 1167–1173 (2010).
[CrossRef] [PubMed]

Laser Photon. Rev. (1)

M. C. Estevez, M. Alvarez, and L. M. Lechuga, “Integrated optical devices for lab-on-a-chip biosensing applications,” Laser Photon. Rev.6, 463–487 (2012).
[CrossRef]

Microfluid. Nanofluid. (1)

A. A. Kayani, A. F. Chrimes, K. Khoshmanesh, V. Sivan, E. Zeller, K. Kalantar-Zadeh, and A. Mitchell, “Interaction of guided light in rib polymer waveguides with dielectrophoretically controlled nanoparticles,” Microfluid. Nanofluid.11, 93–104 (2011).
[CrossRef]

Nat. Photonics (3)

H. Schmidt and A. R. Hawkins, “The photonic integration of non-solid media using optofluidics,” Nat. Photonics5, 598–604 (2011).
[CrossRef]

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics1, 106–114 (2007).
[CrossRef]

X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics5, 591–597 (2011).
[CrossRef] [PubMed]

Nature (2)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature442, 381–386 (2006).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

F. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun.283, 4069–4073 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rep. (1)

I. Garanovich, S. Longhi, A. Sukhorukov, and Y. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep.518, 1–79 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85, 1863–1866 (2000).
[CrossRef] [PubMed]

Sens. Actuators B Chem. (2)

J. Dostálek, J. Ctyroky, J. Homola, E. Brynda, M. Skalsky, P. Nekvindová, J. Špirková, J. Škvor, and J. Schröfel, “Surface plasmon resonance biosensor based on integrated optical waveguide,” Sens. Actuators B Chem.76, 8–12 (2001).
[CrossRef]

S. Grego, K. Gilchrist, J. Carlson, and B. Stoner, “A compact and multichannel optical biosensor based on a wavelength interrogated input grating coupler,” Sens. Actuators B Chem.161, 721–727 (2012).
[CrossRef]

Other (2)

J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed (Wiley-IEEE, 2002).

A. Yariv, Optical Electronics in Mordern Communications, 5th ed (Oxford University, 1997).

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Figures (7)

Fig. 1
Fig. 1

Operation principle: a) Increase in defect waveguide’s output power due to a change from discrete diffraction to defect trapping upon increase of fluid index, b) drop in defect output power upon a change in wavelength, c) shift of wavelength dependent transition due to change in fluid index.

Fig. 2
Fig. 2

a) Top view schematic of the waveguide array. The blue sections indicate 15 mm long fluid inlets to the waveguide cores. At the 3 mm long endings the cores are covered with a KMPR cladding. b) Cross-section of the shielded defect waveguide in the centre of the array and the array waveguide cores with fluid openings.

Fig. 3
Fig. 3

Results of the fluid index dependent simulations: a) Effective index (neff) and propagation constant of isolated defect (black line), and propagation band of waveguide array at varying fluid index. b) Normalised output power of centre defect waveguide after propagation through array. The triangles illustrate the fluid indices of the Cargille liquids. Insets: Plan view of array as light propagates along for fluid indices of (I) 1.30, (II) 1.33, (III) 1.40.

Fig. 4
Fig. 4

Results of wavelength dependent simulations: a) Effective index (neff) of isolated defect waveguide (black), and propagation band of array covered with Cargille Series AAA 1.37 liquid (nfl = 1.363) at varying wavelength. Inset: Wavelength at which the isolated defect waveguide index intersects the lower band edge for the Cargille fluid indices. b) Normalised output power of defect waveguide for the array covered with Cargille fluids.

Fig. 5
Fig. 5

a) SEM cross-section micrograph of openings to the waveguide array cores and shielded defect in the centre. Rectangles indicate location of SU-8 waveguide cores b) close up of the defect core and c) array core.

Fig. 6
Fig. 6

Measured output power of waveguide array covered with Cargille refractive index liquids, normalised to the laser output spectrum.

Fig. 7
Fig. 7

a) Infrared camera image of output cross-section of waveguide array covered with Cargille Series AAA 1.37 refractive index liquid (nfl = 1.363) at varying wavelengths and b) corresponding BPM simulations.

Equations (3)

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

( i d d z + β 0 ) A n + C ( A n 1 + A n + 1 ) = 0
β U = β 0 + 2 C β L = β 0 2 C
C = ω ε 0 4 E w g l Δ ε E w g 2 d A

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