Abstract

We present a simple refractive index sensor based on a step-index fiber with a hollow micro-channel running parallel to its core. This channel becomes waveguiding when filled with a liquid of index greater than silica, causing sharp dips to appear in the transmission spectrum at wavelengths where the glass-core mode phase-matches to a mode of the liquid-core. The sensitivity of the dip-wavelengths to changes in liquid refractive index is quantified and the results used to study the dynamic flow characteristics of fluids in narrow channels. Potential applications of this fiber microstructure include measuring the optical properties of liquids, refractive index sensing, biophotonics and studies of fluid dynamics on the nanoscale.

©2011 Optical Society of America

1. Introduction

The accurate measurement of liquid refractive index (RI) is important in many fields of research and development. A wide variety of different structures have been proposed as RI sensors, for example long-period and Bragg fiber gratings [13], fiber-assisted surface plasmon resonances [4,5], photonic crystal structures [68], fiber couplers [9] and capillary ring and microdisk resonators [10, 11]. Silica-air photonic crystal fibers [12] have also been used in RI sensing, the hollow channels being filled with the sample fluid [1317].

In this paper we report a relatively simple, highly sensitive and low-cost RI sensor based on a step-index fiber with a single hollow channel running parallel to its core (Fig. 1a ). When a fluid of index greater than silica is introduced, a waveguide forms in the liquid-filled channel. Light couples from the glass-core to the resulting liquid-core at certain resonant wavelengths. Sensitivities of 3000 nm per unit refractive index (RIU) are achieved. The device also provides a convenient means of following the dynamics of fluid-flow in nanoscale channels, allowing accurate and rapid measurements of viscosity as well as RI. The device is fabricated by a comparably a simple fiber drawing procedure, does not require any post-processing steps, can be calibrated experimentally (i.e., it does not require any sophisticated numerical modelling techniques), works over a broad wavelength range and is straightforward to use.

 

Fig. 1 (a) Sketch of the device. The hollow channel (radius a F) close to the core (radius a C) is filled with fluid (green, filling length L F) and the centre-centre spacing between core and channel is d. (b) Representative modal indices (nn R) in the vicinity of an anti-crossing (the red and blue lines represent the uncoupled glass and liquid core modes and the black curves the supermodes), calculated using coupled mode theory [18]. (c) Typical plot of transmitted power P(L F) (linear scale) plotted versus wavelength at z = z C for Δn g = 0.0878. For a minimum detectable change δP min in P(L F) the minimum measurable wavelength change (caused by a change in liquid refractive index) is δλ min. The inset is a photograph of the near-field distribution at the output of the fluid-filled structure when broad-band supercontinuum light is launched into the glass core from the unfilled fiber end. The formation of a liquid meniscus distorts the image.

Download Full Size | PPT Slide | PDF

2. Device analysis

2.1 Operating principle

At a resonance the glass-core mode becomes synchronous with one of the guided modes in the liquid-core. Within a narrow wavelength range around this point the phase indices (i.e., effective modal indices) of the uncoupled modes can be linearised to a good approximation as follows:

nk(λ)nRngk(λλR)/λR=nRngkδλ/λR
where n R is the modal phase index at resonance, n gk are the group indices for the glass core (k → C) and fluid-filled channel (k → F), λ R is the resonant wavelength and δλ = λλ R is the detuning from resonance (see Fig. 1b). The intermodal dephasing rate ϑ (the rate at which the uncoupled glass and liquid core modes accumulate a phase difference) is then:
ϑ=βFβCΔng2πδλ/λR2
where βk = n k k 0 (k 0 is the vacuum wavevector) are the modal propagation constants for the uncoupled fluid and glass cores and Δn g = n gCn gF. The beat length between the two modes in the vicinity of the resonance is given by L B = 2π/ϑ. In the presence of coupling an anti-crossing forms at resonance, as illustrated in Fig. 1b where nk(λ) – n R are plotted against δλ, along with the refractive indices of the even and odd eigenmodes (“supermodes”) of the coupled system [18].

The normalized power p(z) in the glass core takes the well-known form [19]:

p(z)=P(z)/P0=1sin2(κz1+(ϑ/2κ)2)1+(ϑ/2κ)2
where P 0 is the launched power, z the distance from the input and κ the coupling constant. At zero dephasing (ϑ = 0) power is fully transferred to the liquid core after one coupling length z C = π/2κ. The spectral dependence of the transmitted power under these conditions is plotted in Fig. 1c.

2.2 Limits of measurement technique

Taking S = ∂λ R/∂n fl in nm per RIU to be the sensitivity of the resonant wavelength to changes in fluid index n fl is (S will be determined experimentally later), the minimum detectable refractive index change δn fl can be found by expanding p(z) in δλ up to second order, leading to the following expression (see Fig. 1c):

δnfl=δλminS=λR22LSπΔngD(2κL)δPminP0
where L is the total device length (L = L F/2 for a single pass, L = L F for a double pass), δP min is the smallest detectable change in transmitted power (limited by the detection system), and the detection function D, defined as:
D(2κL)=1/|(sinc2κLcos2κL)sinc2κL|,
reaches a global minimum D min = 1.55 at κL = 0.876 (Fig. 2 ). This is a consequence of the shape of the sinc functions at small values of argument. Equation (4) suggests the following figure-of-merit (smaller values being better) for the RI sensor:
FoM=λR22LSπΔng
showing that higher values of S and group index difference are advantageous. Larger channel diameters yield higher values of S because the fractional modal overlap with the liquid increases (inset of Fig. 4b ). Δn g can be increased by making the glass and fluid waveguides more dissimilar. For the device parameters in this paper, FoM = 2.9 × 10−7 yielding δn fl = 6.1 × 10−7 per square-root of the minimum detectable fractional power change at D = D min.

 

Fig. 2 Plot of the detection function D. It has a minimum value of 1.55 at κL = 0.876 and diverges at integral multiples of twice the coupling length. The yellow shaded region indicates over-coupling cycles. The device is most sensitive in the first coupling cycle. Inset: (red) wavelength dependence of the transmitted power in the vicinity of the HE21 resonance, calculated using Eq. (3) at the filling length of minimum detectable index change; (blue) corresponding quadratic approximation.

Download Full Size | PPT Slide | PDF

 

Fig. 4 (a) Measured resonance wavelengths (dip positions) versus the refractive index of the individual liquid for the three different modes (black/grey circles: HE21, red circles: TM01, green circles: TE01). The lines are linear fits to the experimental data using Eq. (1). (b) Comparison between the experimental points and the results calculated using the resonance condition. The inset is a theoretical plot of the sensitivity as a function of liquid core diameter for the HE21 mode at a fixed liquid RI of 1.6. The blue dot indicates the sensitivity in the experiments reported.

Download Full Size | PPT Slide | PDF

For optimum sensor performance, κL = 0.876 yields a device length of 7.4 mm using the experimentally determined value of κ = 119 m−1 (Sec. 3.4). Note that although the function D has multiple minima, one in each cycle of over-coupling, the best performance occurs in the first coupling cycle. The temperature dependence of the resonant wavelength is dominated by changes in the refractive index of the fluid (–4.26×10−4 K−1 for the Cargille liquid used compared to ~12.5×10−6 K−1 for silica) and works out at ~1.2 nm/K for the HE21 resonance (S = 2956 nm/RIU).

3. Experimental procedure

3.1 Sample fabrication

The fibre used in the experiments had a 17 mol% GeO2-doped silica core with a single hollow channel next to it. It was drawn from a 19-cell hexagonal array of silica capillaries [20], a commercial multimode graded-index fiber (core diameter 200 µm, cladding diameter 280 µm) being inserted into the central capillary. The entire stack was then drawn down to a preform cane, the conditions being chosen so that all the capillaries except one collapsed to solid rods. The remaining capillary, in the ring adjacent to the core, was kept open by judicious application of pressure during drawing. The preform cane was then placed inside a jacket tube and the whole structure drawn down to fiber. The radius of the core (aC) and the center-center core-channel spacing d are mainly controlled by the outer diameter of the fiber. The channel radius a F, on the other hand, could be independently adjusted by applying pressure during drawing. The sample used in the experiments had a F = 489 nm and d = 5.8 µm and the glass core (a C = 550 nm) remained single-mode down to 500 nm, which represents the shortest wavelength where reliable measurements could be made. Its long wavelength bend-edge was located at ~1000 nm. The total length of the fiber was 30 cm, so that one end could easily be dipped into liquid. Fluids (Cargille Laboratory) of RI between 1.5 and 1.66 (at a reference wavelength of 589.3 nm) were introduced into the hollow channel by suction. Lengths of L F ~1 cm could be filled within 10 minutes.

3.2 Optical measurements

Light from a PCF-based supercontinuum (SC) source (emitting from 400 to 1700 nm) was launched the fluid-filled fibre and the output recorded with an optical spectrum analyzer. A polarizer (P) and a half-wave plate (HW) were inserted between the SC source and the sample to allow excitation of one of the two orthogonally polarized eigenmodes. A typical optical near-field image of the modal intensity distribution is shown in Fig. 1c, showing light guided in the fluid-filled channel.

At 500 nm for n fl = 1.66 (the highest used) the fluid-filled core supports sixteen guided modes. Figure 3a shows the transmission spectra for two orthogonal states of polarization and n fl = 1.58. A transmission dip is observed at 690.8 nm for both polarizations. Two further dips (in the vicinity of 710 nm) appear, each associated with a different polarization state (y-polarization: 708 nm, x-polarization: 710.6 nm). A comparison of the uncoupled modes of the waveguide system, calculated assuming cylindrical step-index glass and liquid cores, reveals that as expected these dips occur at anti-crossings between the HE21, TM01 and TE01 modes of the fluid core and the glass core mode (silica refractive index taken from [21] for a Ge concentration of 17 mol%). In the vicinity of these anti-crossings, supermodes form through the overlap of the evanescent fields of the glass and liquid core modes, the coupling strength depending on polarization state. Considering the symmetries of the electric field patterns (Fig. 3b), the radially-polarized TM01 mode will couple most strongly to the x-polarized glass-core mode, while being decoupled from the y-polarized mode. The opposite is the case for the azimuthally-polarized TE01 mode. These predictions are confirmed by experiment, almost no polarization cross-coupling being observed. In contrast, the HE21 mode couples to both polarization states of the glass-core mode, with only a very weak dependence on input polarization. The insensitivity to polarization makes this mode ideal for RI sensing since no polarizing element is needed and the device will be insensitive to accidental birefringence.

 

Fig. 3 (a) Measured transmission spectra for a liquid of refractive index n fl = 1.58 at 598 nm for x (green curve) and y (blue curve) input polarization states. The inset shows the coordinate system (fluid core in yellow). Lower three dashed curves: Calculated dispersion of three modes of the fluid waveguide. The black solid line represents the glass core mode. (b) Calculated axial Poynting vector distributions (at the resonant wavelengths) for the HE21, TM01, and TE01 modes of the isolated (i.e., uncoupled) fluid-filled channel. The dashed white circle indicates the edge of the channel and the arrows the instantaneous local electric field.

Download Full Size | PPT Slide | PDF

3.3 Refractive index sensitivity

In Fig. 4a the wavelengths λ R of the transmission dips for 8 different fibres are plotted against the fluid index n fl (Fig. 3a). The dependence is approximately linear with n fl, allowing estimation of the sensitivity parameter S. The values of S for the three modes are 3183 nm/RIU (TE01), 3259 nm/RIU (TM01) and 2956 nm/RIU (HE21). These compare favorably with the values reported for micro-disk resonators (182 nm/RIU) [11], capillary ring resonators (800 nm/RIU) [10], photonic crystal fiber-based long period gratings (1500 nm/RIU) [15] and fiber-based surface plasmon devices (3365 nm/RIU) [4] (Higher values of S (30100 nm/RIU [16], 32400 nm/RIU [17]) have been observed in more sophisticated fiber devices.). Although the HE21 mode has a slightly lower sensitivity than the TE01 and TM01 modes, its polarization insensitivity makes it preferred for applications.

Figure 4b compares the experimental data (circles) with the calculated positions of the anti-crossing wavelengths (curves). For n fl less than ~1.59, theory and experiment agree quite well. At higher values of n fl, however, the predicted wavelengths diverge to larger values. This we attribute to a rounding off of the step-index profile of the Ge-doped glass core during redrawing, the result of dopant diffusion at high temperature [22]. Although this does not change the V-parameter of the waveguide (which remains single-mode [23]), it does increase the modal area, especially at longer wavelengths. This will not however affect device operation in practice, because the sensor must always be calibrated using reference liquids before making measurements on unknown liquids.

3.4 Dynamic measurements

When one fibre end is dipped into liquid, the channel gradually fills up by capillary action, causing the transmitted optical signal to change dynamically at each spectral dip. If the dipped fiber end is further coated with a reflecting gold layer (Fig. 5a ), the transmitted light can be reflected back to the input face, where a beam-splitter (BS) can be used to divert it, via a multi-mode fiber (MMF), into a fast optical spectrometer (SM) (Ocean Optics HR4000). The flow of liquid into the micro-channel can then be monitored as function of time, allowing estimation of parameters such as the viscosity and slip coefficient. If the liquid parameters are known, the coupling constant of the coupled waveguide system can be obtained directly from these dynamic measurements, allowing full calibration of the sensor without need for sophisticated numerical simulations. Spectra in the vicinity of the resonances were recorded at regular intervals (acquisition time < 60 ms) and the dip-depths were determined. Figure 5b shows the resulting time-dependence of (1−p(2LF)) for the HE21-mode in y-polarization for n fl = 1.58, each point corresponding to one spectral measurement. The dip-depth reaches a maximum after ~5 minutes.

 

Fig. 5 (a) Optical set-up for investigating the filling dynamics of liquids (b) Dip depth (1 − p(2L F)) as a function of filling time for the HE21 resonance (y-polarization, n fl = 1.58). The solid blue line is a fit using Eqs. (3) and 7. The inset depicts the simulated filling speed of the silica hole as function of time and the corresponding experimental data (green dots). The filling time in the experiment was 10 minutes (grey shaded region).

Download Full Size | PPT Slide | PDF

Since the Reynolds number is very small (~10−4), the flow of liquid in the hollow channel is laminar, obeying the Hagen-Poiseuille law [24]. Based on the assumptions in [25], it is straightforward to show that, in the absence of any external pressure, the filling length L F will follow the relationship as a function time t:

LF=RFσcosθ2ηt=CFt
where σ is the surface tension, θ the contact angle and η the dynamic viscosity. For the liquid used σ = 0.037 N/m, η = 0.0672 Pa·s and θ ≈15°, yielding C F = 0.36 mm·s−0.5 for a F = 489 nm. Equation (7) can be used to calculate the filling speed v F = dL F/dt for the test sample, as plotted in the inset of Fig. 5b. The channel fills up quickly within the first few minutes, the filling rate falling to 0.5 mm/min after 10 minutes. The filling rate needs to be slow enough to allow acquisition of a sufficient number of data points. This can be arranged by choosing a channel diameter that is neither too small (very slow filling) nor too large (too fast filling). For the liquid investigated, a diameter of ~1 μm (close to that used in the experiments) is a good compromise, yielding clear transmission resonances at high values of S.

Since the light makes a double-pass along the liquid-filled section, the filling length for full power transfer to the liquid core is π/2κ. With κ as a free parameter, the data points can be fitted to Eq. (7) using Eq. (3), as shown by the solid blue curve in Fig. 5b. There is fair agreement between experiment and theory for κ = 119 m−1, yielding a filling length for full coupling of 6.6 mm. The deviations we attribute to wavelength dependence in κ, which is not taken account of in this simple theory.

4. Conclusions

A step-index fiber with a parallel hollow micro-channel provides a versatile means of accurately measuring the RI of liquids. Its wide reconfigurability (glass core-cladding index-step, diameter of the glass core, width of micro-channel and its spacing from glass core) means that it can be used for a wide range of different refractive indices and fluid viscosities. The structure also provides a unique way of monitoring the flow of liquid in nanochannels (radii < 100 nm, much smaller than previously reported [1417], [26]), where direct observation of the liquid column is very difficult. It can be used to determine the refractive indices, the viscosity and slip coefficients of liquids. It is easy to use, as no fiber post-processing is needed and can be directly inserted into the liquid [27]. The sensor can be calibrated without any sophisticated simulation techniques, simply using the dispersions obtained from the fiber-step index model. The analysis does not require any sophisticated simulation techniques such as finite-element modelling, it simply relies on coupled step index waveguides. We anticipate applications in areas such as non-aqueous optofluidics (flow dynamics in nm-scale channels) and refractive index calibration. The device could also be used to measure very small thermo-optical coefficients by tracking the resonance detuning at the point of highest slope when changing the ambient temperature. The lowest measurable liquid RI in this simple single-channel device is ~1.45, limited by the need for a guided mode in the liquid core. A PCF with a glass core small enough to ensure that the modal index is less than 1.33 could be used for aqueous liquids, provided that only a single channel close to the core is filled with liquid.

References and links

1. W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005). [CrossRef]  

2. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996). [CrossRef]   [PubMed]  

3. X. Fang, C. R. Liao, and D. N. Wang, “Femtosecond laser fabricated fiber Bragg grating in microfiber for refractive index sensing,” Opt. Lett. 35(7), 1007–1009 (2010). [CrossRef]   [PubMed]  

4. T. Allsop, R. Neal, S. Rehman, D. J. Webb, D. Mapps, and I. Bennion, “Generation of infrared surface plasmon resonances with high refractive index sensitivity utilizing tilted fiber Bragg gratings,” Appl. Opt. 46(22), 5456–5460 (2007). [CrossRef]   [PubMed]  

5. L. Ma, T. Katagiri, and Y. Matsuura, “Surface-plasmon resonance sensor using silica-core Bragg fiber,” Opt. Lett. 34(7), 1069–1071 (2009). [CrossRef]   [PubMed]  

6. J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008). [CrossRef]  

7. T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009). [CrossRef]  

8. J. Jágerská, H. Zhang, Z. Diao, N. Le Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35(15), 2523–2525 (2010). [CrossRef]   [PubMed]  

9. H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007). [CrossRef]  

10. M. Sumetsky, R. S. Windeler, Y. Dulashko, and X. Fan, “Optical liquid ring resonator sensor,” Opt. Express 15(22), 14376–14381 (2007). [CrossRef]   [PubMed]  

11. J. Hu, N. Carlie, N. N. Feng, L. Petit, A. Agarwal, K. Richardson, and L. Kimerling, “Planar waveguide-coupled, high-index-contrast, high-Q resonators in chalcogenide glass for sensing,” Opt. Lett. 33(21), 2500–2502 (2008). [CrossRef]   [PubMed]  

12. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006). [CrossRef]  

13. R. Jha, J. Villatoro, G. Badenes, and V. Pruneri, “Refractometry based on a photonic crystal fiber interferometer,” Opt. Lett. 34(5), 617–619 (2009). [CrossRef]   [PubMed]  

14. M. C. Phan Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J. L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007). [CrossRef]   [PubMed]  

15. L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008). [CrossRef]   [PubMed]  

16. D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009). [CrossRef]   [PubMed]  

17. T. Han, Y. G. Liu, Z. Wang, B. Zou, B. Tai, and B. Liu, “Avoided-crossing-based ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 35(12), 2061–2063 (2010). [CrossRef]   [PubMed]  

18. A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 6th edition, 2006).

19. A. W. Snyder, and J. Love, Optical Waveguide Theory (Springer, 1st edition, 1983).

20. H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010). [CrossRef]   [PubMed]  

21. J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. 23(24), 4486–4493 (1984). [CrossRef]   [PubMed]  

22. J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

23. K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990). [CrossRef]  

24. E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921). [CrossRef]  

25. N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010). [CrossRef]  

26. K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

27. M. Vieweg, T. Gissibl, S. Pricking, B. T. Kuhlmey, D. C. Wu, B. J. Eggleton, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18(24), 25232–25240 (2010). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
    [Crossref]
  2. V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
    [Crossref] [PubMed]
  3. X. Fang, C. R. Liao, and D. N. Wang, “Femtosecond laser fabricated fiber Bragg grating in microfiber for refractive index sensing,” Opt. Lett. 35(7), 1007–1009 (2010).
    [Crossref] [PubMed]
  4. T. Allsop, R. Neal, S. Rehman, D. J. Webb, D. Mapps, and I. Bennion, “Generation of infrared surface plasmon resonances with high refractive index sensitivity utilizing tilted fiber Bragg gratings,” Appl. Opt. 46(22), 5456–5460 (2007).
    [Crossref] [PubMed]
  5. L. Ma, T. Katagiri, and Y. Matsuura, “Surface-plasmon resonance sensor using silica-core Bragg fiber,” Opt. Lett. 34(7), 1069–1071 (2009).
    [Crossref] [PubMed]
  6. J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
    [Crossref]
  7. T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
    [Crossref]
  8. J. Jágerská, H. Zhang, Z. Diao, N. Le Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35(15), 2523–2525 (2010).
    [Crossref] [PubMed]
  9. H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
    [Crossref]
  10. M. Sumetsky, R. S. Windeler, Y. Dulashko, and X. Fan, “Optical liquid ring resonator sensor,” Opt. Express 15(22), 14376–14381 (2007).
    [Crossref] [PubMed]
  11. J. Hu, N. Carlie, N. N. Feng, L. Petit, A. Agarwal, K. Richardson, and L. Kimerling, “Planar waveguide-coupled, high-index-contrast, high-Q resonators in chalcogenide glass for sensing,” Opt. Lett. 33(21), 2500–2502 (2008).
    [Crossref] [PubMed]
  12. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
    [Crossref]
  13. R. Jha, J. Villatoro, G. Badenes, and V. Pruneri, “Refractometry based on a photonic crystal fiber interferometer,” Opt. Lett. 34(5), 617–619 (2009).
    [Crossref] [PubMed]
  14. M. C. Phan Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J. L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007).
    [Crossref] [PubMed]
  15. L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008).
    [Crossref] [PubMed]
  16. D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009).
    [Crossref] [PubMed]
  17. T. Han, Y. G. Liu, Z. Wang, B. Zou, B. Tai, and B. Liu, “Avoided-crossing-based ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 35(12), 2061–2063 (2010).
    [Crossref] [PubMed]
  18. A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 6th edition, 2006).
  19. A. W. Snyder, and J. Love, Optical Waveguide Theory (Springer, 1st edition, 1983).
  20. H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
    [Crossref] [PubMed]
  21. J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. 23(24), 4486–4493 (1984).
    [Crossref] [PubMed]
  22. J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).
  23. K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
    [Crossref]
  24. E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
    [Crossref]
  25. N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
    [Crossref]
  26. K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).
  27. M. Vieweg, T. Gissibl, S. Pricking, B. T. Kuhlmey, D. C. Wu, B. J. Eggleton, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18(24), 25232–25240 (2010).
    [Crossref] [PubMed]

2010 (6)

2009 (4)

2008 (3)

2007 (5)

2006 (1)

2005 (2)

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

1996 (1)

1990 (1)

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

1984 (1)

1921 (1)

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[Crossref]

Agarwal, A.

Aizawa, Y.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Allsop, T.

Auguste, J. L.

Badenes, G.

Bang, O.

Bennion, I.

Bhatia, V.

Bjarklev, A.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Blanc, W.

Carlie, N.

Da, N.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

Day, D.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[Crossref]

Dellith, J.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Dewynter, V.

Diao, Z.

Dulashko, Y.

Dussardier, B.

Eggleton, B. J.

Fan, X.

Fang, X.

Feng, N. N.

Ferdinand, P.

Fleming, J. W.

Giessen, H.

Gissibl, T.

Granzow, N.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

Gu, M.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[Crossref]

Han, T.

Hansen, T. P.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Ho, W. D.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[Crossref]

Houdré, R.

Hsiao, Y. H.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[Crossref]

Hu, J.

Huang, Y.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Jágerská, J.

Jha, R.

Joly, N.

Kanie, T.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[Crossref]

Katagiri, T.

Katayama, M.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[Crossref]

Kawakawi, S.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Kimerling, L.

Kirchhof, J.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Knappe, B.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Kuhlmey, B. T.

Laffont, G.

Le Thomas, N.

Lee, H. W.

Lee, P. T.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[Crossref]

Lee, R. K.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Liang, W.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Liao, C. R.

Liu, B.

Liu, Y. G.

Lu, T. W.

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[Crossref]

Ma, L.

Mapps, D.

Matsuura, Y.

Neal, R.

Nielsen, K.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Noordegraaf, D.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Pagnoux, D.

Petit, L.

Phan Huy, M. C.

Pricking, S.

Pruneri, V.

Rehman, S.

Richardson, K.

Rindorf, L.

Roy, P.

Russell, P. St. J.

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[Crossref] [PubMed]

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
[Crossref]

Scharrer, M.

Schmidt, M. A.

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[Crossref] [PubMed]

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

Shiraishi, K.

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

Sørensen, T.

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Sumetsky, M.

Tai, B.

Tazawa, H.

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[Crossref]

Tyagi, H. K.

Uebel, P.

Unger, S.

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Vengsarkar, A. M.

Vieweg, M.

Villatoro, J.

Wang, D. N.

Wang, Z.

Washburn, E. W.

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[Crossref]

Webb, D. J.

Windeler, R. S.

Wondraczek, L.

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

Wu, D. C.

Wu, D. K. C.

Wu, J.

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[Crossref]

Xu, Y.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Yariv, A.

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

Zhang, H.

Zou, B.

Appl. Opt. (2)

Appl. Phys. Lett. (4)

W. Liang, Y. Huang, Y. Xu, R. K. Lee, and A. Yariv, “Highly sensitive fiber Bragg grating refractive index sensors,” Appl. Phys. Lett. 86(15), 151122 (2005).
[Crossref]

J. Wu, D. Day, and M. Gu, “A microfluidic refractive index sensor based on an integrated three-dimensional photonic crystal,” Appl. Phys. Lett. 92(7), 071108 (2008).
[Crossref]

T. W. Lu, Y. H. Hsiao, W. D. Ho, and P. T. Lee, “Photonic crystal heteroslab-edge microcavity with high quality factor surface mode for index sensing,” Appl. Phys. Lett. 94(14), 141110 (2009).
[Crossref]

H. Tazawa, T. Kanie, and M. Katayama, “Fiber-optic coupler based refractive index sensor and its application to biosensing,” Appl. Phys. Lett. 91(11), 113901 (2007).
[Crossref]

J. Lightwave Technol. (2)

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
[Crossref]

K. Shiraishi, Y. Aizawa, and S. Kawakawi, “Beam expanding fiber using thermal diffusion of the dopant,” J. Lightwave Technol. 8(8), 1151–1161 (1990).
[Crossref]

J. Non-Cryst Solid. (1)

N. Da, L. Wondraczek, M. A. Schmidt, N. Granzow, and P. St. J. Russell, ““High index-contrast all-solid photonic crystal fibers by pressure-assisted melt infiltration of silica matrices,” J. Non-Cryst Solid. 356, 1829–1836 (2010).
[Crossref]

J. Opt. B (1)

K. Nielsen, D. Noordegraaf, T. Sørensen, A. Bjarklev, and T. P. Hansen, “Selective filling of photonic crystal fibres,” J. Opt. B 7, L13–L20 (2005).

Opt. Express (2)

Opt. Lett. (11)

J. Hu, N. Carlie, N. N. Feng, L. Petit, A. Agarwal, K. Richardson, and L. Kimerling, “Planar waveguide-coupled, high-index-contrast, high-Q resonators in chalcogenide glass for sensing,” Opt. Lett. 33(21), 2500–2502 (2008).
[Crossref] [PubMed]

J. Jágerská, H. Zhang, Z. Diao, N. Le Thomas, and R. Houdré, “Refractive index sensing with an air-slot photonic crystal nanocavity,” Opt. Lett. 35(15), 2523–2525 (2010).
[Crossref] [PubMed]

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21(9), 692–694 (1996).
[Crossref] [PubMed]

X. Fang, C. R. Liao, and D. N. Wang, “Femtosecond laser fabricated fiber Bragg grating in microfiber for refractive index sensing,” Opt. Lett. 35(7), 1007–1009 (2010).
[Crossref] [PubMed]

L. Ma, T. Katagiri, and Y. Matsuura, “Surface-plasmon resonance sensor using silica-core Bragg fiber,” Opt. Lett. 34(7), 1069–1071 (2009).
[Crossref] [PubMed]

R. Jha, J. Villatoro, G. Badenes, and V. Pruneri, “Refractometry based on a photonic crystal fiber interferometer,” Opt. Lett. 34(5), 617–619 (2009).
[Crossref] [PubMed]

M. C. Phan Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J. L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007).
[Crossref] [PubMed]

L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008).
[Crossref] [PubMed]

D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009).
[Crossref] [PubMed]

T. Han, Y. G. Liu, Z. Wang, B. Zou, B. Tai, and B. Liu, “Avoided-crossing-based ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 35(12), 2061–2063 (2010).
[Crossref] [PubMed]

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. St. J. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010).
[Crossref] [PubMed]

Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B (1)

J. Kirchhof, S. Unger, B. Knappe, and J. Dellith, “Diffusion in binary GeO2–SiO2 glasses,” Phys. Chem. Glasses: Eur. J. Glass Sci. Technol. B 48, 129–133 (2007).

Phys. Rev. (1)

E. W. Washburn, “The dynamics of capillary flow,” Phys. Rev. 17(3), 273–283 (1921).
[Crossref]

Other (2)

A. Yariv, and P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 6th edition, 2006).

A. W. Snyder, and J. Love, Optical Waveguide Theory (Springer, 1st edition, 1983).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 (a) Sketch of the device. The hollow channel (radius a F) close to the core (radius a C) is filled with fluid (green, filling length L F) and the centre-centre spacing between core and channel is d. (b) Representative modal indices (nn R) in the vicinity of an anti-crossing (the red and blue lines represent the uncoupled glass and liquid core modes and the black curves the supermodes), calculated using coupled mode theory [18]. (c) Typical plot of transmitted power P(L F) (linear scale) plotted versus wavelength at z = z C for Δn g = 0.0878. For a minimum detectable change δP min in P(L F) the minimum measurable wavelength change (caused by a change in liquid refractive index) is δλ min. The inset is a photograph of the near-field distribution at the output of the fluid-filled structure when broad-band supercontinuum light is launched into the glass core from the unfilled fiber end. The formation of a liquid meniscus distorts the image.
Fig. 2
Fig. 2 Plot of the detection function D. It has a minimum value of 1.55 at κL = 0.876 and diverges at integral multiples of twice the coupling length. The yellow shaded region indicates over-coupling cycles. The device is most sensitive in the first coupling cycle. Inset: (red) wavelength dependence of the transmitted power in the vicinity of the HE21 resonance, calculated using Eq. (3) at the filling length of minimum detectable index change; (blue) corresponding quadratic approximation.
Fig. 4
Fig. 4 (a) Measured resonance wavelengths (dip positions) versus the refractive index of the individual liquid for the three different modes (black/grey circles: HE21, red circles: TM01, green circles: TE01). The lines are linear fits to the experimental data using Eq. (1). (b) Comparison between the experimental points and the results calculated using the resonance condition. The inset is a theoretical plot of the sensitivity as a function of liquid core diameter for the HE21 mode at a fixed liquid RI of 1.6. The blue dot indicates the sensitivity in the experiments reported.
Fig. 3
Fig. 3 (a) Measured transmission spectra for a liquid of refractive index n fl = 1.58 at 598 nm for x (green curve) and y (blue curve) input polarization states. The inset shows the coordinate system (fluid core in yellow). Lower three dashed curves: Calculated dispersion of three modes of the fluid waveguide. The black solid line represents the glass core mode. (b) Calculated axial Poynting vector distributions (at the resonant wavelengths) for the HE21, TM01, and TE01 modes of the isolated (i.e., uncoupled) fluid-filled channel. The dashed white circle indicates the edge of the channel and the arrows the instantaneous local electric field.
Fig. 5
Fig. 5 (a) Optical set-up for investigating the filling dynamics of liquids (b) Dip depth (1 − p(2L F)) as a function of filling time for the HE21 resonance (y-polarization, n fl = 1.58). The solid blue line is a fit using Eqs. (3) and 7. The inset depicts the simulated filling speed of the silica hole as function of time and the corresponding experimental data (green dots). The filling time in the experiment was 10 minutes (grey shaded region).

Equations (7)

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

n k ( λ ) n R n g k ( λ λ R ) / λ R = n R n g k δ λ / λ R
ϑ = β F β C Δ n g 2 π δ λ / λ R 2
p ( z ) = P ( z ) / P 0 = 1 sin 2 ( κ z 1 + ( ϑ / 2 κ ) 2 ) 1 + ( ϑ / 2 κ ) 2
δ n fl = δ λ min S = λ R 2 2 L S π Δ n g D ( 2 κ L ) δ P min P 0
D ( 2 κ L ) = 1 / | ( sin c 2 κ L cos 2 κ L ) sin c 2 κ L | ,
FoM = λ R 2 2 L S π Δ n g
L F = R F σ cos θ 2 η t = C F t

Metrics