
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71900
10.1038/s41598-024-71900-7
Article
Purcell gain equalized zero-mode waveguide
Liu Tang-Chun
Yu Wen-Hsiang
Tseng Chung-Kai
Thakur Diksha
Tai Chao-Yi cytai@dop.ncu.edu.tw

https://ror.org/00944ve71 grid.37589.30 0000 0004 0532 3167 Department of Optics and Photonics, National Central University, No. 300, Zhongda Rd., Zhongli, Taoyuan, 320317 Taiwan
6 9 2024
6 9 2024
2024
14 208514 6 2024
2 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
We refresh the design of zero-mode waveguides (ZMWs) by introducing metamaterials that makes the zeroth order resonant mode existence. Of particular importance, the resulting electromagnetic field exhibits nearly constant distribution but not a trivial solution of Maxwell’s equation, showing great advantage to equalize the excitation rate of molecules throughout the waveguides. A closed form expression for the wave impedance is derived which is verified by the finite-difference time-domain simulations. Benefitted from the cavity Purcell effect which is lacking in existing ZMWs, fluorescence amplification and lifetime reduction are simultaneously enhanced. A practical design where the excitation volume reduced down to sub-zeptoliter and the fluorescence lifetime shortened to picosecond scale is illustrated. This result makes single molecule real time (SMRT) sensing of biochemical reactions at biophysically relevant concentrations (~ μM) possible, combining off-the-shelf ultrafast lasers.

Subject terms

Nanophotonics and plasmonics
Nanophotonics and plasmonics
Biosensors
http://dx.doi.org/10.13039/100020595 National Science and Technology Council NSTC 112-2112-M-008-022 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Zero mode waveguide (ZMW) plays a key role in today’s sensing platforms, based on which, single molecule kinetic processes such as DNA polymerase, sequencing, enzyme activity, electrochemical and protein interactions at high concentrations become possible1. In view of the urgent demands for rapid screening test of virus causing disease such as COVID-19, upgrading the state-of-the-art ZMWs is substantial. So far, the effective excitation volume and the fluorescence lifetime can be reduced, respectively, down to several zeptoliter scale and hundred picoseconds exploiting rectangular shaped ZMWs2. To cope with ultrafast biomolecular dynamics and bioaerosols detection, a further enhancement of fluorescence and decay rate is desired. In parallel to the development of ZMWs, alternative approach exploiting resonant structures is lacking. Seeking for resonators with the highest possible quality factor and the smallest possible volume is non-stop research owing to wide applications in nonlinear optics3, quantum electromagnetic dynamics4, and enhanced light and matter interactions5. Due to the intrinsic mismatch between the wavelength of light and the size of molecules, their mutual interactions are extremely weak. To bridge the gap, various resonator designs have been proposed. Photonic crystal based resonators can surely promise high quality factors, but the smallest mode volume can only be made to approach the diffraction limit (λ/2n)3. With the advent of metallodielectrics, electromagnetic fields redistribute themselves, peaking at the interface between metal and dielectrics. As a result, light energy can be confined within a volume well below the diffraction limit. Convincing examples include metal-dielectric-metal (MDM) slot waveguides6, metallic disk cavities7, cylinders on slab8, nanoparticles on mirror (NPoM)9, and side-by-side single crystalline Au nanorods which achieves atomic level confinement10. Zhang et al. further imported the concept of indefinite cavity by taking advantage of the non-closed hyperbolic dispersion relations. With this approach, the wavevector can be made extremely large, corresponding to a very large effective index. As a result, the modal volume can be made extremely small11. Nevertheless, in the longitudinal direction, resonances are still governed by the Fabry-Pérot (FP) condition which requires a minimal length about half of the wavelength.

To be able to isolate a single molecule at high concentrations, conventional ZMWs operate at wavelengths beyond cut-off. In this way, the excitation volume is localized at the incident entrance. The decay of the electromagnetic field in the direction of propagation is governed by the attenuation distance12. Upon detection in the epi-direction, only very limited number of molecules are sampled even at concentrations as high as several μM. However, without manipulating the density of states (DOSs), further boost of photon-molecule interactions is never possible. Most ZMWs so far adopting metallic cladded waveguide structures to confine optical fields laterally, and the typical size is on the order of several tens nanometers. The problem is that when the separation between metallic walls gets smaller, coupled plasmonic wave forms which can channel the waveguide through. This spoils the otherwise cut-off condition, posing a lower bound for the transverse dimension. Worse, the abovementioned phenomenon was completely ignored and perfect electric conductor (PEC) approach was made, yielding impractical results. Apart from this, the lateral modal structures, particularly in rectangular ZMWs, cause inhomogeneous response of fluorophores that poses another difficulty in signal processing. To resolve the abovementioned drawbacks, a ZMW with nearly zero length and meanwhile supporting the 0th order resonant mode with featureless structures is proposed.

Previous research, stimulated by Bethe’s theory13, put emphasis on the so-called extraordinary transmission (EOT) through tiny holes drilled in ultrathin metallic films. For a long time, it is widely believed that EOT is a consequence of resonant coupling, where FP resonance of surface plasmons within the hole is considered as a necessary condition14,15. Since then, it has been taken for granted that a minimal thickness (~ λ/2n) that is capable of accommodating resonance of the fundamental mode (mode order N = 1) is a must for EOT. It was not until F. Pardo et al.16 who unveiled that funnelling of light originates from the magnetoelectric interference between the incident and evanescent waves. Since evanescent wave does not carry energy channeling into nanoapertures, its role needs further clarification, particularly for the effect on phase change upon reflection. This motivates us to revisit Bethe’s study seeking opportunities to develop a ZMW with minimal length and significant Purcell enhancement. In fact, resonant transmission on conditions that deviate from FP predictions were ever discussed for nanoslits surrounded by PECs17. There were also experimental verifications for slits surrounded by real metals at microwave frequencies18,19. However, at optical frequencies, metals response to electromagnetic waves distinctly, in particular, with the presence of metamaterials.

In this study, realistic situations are considered where constant field distribution, lossless, and dispersion-free assumptions are dropped. In addition, coupling between surface plasmon waves are also taken into account. A semi-analytical expression for the phase change upon reflection off a negative index metasurface is derived. We unravel this phase shift is analogous to the Goos-Hänchen (GH) shift, sharing the same origination as the nature of diffraction, and is characterized by the degree of coupling between the resonant mode and the evanescent waves. Of particular interest, by cladding a negative-index material at one end, the ZMW is capable of accommodating the N = 0 mode: a very peculiar resonance with high interface reflectivity but no phase accumulation in a round trip; with nearly zero mode volume and giant Purcell factor; with featureless field distributions so that any fluorescence is auto-equalized which can be precisely modeled by Fermi’s golden rule20 regardless of positions. These properties upgrade current ZMWs largely and are applicable to any wavelength, posing a big leap in sensing technology.

Results and Discussion

Figure 1a schematically depicts the cross-section of a typical ZMW structure. The case considered here is an air-core waveguide surrounded laterally by silver. Its width and length are denoted by w and L, respectively. The dielectric function of silver, characterized by the Lorentz-Drude model, is expressed as ε(ω) = εm − ωp2/(ω2 + iωγd). The plasma and damping frequency of silver are extracted from the experimental result of Johnson and Christy21. For comparison purpose, ωp = 1.38 × 1016 rad/s, γd = 2.73 × 1013 rad/s, and the relative permittivity of silver at infinite frequency εm = 3.7 are set as in22. Exploiting realistic material parameters, the additional phase shift, corresponding to the penetration depth in the lateral direction of the gap plasmon mode (GPM) or the so-called coupled surface plasmon mode (CSPM) termed here is considered. Unlike PECs, at optical frequencies, Ag/air/Ag plasmonic waveguide supports coupled surface plasmons mode (CSPM) which propagates along the inner walls of Ag claddings in the + z direction. As soon as the wave strikes the upper interface at z = 0, back reflection occurs. Due to large index contrast (or wave impedance) at the air/metamaterial interface, the back-reflected wave maintains the same field profile as the incidence, and the total field within the waveguide is thereby described as a superposition of the forward and backward traveling waves, as in Eq. (1).1 Etot⇀=Einc⇀+Esca⇀

Fig. 1 (a) The structure of a typical ZMW. (b) FDTD simulated modal profiles of the incident (black) and reflected waves (red). The shift between the two curves depicts the phase difference. The triangular and star symbols are the data points extracted from22 and those in the present study, respectively.

To verify our finite-difference time-domain (FDTD) result, the electric field of the incident (black curve) and back-scattered (red curve) waves are calculated, as shown in Fig. 1b. To effectively excite the CSPM, the incidence includes a total-field scattered-field (TFSF) source which resembles an ideal plane wave polarized in the x-direction propagating in the z-direction. The spatial–temporal grids used here are 0.25 nm and 4.7 × 10−4 fs, respectively. For comparison purpose, we extracted some data points following the calculation in22, as shown by the triangular symbols. As well, we show data points (the star symbols) in our calculation (on the red curve) at the same position z, verifying our result is trustworthy. It is clear in evidence that the phase shift upon reflection does not equal to zero or π as the case of a plane wave strikes normally onto a lossless material.

To clarify the origin, the reflection coefficient was calculated under the condition of w << λ, namely, the slit width is very much smaller than the wavelength. By matching boundary conditions at z = 0, the reflection coefficient can be expressed as Eq. (2),2 r=1-I01+I0

3 I0=w′∫01sinc2uw′1-u2du+∫1∞sinc2uw′1-u2du

where I0 is analogous to the effective wave impedance. Note here the result is obtained assuming that the magnetic field inside the core has a rectangular profile Rect(x/w), where w′ = w/λ corresponds to the normalized width, and u = kx/k0 denotes the normalized transverse wavenumber. By decomposing I0 into two terms, as opposed to previous studies22,23, the origin of the phase shift upon scattering is uncovered. The first term in the parentheses stands for the coupling between the CSPM and the out-scattered plane waves (kx < k0), thus the integral is a real number. The second term represents the coupling between the CSPM and the scattered evanescent waves (kx > k0) at the interface z = 0, thus the integral is a pure imaginary number. Overall, I0 is a complex number and the degree of phase shift is governed by the relative ratio between the coupling of the CSPM to outgoing plane waves and that to evanescent waves. This process is very similar to the Goos-Hänchen shift, i.e., the origin of phase shift is due to the diffraction of a finite-sized beam that mimics the extra optical path that evanescent wave travels into the overcladding upon reflection.

At optical frequencies, neither modal dispersion nor field distribution of CSPMs can be approximated by a constant and rectangular profile. To solve the reflection coefficient with realistic field distributions, the electromagnetic fields outside the waveguide are expressed as superpositions of plane and evanescent waves. At z = 0, the magnetic field is expressed as:4 Hout⇀x=y^∫-∞∞Tkxejkxxdkx

Here T(kx) represents the amplitude of each component wave. Inside the waveguide, CSPMs are composed of forward and backward traveling waves with propagation loss characterized by the attenuation coefficient. At z = 0, the magnetic field can be expressed as:5 HCSPM⇀x=y^A1-r×eκx/2+e-κx/2eγx+w/2,x<-w/2eκx+e-κx,-w/2≤x≤w/2eκx/2+e-κx/2e-γx-w/2,x>w/2

where A is the normalized amplitude that makes unity magnetic intensity over the cross section of the waveguide, γ=k0εm-neff21/2 and κ=k01-neff21/2 are the complex attenuation coefficient in metallic claddings and the transverse wavenumber in the air core, respectively. neff is the modal index of the CSPM. By matching the tangential components of the electric and magnetic fields at z = 0, wave impedance I0 can be expressed analytically by an integration equation, as in (6), here F stands for Fourier transformation.6 I0=neff2π∫-∞∞FHCSPM∗xFHCSPMx/εx1-u2du

In order to verify the accuracy of our derivation, the results calculated from Eq. (3) and Eq. (6) are compared with that obtained by the FDTD method. The spatial and temporal steps in the FDTD simulations are Δx = Δz = 0.7 nm and Δt = 1.17 × 10−18 s, respectively. The recursive convolution method is applied to transform the frequency dependent permittivity into its temporal counterpart. Fifteen convolutional perfectly matched layers (CPMLs) are allocated around the simulation boundaries to diminish reflections.

The reflected amplitude and phase simulated by the FDTD method are presented by curves marked with dash-star and dash-dot in Fig. 2a, respectively. With considerations of actual field distribution of the CSPMs, more accurate results (solid curves) than those obtained with constant field assumption (hollow-star and hollow-dot) are achieved. It should be noted that Eq. (6) can also be reduced to Eq. (3) under constant field assumption, verifying the result trustworthy. It is also found that when the normalized waveguide width becomes narrower, the reflected amplitude increases and the associated phase decreases. This can be understood intuitively by considering the coupling strength between the surface plasmon waves propagating along the inner walls of the waveguide. Namely, the stronger the coupling, the narrower the angular width of the outgoing radiation at z > 0. This results in a relatively higher reflection amplitude and a relatively lower phase shift at z = 0, mimicking the same behavior of the GH shift as a function of incident angles. Similar argument applies well to cases with different operational wavelengths, e.g., at longer operational wavelength λ = 1550 nm as shown in Fig. 2b. To complete the simulation, several commonly used wavelengths: λ = 532 nm, 800 nm, 1064 nm, and 1550 nm are selected. As expected, monotonically increase of the reflection phase with decreasing of the operational wavelength is obtained, as shown in Fig. 2c. The insets also depict the directivity of the output radiation as a function of wavelength.Fig. 2 Back-reflected amplitude (black) and phase (colored) as a function of the normalized width. (a) For λ = 800 nm, and (b) λ = 1550 nm. (c) The reflection phase for several typical wavelengths. The insets in (a, b) schematically depict the angular spreading of the outgoing radiations. Their correlation with the normalized transverse wavenumber u = kx/k0 and the operational wavelength is also depicted in the inset of figure (c). Briefly, the longer the wavelength, the better the directivity (the smaller u), and the better the modal confinement in the waveguide.

The difference between the reflected phase obtained from Eq. (3) and (6) reveals the origin of deviation is due to constant field and lossless assumptions. This result is also confirmed by the FDTD method. Undoubtedly, this error may as well affect, as described in Eq. (7), the longitudinal resonant condition which is substantial particularly in the development of ZMWs with a minimal length. Here, kCSPM is the propagation constant of the excited CSPM, L is the length of the waveguide, φr is the reflected phase, and N is an integer standing for the mode order.7 kCSPML+φr=Nπ

Generally, the zeroth-order mode (N = 0 mode) does not exist in Fabry-pérot resonators because of the absence of negative phase shift in normal materials. However, the success of fabricating negative index metamaterials in the visible wavelength range24 makes negative phase shift possible. This enables one to revolutionize current ZMW designs, making them able to support the 0th order mode so as to reduce the excitation volume further and meanwhile achieve equalized response throughout the waveguide. In case the surrounding medium (air) is replaced by a metamaterial with effective refractive index n =  − 1, the sign of the reflected phase is flipped, i.e., described by Eq. (8).8 φr,meta=-φr,air

The accumulated positive optical phase associated with the optical path traveled by the CSPMs can be now balanced by a suitably designed negative phase shift, making the total length (so as the excitation volume) of the ZMW to be infinitesimally short while still keeping resonance for the N = 0 mode. Under this circumstance, zero accumulated phase in a round trip can be realized. Hence, the fundamental mode should be redefined and the structureless modal feature should be highlighted as opposed to conventional waveguide.

Figure 3a shows the cut-off height for the N = 0, 1, 2 modes as the waveguide width decreases. Unlike higher order modes (N ≥ 1), where a minimum threshold height is required for resonance, the N = 0 mode preserves resonance no matter how small the width or height of the waveguide are. Extending to extreme conditions, namely, nearly zero height (0 < L < 0.001λ0), the N = 0 mode is able to retain a high interface reflectivity (|r|> 0.99) and a significantly high Purcell factor (Q/Aeff) which exhibits an extraordinarily rapid rise with the decrease of the waveguide width, as shown in Fig. 3b. Note here the effective area is nearly equal to the real area Aeff ~ A due to a constant field distribution.Fig. 3 (a) Cutoff height as a function of the normalized width for different resonant modes N = 0,1, and 2. The solid/dashed curves denote conditions for microwave/near infrared wavelengths, and the reflection coefficients for the N = 0 mode are marked by squares. Note that the cutoff height exhibits distinct trend for the N = 0 mode showing great advantage to reduce the excitation/detection volume. (b) The associated Purcell factor that increases exponentially as the width is getting narrower.

To illustrate the upgraded ZMW is applicable for SMRT sensing, a practical structure consisting of a metamaterial with effective refractive index neff =  − 1.3 atop of a water-filled ZMW is designed, as schematically shown in Fig. 4a,b. It is noted here that the metamaterial layer is based on an experimentally feasible structure24 with redefined parameters that zeros the total accumulated phase in a round trip at λ = 532 nm. Here, the FDTD simulator, Lumerical, is used to calculate the modal field intensity distribution of the 0th order mode, as shown in Fig. 4c. The normalized field intensity here takes the ratio between the local electric field intensity and that of a homogeneous water reference. Meanwhile, a fluorescent molecule is modeled as a radiative dipole25 whose decay characteristic is modified accordingly with the presence of the ZMW. According to Eq. (9) in 26, the modified lifetime Γr⇀ can be estimated with the calculated electromagnetic fields E⇀r⇀ and B⇀r⇀. In the simulation, the magnitude of the electric and magnetic dipole moments p and m are assumed to be constants, and typically m ≪ p. As a result, the Purcell enhancement factor F = Γ/Γ0, defining as the ratio between the lifetimes with and without the resonator, is dominated by the enhanced electric field.9 Γr⇀=2ħImp⇀∗·E⇀r⇀+m⇀∗·B⇀r⇀

Fig. 4 (a) The cross-section of the designed ZMW in this study. (b) The structure of the unit cell of the overclad metamaterial in (a). (c) Modal field intensity distribution of the 0th order mode in linear scale (top) and logarithm scale (bottom). (d) Position dependent lifetime within the core of the ZMW. It is noted that the modal field intensity and the lifetime are enhanced by nearly two orders of magnitude and both of them are nearly equally distributed throughout the waveguide.

The position dependent lifetime is calculated, as shown in Fig. 4d. We now compare our results with the state-of-the-art achievements2,27 from three aspects: field uniformity, fluorescence enhancement and lifetime reduction. In previous studies, CSPMs propagate along the inner walls of the ZMWs can be clearly seen even the waveguides are cut-off. This phenomenon becomes more obvious as the lateral dimensions decrease. In fact, this is the main cause of field inhomogeneity. Surprisingly, the effect has been ignored completely that poses significant difficulties in isolating a single molecule and signal readout. Since the origin is intrinsically from metallic boundaries, it is unavoidable in our case as well, especially in the corner regions. However, due to the establishment of the 0th order resonant mode, i.e., a standing wave without nodes in the longitudinal direction, not only the exponentially decayed electric field is erased but the inhomogeneity caused by the CSPM along the inner walls are largely mitigated. Together with the Purcell effect, the fluorescence intensity and lifetime are largely enhanced, exhibiting nearly equalized response throughout the waveguide. Overall, we obtained more than 100 times enhancement for the field intensity, at least 10 times shortening for the fluorescence lifetime, and a sub-zeptoliter excitation/detection volume. Compare to the conventional rectangular ZMW structure2 where a maximum of 30-fold local intensity enhancement was obtained, our result is three times larger regardless of positions, showing significant upgrade with respect to the existing ZMWs.

A final comment to the issue of fluorescence quenching should be made despite the dimensions of the ZMW proposed in this study are capable of decreasing further, say, down to quantum limit. There are many passivation techniques so far to avoid fluorescence quenching though27, the nano-sized metallic walls themselves present much fewer density of states (DOS)28 as oppose to that in bulk metals. This property reduces the probability of non-radiative transitions so as to prevent the quenching effect.

Conclusion

In conclusion, introducing metamaterials in the design of ZMWs enables the zeroth order resonant mode. The structureless field distribution, nearly zero cavity length, and much enhanced Purcell effect altogether upgrades the existing ZMWs. Compare to the state-of-the-art ZMWs, our result exhibits more than three times enhancement in field intensity and a further reduction of the excitation/detection volume, down to sub-zeptoliter scale. Besides, the homogeneous Purcell effect acts as a gain equalizer wherever the fluorescent molecule lies in the waveguide. This also causes a further reduction of the fluorescence lifetime, down to sub-picosecond range. This study sheds light on SMRT detection using off-the-shelf ultrafast laser systems.

Acknowledgements

The authors would like to thank the National Science and Technology Council for the Grant support under contract number NSTC 112-2112-M-008-022 -.

Author contributions

Chao-Yi Tai initiated and directed the project, analyzed the data, and wrote the manuscript. Wen-Hsiang Yu, Tang-Chun Liu, and Chung-Kai Tseng. carried out the simulations and format the manuscript. Diksha Thakur provides fruitful information and helps edit and proofread this manuscript.

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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