## Abstract

We report near- to far-field measurement of optical wavefronts emanating from a nanoslit formed in a thin (50 nm thick) Ag film. The evolution of optical phases is imaged using a self-interference technique in conjunction with a scanning probe method. The phase relationship of the slit-transmitted waves with respect to the direct transmission through the thin metal film is quantitatively established. The singular-phase points resulting from the interplay of slit diffraction and surface plasmons are identified in the intermediate-field region.

©2008 Optical Society of America

Interaction of light with a nanoapertured metal film has been the subject of extensive study because it produces many interesting phenomena, such as “enhanced” transmission of light through a nanohole array or shaping the spatial or spectral profiles of the transmitted light [1–11]. The richness of the phenomena stems from the complexity of the way that light interacts with the nanostructures formed in the metal film. Surface plasmons (SPs), collective oscillation of electrons carrying the electromagnetic energy in the form of photons trapped at a meta/dielectric interface, can effectively mediate the interactions between metal nanostructures [12]. Unlike the dielectric case, a metal nanostructure can also efficiently interact with free-space radiation, diffracting an incident light and/or coupling the light into surface plasmons (vice versa, decoupling surface plasmons into free-space radiation). This implies the multiplicity of the interaction pathways available on nanostructured metal surface. When properly designed, the effects of near-field interactions between nanostructures can also reach the far-field region through diffraction and constructive interference among them. The spatial and/or spectral profiles of far-field optical transmission through a metal nanoslit array, for example, are known to be governed by various resonances occurring on different sections of metal surface [13]. The roles played by surface plasmons and free-space radiation in the interaction of metal nanostructures, however, have not been clearly understood and have been a subject of debate. The surface-bound wave and free-space radiation behave differently in terms of propagation constant (wavelength and attenuation), field distribution, etc., and analyzing individual roles and their interplay in an arbitrary structure is considered a challenge.

A single nanoaperture formed in a metal film is a simple and yet the most fundamental structure that can be viewed as a basic building block of aperture-based nano-plasmonic structures. Evolution of optical wavefronts emanating from a metal nanoaperture is of essential interest in studying the plasmonic structures, and yet a detailed understanding is not fully established on how the different wave components (free-space diffraction and surface bound waves) interplay and evolve over the near- to far-field regime [3–10]. In this paper we report near- to far-field imaging of optical wavefronts emanating from a single nanoslit formed on a thin Ag film.

A 80-nm-wide slit was formed with focused-ion-beam etching in a 50-nm-thick Ag film deposited on a fused-silica substrate (Fig. 1(a), left). A TM-polarized laser light (633 nm wavelength) was incident to the substrate side, and the transmitted light was imaged by scanning a nanoprobe (Veeco Aurora NSOM probe 1720-00: 100-nm-thick Al coated; 80-nm aperture diameter) in the horizontal direction at the exit side (Fig. 1(a), right). Figure 1(b) shows some of the scan profiles obtained in the near- to far-field regime using a metal nanoapertured probe. Fringes form and evolve over the entire regime, since the light partially (and directly) transmitted through the thin Ag film interferes with the waves transmitted through the nanoslit [14–16]. The fringe spacing increases for larger probe-to-surface distance and/or towards the central region on a given scan. Figure 1(c) shows a two-dimensional (2D) map of the scan profiles over the entire regime (near- to far-fields): 71 scan profiles are displayed with the fringe amplitude color-coded.

In order to elucidate the interference nature of fringe formation, a schematic of two propagating waves is shown in Fig. 2(a), one emanating from a nanoslit with cylindrical wavefronts and the other directly transmitted through a thin metal film for a planar wave incident from the bottom side. The fringes resulting from constructive interference of the two waves are marked on the cross points of the wavefronts. In this diagram, the (*m,n*)-th cross point corresponds to the interference of the *m*-th cylindrical wavefront (*m*=1, 2, 3,…) radiating from the nanoslit and the *n*-th planar wavefront (*n*=0, 1, 2, 3,…) directly transmitted through a metal film. After a simple analysis, the coordinates of the (*m,n*)-th cross points, (*x _{m,n,} y_{m,n}*) can be determined as follows.

Here *λ* is the free-space wavelength of the transmitted light. *ϕ* is the phase difference between the directly-transmitted planar wave (*ϕ*
_{1}) and the nanoslit-transmitted cylindrical wave (*ϕ*
_{2}), that is, *ϕ=ϕ _{1}ϕ_{2}*, and |

*φ*|<

*π*. In the region far from the slit and yet with relatively small probe-surface distance (

*m*≫

*n*), the fringe location can be expressed as ${x}_{m,n}\cong (m+\frac{\phi}{2\pi})\lambda ;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}_{m,n}=n\lambda $. This tells the fringe spacing asymptotically approaches the free-space wavelength, and the fringe locations are off-shifted from the integer multiple of wavelength positions by the amount proportional to phase retardation

*ϕ*.

Interference of two waves of different symmetry (cylindrical versus planar) results in a characteristic fringe pattern (Fig. 1(c) and Fig. 2). The first track on either side from the center corresponds to the cross-points of the *n*-th planar wavefront and the (*n*+1)-th radial wavefront, i.e., the case of *m*=*n*+1 in Eq. (1). In general, the *l*-th track comprises the cross-points, (*x _{m,n,} y_{m,n}*) with

*m=n+l*. On a given track and in the region far from the metal surface (

*n*≫

*l*), the fringe location traces approximately a parabolic profile, i.e., ${x}_{l}(n)\cong \lambda \sqrt{2nl}$. Figure 2(c) shows a Poynting vector (the

*y*-component) distribution calculated from the finite-difference-time-domain (FDTD) analysis of wave transmission through a nanoslit [17]. [The vertical (

*y*-) component of the Poynting vector is the main contributor to the scanning probe output, since the probe is aligned normal to the sample surface [3].] Overall the fringe pattern obtained from a FDTD simulation shows a good agreement with the measurement result (Fig. 1(c) and Fig. 2(c)). Figure 2(b) shows a detailed comparison between the cylindrical-and-planar-wave-interference model (blue dashed curves) and the FDTD simulation (red dashed curves) results. While both agree well in the far-field regime, a clear difference is observed in the near- to intermediate-field regime: the FDTD simulation predicts a fringe track shifted inward from that of the two-wave-model. This discrepancy is ascribed to stronger presence of surface plasmon waves near the metal surface compared with the slit-transmitted cylindrical wave (Fig. 2(c)). The fringe spacing near the metal surface is significantly smaller than the free space wavelength as can be seen in the mismatch of the fringe patterns in the near and far-fields. The fuzzy area corresponds to the transition region between the two distinct regimes where either slit radiation or surface plasmon field is dominant over the other. In this region, phase singularity exists at the points where the two fringes completely mismatch [6,15] (see the inset of Fig. 2(c)). The phase singularity indicates that the optical fields of slit-diffraction and surface plasmons cancel each other and the phase becomes undefined in the local area. (It should be mentioned that the surface plasmon and free-space radiation waves have different wavelengths, therefore no absolute phase value can be defined for both waves in this region.)

For detailed analysis, the three wave components that are expected to have significant presence in the near- to far-fields are expressed as follows. 1) a partial, directly-transmitted wave (TM polarized) through a thin metal layer: *H⃗ _{1}=ẑH_{1}e
^{iky}* and

*E⃗*

_{1}= −

*x̂ηH*

_{1}

*e*, where $\eta =\sqrt{\mu /\epsilon}$. 2) a free-space radiation wave emanating from the slit:

^{iky}*H⃗*

_{2}=

*ẑH*

_{2}

*e*and

^{ik⃗·r⃗}*E*⃗

_{2}= (-1/

*iωε*)∇×

*H*⃗

_{2}. In the regime where a slowly varying condition (i.e., $\left|{H}_{2}\right|\gg \left|\frac{\partial {H}_{2}}{\partial x}\right|,\left|\frac{\partial {H}_{2}}{\partial y}\right|$) is satisfied, the electric field vector can be approximated as follows.

*E*⃗

_{2}=(-

*x*̂sin

*θ*+

*ŷ*cos

*θ*)η

*H*

_{2}

*e*

^{ik(xcosθ+ysinθ)}. 3) surface plasmons generated at the slit edges and propagating away from the slit: ${\overrightarrow{H}}_{3}=\widehat{z}{H}_{3}{e}^{-{\gamma}_{sp}y}{e}^{\pm i{k}_{sp}x}$ and

Here the ± sign corresponds to the surface plasmons propagating along the positive (+) or negative *x*-direction (-), respectively. Regarding the phase relationship of slit-transmitted waves, it should be noted that the free-space radiation and SP waves (*H*
_{2} and *H*
_{3}) are in phase at the slit exit, since they originate from the same surface plasmon wave transmitted through a nanoslit. The region proximal to the slit and the intermediate-to-far-field regime (where the SP fields are negligible) can be well described in terms of interplay of the two wave components, direct transmission through a film (*E*⃗_{1},*H*⃗_{1}) and nanoslit radiation (*E*⃗_{2},*H*⃗_{2}). The time-averaged energy flow can then be expressed with the Poynting vector as follows. 〈*S*⃗〉=(*E*⃗_{1}+*E*⃗_{2})×(*H*⃗_{1}+*H*⃗_{2})*. As the probe scans away from the slit (*θ*→0) at a constant probe-surface distance (*y=y*
_{0}), the y-component of the Poynting vector (the main contributor to the probe output) asymptotically approaches the following expression:

Here *ϕ* is the phase difference between the directly-transmitted and slit-transmitted waves as defined above with Eq. (1). The real part of the Poynting vector component, Re(〈*S _{y}*〉) corresponds to the scan profile measured with a nanoapertured probe, and the fringe peaks occur at $x=\lambda (m+\frac{{y}_{0}}{\lambda}+\frac{\phi}{2\pi})$. Note that at

*y*

_{0}=

*nλ*this formula reduces to the one derived from the two-wave-interference diagram shown in Fig. 2(b). The phase relationship (

*ϕ*) of the slit transmitted wave and the direct transmission determines the exact location of fringes (the offset from the positions at integral multiple of free-space wavelength).

In the vicinity of the slit, the radial wave (*H*
_{2}) shows faster damping (1/*r*
^{1/2}) compared with the SP’s exponential decay along the *x*-direction. In the region distant from the slit but near the metal surface, the wave interaction can be described in terms of SP and direct transmission, and the probe output at *y*=0 can be expressed as follows.

$$=\eta {\mid {H}_{1}\mid}^{2}+\eta \mid {H}_{1}\mid \mid {H}_{3}\mid A\mathrm{cos}({k}_{\mathrm{sp}}x\mp \varphi \pm \alpha )$$

Here tan *α*=tan^{-1}(*γ _{sp}/k*). The fringe peaks occur at

*k*=2

_{sp}x*πm*±

*ϕ*∓

*α*with the spacing equal to the plasmon wavelength

*λ*. Compared with the far-field case, the fringe location is affected by an extra term

_{sp}*α*, which is separate from the shift caused by the phase retardation

*ϕ*. This extra phase shift originates from the horizontal component of the SP electric field (

*E*), which is out-of-phase (by

_{3x}*π*/2) with its normal component (

*E*

_{3y}) [12,20].

Figure 3 shows a detailed comparison of the scan profiles measured in the near- to intermediate- to far-field regimes (*y*=200 nm, 630 nm, and 7.6 µm, respectively) with the corresponding FDTD simulation results. Overall a good agreement is observed in all three regimes. The slight asymmetry (unequal sidelobe intensities around the center) observed in the near- and far-field scans is ascribed to a possible non-ideal tip profile of the NSOM probe used in this work: A slight asymmetry caused by local roughness (protrusion) in the Al-coated aperture area was observed. According to the FDTD simulation (Fig. 2(c)), the region outside ~1 µm distance from the metal surface is free from the effect of surface plasmon presence. The fringe locations in this regime are governed by the interference of direct transmission and slit radiation, and can be described by Eq. (1). From the measured peak locations and referring to Eq. (1) with *λ*=633 nm, *n*=12, and *m*=13 to 19, the phase retardation *ϕ* is estimated to be -86±10 degrees. This negative retardation depicts a relationship that the phase of the nanoslit-transmitted wave leads the directly transmitted wave by ~86 degrees (or equivalently, in terms of wavefronts, trails by ~86 degrees). In the case of near-field scan, the measured fringe spacing is 605±10 nm, showing a reasonable agreement with the surface plasmon wavelength *λ _{sp}* (613 nm) as expected from Eq. (3).

Next we elucidate the nature of phase evolution of each wave component during transmission through a nanoslit or a thin metal film. For a given planar wave incident from the substrate side, the three waves take different paths, accumulating different amount of phase when measured at the exit surface of nanoslit.

For the case of direct transmission through a metal film (of thickness *d*), the total phase retardation can be calculated from the transmission coefficient of the metal layer: $t=\frac{{t}_{12}{t}_{23}{e}^{i{k}_{2}d}}{1-{r}_{21}{r}_{23}{e}^{i2{k}_{2}d}}$. Here *t _{ij}* is the transmission at the interface of the

*i*-th and the

*j*-th layers, and can be expressed as ${t}_{ij}=2\sqrt{{\epsilon}_{i}}/(\sqrt{{\epsilon}_{i}}+\sqrt{{\epsilon}_{j}})$. i=1, 2, and 3 represents substrate (quartz), metal (Ag) and air side, respectively, and

*ε*is the corresponding dielectric constant. Similarly,

_{i}*r*is the reflection coefficient at the

_{ij}*i-j*layer interface with the

*i*-th layer as the incidence side, and is expressed as ${r}_{ij}=(\sqrt{{\epsilon}_{j}}-\sqrt{{\epsilon}_{i}})/(\sqrt{{\epsilon}_{i}}+\sqrt{{\epsilon}_{j}})$. The total amount of phase change of direct transmission is calculated to be -50 degrees. [For the metal thickness (50 nm) studied in this work, the single pass transmission in the metal layer is 0.14, therefore the multiple internal reflection effect is insignificant. The main contributions are from the two interfacial transmissions, i.e., from the phases of

*t*

_{12}and

*t*

_{23}]. The phase change of a slit-transmitted wave comes from the phase accumulation during SP propagation through a slit, Re(

*k*)

_{sp}*d*. For the case of a 80-nm-wide and 50-nm-deep slit, the phase retardation is estimated to be +35 degrees. Combining the two phase change components, overall the nanoslit transmitted wave leads the directly transmitted wave by 85 degrees of phase, that is

*ϕ*=-85 degrees. Figure 4 shows a FDTD simulation of phase relationship of the two wave components. The wave transmitted through a nanoslit leads the direct transmission by 90 degrees of phase as can be seen from the comparison of wavefront locations at the exit side for the same incident wave. The analytical and simulation results clearly confirm the phase retardation (

*ϕ*=-86±10 degrees) extracted from the measurement data discussed above.

In summary, we have experimentally characterized the phase evolution of optical wavefronts emanating from a nanoslit (80-nm width) formed in a 50-nm-thick Ag film. A planar wave, directly transmitted through the thin metal film, was used as a reference in forming an interference pattern with the slit-transmitted free-space radiation and surface plasmons in the near- to far-field regimes. The phase relationship of the slit-transmitted waves with respect to the direct transmission was quantitatively established by comparing the measurement result with the analytical and FDTD simulation results. Imaging of optical wavefronts from a metal nanoaperture structure in the near- to far-fields is expected to be important in designing advanced nano-optic and plasmonic structures where precise control of optical phase is essential.

## Acknowledgments

The authors gratefully acknowledge support from the NSF grants (NIRT-ECS-0403865/ECE0424210).

## References and links

**1. **C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature **445**, 39–46 (2007). [CrossRef] [PubMed]

**2. **H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming Light from a Subwavelength Aperture,” Science **297**, 820–822 (2002). [CrossRef] [PubMed]

**3. **L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S.-H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. **85**, 467–469 (2004). [CrossRef]

**4. **G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nanostructured surfaces and the composite diffracted evanescent wavemodel,” Nature Phys. **2**, 262–267 (2006). [CrossRef]

**5. **P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nature Phys. **2**, 551–556 (2006). [CrossRef]

**6. **L. Chen, J. T. Robinson, and M. Lipson, “Role of radiation and surface plasmon polaritons in the optical interactions between a nano-slit and a nano-groove on a metal surface,” Opt. Express **14**, 12629–12636 (2006). [CrossRef] [PubMed]

**7. **F. López-Tejeira, G. Sergio, L. Rodrigo, F. J. Martín-Moreno, E. García-Vidal, T. W. Devaux, J. R. Ebbesen, I. P. Krenn, S. I. Radko, M. U. Bozhevolnyi, J. C. González, A. Weeber, and Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nature Phys. **3**, 324–328 (2007). [CrossRef]

**8. **K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nature Photon. **1**, 53–56 (2007). [CrossRef]

**9. **L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julié, V. Mathet, and M. Mortier, “Near-field analysis of surfacewaves launched at nanoslit apertures,” Phys. Rev. Lett. **98**, 153902(1)–(4) (2007). [CrossRef]

**10. **Y. S. Jung, J. Wuenschell, T. Schmidt, and H. K. Kim, “Near- to far-field imaging of free-space and surface-bound waves emanating from a metal nanoslit,” Appl. Phys. Lett. **92**, 02310(1)–(3) (2008). [CrossRef]

**11. **T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**, 667–669 (1998). [CrossRef]

**12. **H. Raether, *Surface plasmons on smooth and rough surfaces and on gratings*, (Springer, Berlin, 1988).

**13. **Z. Sun, Y. S. Jung, and H. K. Kim, “Role of surface plasmons in the optical interaction in metallic gratings with narrow slits,” Appl. Phys. Lett. **83**, 3021–3023 (2003). [CrossRef]

**14. **J. N. Walford, K. A. Nugent, A. Roberts, and R. E. Scholten, “Three-dimensional phase imaging with a scanning optical-fiber interferometer,” Appl. Opt. **38**, 3508–3515 (1999). [CrossRef]

**15. **M. L. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. **85**, 294–297 (2000). [CrossRef] [PubMed]

**16. **A. Nesci, R. Dändliker, and H. P. Herzig, “Quantitative amplitude and phase measurement by use of a heterodyne scanning near-field optical microscope,” Opt. Lett. **26**, 208–210 (2001). [CrossRef]

**17. **D. M. Sullivan, *Electromagnetic Simulation Using the FDTD Method*, (IEEE Press, New York, 2000). [CrossRef]

**18. **E. D. Palik (ed.), *Optical constants of solids*, (Academic Press, New York, 1998).

**19. **P. B. Johnson and R. W. Christy, “Optical Constants of Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

**20. **J. Wuenschell and H. K. Kim, “Surface plasmon dynamics in an isolated metallic nanoslit,” Opt. Express **14**, 10000–10013 (2006). [CrossRef] [PubMed]