## Abstract

We report surface-plasmon mediated total absorption of light into a silicon substrate. For an Au grating on Si, we experimentally show that a surface-plasmon polariton (SPP) excited on the air/Au interface leads to total absorption with a rate nearly 10 times larger than the ohmic damping rate of collectively oscillating free electrons in the Au film. Rigorous numerical simulations show that the SPP resonantly enhances forward diffraction of light to multiple orders of lossy waves in the Si substrate with reflection and ohmic absorption in the Au film being negligible. The measured reflection and phase spectra reveal a quantitative relation between the peak absorbance and the associated reflection phase change, implying a resonant interference contribution to this effect. An analytic model of a dissipative quasi-bound resonator provides a general formula for the resonant absorbance-phase relation in excellent agreement with the experimental results.

©2011 Optical Society of America

## 1. Introduction

A dissipative resonator is known to perfectly absorb incident light when light scattering vanishes at the condition of *critical coupling* [1–3] at which the non-radiative loss rate (*γ*_{nr}) equals the radiation coupling rate (*γ*_{rad}). Under critical coupling conditions, leakage radiation from the resonator mode destructively interferes with non-resonantly scattered light. Thus, the incident light is strongly confined to a resonator mode dissipating total incoming power even with weakly absorbing materials. This interference effect, for example, is basis for indirect-bandgap Si Fabry-Pérot cavity as perfect absorber with the critical coupling condition effectively maintained by coherent interplay of multiple incident beams [4].

Another method to achieve perfect absorption of light is to use surface plasmon resonances on metal surfaces. The plasmonic total absorption of light on a periodically textured surface is an important basis of high-sensitivity bio-chemical sensors due to marked distinction in the reflection phase spectrum for small variations in optical constants near the surface [5,6]. The main mechanism of the plasmonic absorption is ohmic damping of collectively oscillating free electrons. Balancing the ohmic damping rate (*γ*_{ohm}) with *γ*_{rad} has shown maximal absorption in multiple-order metallic gratings [1] and omnidirectional total absorption in void-plasmonic templates [3].

Using plasmonic total absorption for applications such as photovoltaic cells is also of special interest [7–9]. Even though the narrow-band response of the SPP resonance may be detoured by employing a quasi-periodic nanostructure on the metal surface or randomly distributed metal nano particles, strong ohmic damping in metals limits efficient energy transfer for photocurrent generation in plasmon-resonant photovoltaic devices.

Here, we report non-ohmic total absorption of light induced by an SPP resonance in a periodically corrugated Au film on an Si substrate. The absorption in the substrate largely dominates the ohmic absorption associated with the SPP. We experimentally demonstrate SPP-resonant total absorption with attendant *γ*_{nr} nearly 10 times larger than *γ*_{ohm}. This remarkable enhancement in *γ*_{nr} is explained by SPP-mediated resonant transfer of light. Rigorous numerical calculations show that the resonant interference due to SPP excitation cancels the reflection while greatly enhancing forward diffraction to multiple orders that experience loss in the Si substrate. We also find that the peak absorbance is quantitatively related to the reflection phase change, showing directly that interference plays a crucial role in the resonant absorption process. An analytic model based on temporal coupled-mode theory [1,10] predicts a general formula for the peak absorbance as a function of the associated phase change in excellent agreement with experimental results.

## 2. Experimental procedures and results

As depicted in Fig. 1
, we fabricated a set of nine linear gratings on a single Si substrate by means of laser interference lithography. All these gratings have a fixed period (Λ) of 609 nm but different heights (*h*) in the range of 47 nm–64 nm. We first obtained photoresist (PR: SEPR701, Shin-Etsu) gratings with constant height and different fill factors by varying the UV-laser (λ = 266 nm, Azure 200, Coherent) exposure time. Reflow process at 210° C for 1 minute before Au deposition produced roughness-free grating lines with different fill factors and heights. Figure 1(b) shows an AFM image of the resulting hemi-cylindrical profiles that have a surface roughness less than ~1 nm. The grating heights are, for example, 64 nm (D1), 57 nm (D5), and 47 nm (D9) as shown in Fig. 1(c). We measure the reflection amplitude and phase spectra at three different Au thicknesses (*t*_{Au}) while keeping the grating profiles constant by consecutively depositing two, 4.5-nm-thick Au films on top of the 27-nm-thick initial deposit; i.e., *t*_{Au} = 27 nm, 31.5 nm, and 36 nm for each measurement, respectively.

By using ellipsometry (VB-400 VASE ellipsometer system, J. A. Woollam Co.) at a fixed angle of incidence (*θ* = 20°), we measured the transverse-magnetic (TM) reflection coefficients, *ρ*_{ΤΜ}, and transverse-electric (TE) reflection coefficients, *ρ*_{ΤΕ}, of the fabricated gratings. From the ellipsometric parameters of phase difference, Δ = arg(*ρ*_{TE})–arg(*ρ*_{ΤΜ}), andmagnitude ratio, Ψ = |*ρ*_{ΤΜ}/*ρ*_{TE}|, we define a complex ratio of the reflection coefficients as Φ ≡ Ψ*e*^{–}^{i}^{Δ}, where the relative TM reflectance to TE reflectance *R* = |Ψ|^{2}. Note that the TE reflectance in the separate reflectance measurement is nearly constant at |*ρ*_{TE}|^{2} ≈95 within ± 1%. Thus, the chief resonant characters of *R* and Δ are caused by a TM resonance in *ρ*_{ΤΜ}.

By using ellipsometry (VB-400 VASE ellipsometer system, J. A. Woollam Co.) at a fixed angle of incidence (*θ* = 20°), we measured the transverse-magnetic (TM) reflection coefficients, *ρ*_{ΤΜ}, and transverse-electric (TE) reflection coefficients, *ρ*_{ΤΕ}, of the fabricated gratings. From the ellipsometric parameters of phase difference, Δ = arg(*ρ*_{TE})–arg(*ρ*_{ΤΜ}), and magnitude ratio, Ψ = |*ρ*_{ΤΜ}/*ρ*_{TE}|, we define a complex ratio of the reflection coefficients as Φ ≡ Ψ*e*^{–}^{i}^{Δ}, where the relative TM reflectance to TE reflectance *R* = |Ψ|^{2}. Note that the TE reflectance in the separate reflectance measurement is nearly constant at |*ρ*_{TE}|^{2} ≈95% within ± 1%. Thus, the chief resonant characters of *R* and Δ are caused by a TM resonance in *ρ*_{ΤΜ}.

Figures 2(a)
–2(c) show the measured Δ-spectra of the nine gratings with *t*_{Au} = 27 nm, 31.5 nm, and 36 nm, respectively, and the *R*-spectra in the insets. Resonance dips in *R* and the abrupt change in Δ near λ = 834 nm are attributed to resonant excitation of an SPP mode on the air/Au interface under phase matching at first order diffraction formulated as

*θ*is the angle of incidence,

*ε*

_{Au}is the dielectric constant of Au, and

*λ*

_{SPP}is SPP resonance wavelength. Using a Drude model for

*ε*

_{Au}with plasma frequency

*ω*

_{p}= 1.13 × 10

^{16}rad/s and collision frequency Γ = 3.37 × 10

^{13}rad/s, Eq. (1) gives λ

_{SPP}= 830.5 nm at

*θ*= 20° and Λ = 609 nm. A slight red-shift of the measured resonance position (834 ± 2 nm) from λ

_{SPP}in Eq. (1) is due to the finite Au film thickness. This is because the field associated with the SPP on the air/Au interface possesses a non-zero evanescent tail that penetrates the high-index substrate (Si), increasing its in-plane momentum. The small variations in the resonance locations in the inset of Fig. 2 are within the experimental parametric tolerances. As

*h*gradually decreases from 64 nm (D1) to 47 nm (D9), the nine gratings reveal similar

*R*spectral shapes but exhibit distinguished Δ behaviors. For example, the phase changes of the D1 grating with

*t*

_{Au}= 27 nm in Fig. 2(a) and the D9 grating with

*t*

_{Au}= 36 nm in Fig. 2(c) are very different from other gratings; the D5 grating exhibits a

*π*-jump when

*t*

_{Au}= 31.5 nm, but it exhibits phase that is less (larger) than

*π*when

*t*

_{Au}= 27 nm (36 nm).

To understand these distinguished Δ-spectral behaviors and Ψ-Δ dependence on Au thickness, we visualize the measured Ψ and Δ spectra on a complex plane of Φ ( = Ψ*e*^{–}^{i}^{Δ}). Figures 2(d)–2(f) show the phasor diagrams of Φ’s for the three Au thicknesses. Each of the Φ's traces nearly an exact circle whose radius (*b*) increases with *h* and *t*_{Au}. The grey disc at the origin represents a forbidden area (R < 0.01) due to background noise received by the photo detector used in the measurement. When the phasor Φ has *b* < 0.5, for example, the circular traces (black color) of the D9 grating in Figs. 2(d)–2(f), have an angular span less than *π* with respect to the origin. When the phasor Φ has *b* > 0.5, all the circular traces (blue color) for D1, exhibit a 2*π* phase change since each of these circles surrounds the origin. On the other hand, when the circular traces exactly pass through the origin with *b* = 0.5, for example, the circular trace (red color) for D5 in Fig. 2(e), the Φ phase abruptly changes from –*π*/2 to + *π*/2 near the origin and *R* goes to zero. Therefore, the measured results in Fig. 2(b), showing nearly *π*-jump and *R* → 0 only for the D5 device, are now clearly described by means of the Φ phasor diagrams.

The total absorption at *b* = 0.5 is closely related to the critical balance of *γ*_{rad} = *γ*_{nr} [1–3], where light reflection at resonance is completely forbidden by destructive interference between leakage radiation from the resonance mode and non-resonantly reflected light. Thus, it is reasonable to estimate *γ*_{nr} by *γ*_{tot} = *γ*_{rad} + *γ*_{nr} ≈2*γ*_{nr} with *γ*_{tot} given by the spectral full-width half-maximum (FWHM) of the measured *R* spectra for the D5 grating. The estimated *γ*_{nr} from the *R* spectra of D5 in Fig. 2 is 0.68 × 10^{13} Hz (Δλ = 10.07 nm); it is 8.8 times larger than *γ*_{ohm} = 0.775 × 10^{12} Hz, where *γ*_{ohm} denotes the ohmic damping rate of an SPP on a flat air/Au interface. The large difference between *γ*_{nr} and *γ*_{ohm} suggests that ohmic damping of collectively oscillating free electrons in the Au film is not a main absorbing channel in our case. Instead, loss in the Si substrate is attributed as the dominant absorbing mechanism even though evanescent penetration of the SPP into the Si substrate is small. Resonant tunneling of light due to SPP excitation is involved in our case.

## 3. Mechanism of total absorption

To clearly understand the SPP-mediated total absorption, we performed a numerical simulation of reflection spectra based on the Chandezon method [11]. In the simulation, we used the grating profile shown in Fig. 3(a)
(solid curve) and *t*_{Au} = 39.5 nm, 43.5 nm, and 47.5 nm. Figures 3(b)–3(e) show excellent agreement of the numerical Δ and *R* spectra (solid curves) with the measured spectra (open squares). Although the grating profile and Au thickness in the simulation deviate from those obtained by the metrology (grating profile by AFM and *t*_{Au} by quartz crystal oscillator), quantitative agreements in Δ and *R* spectra associated with the SPP resonance at λ = 834.1 nm confirm that the numerical situation properly describes the total absorption in the experiment. We believe that there is inaccuracy in the quartz-crystal thickness determination that accounts for the deviation.

Figure 4(a)
shows the tangential magnetic-field distributions near the Au grating layer (sinusoidal horizontal curve on surface) at λ = 834.1 nm in Fig. 3(d). The incident light from air excites an SPP mode propagating leftward on the air/Au interface, followed by five lossy waves (arrows in Si substrate) from multiple-order diffraction of the SPP mode. Due to the complex refractive index of *n*_{Si} = 3.68+*i*0.0108 [12], propagation decay distance of the lossy waves in Si is ~*L* = λ*/* [4*π* Im(*n*_{Si})] = 6.1 μm from the grating layer. The light energy carried by each lossy wave is indicated at the end of the arrow; the total is 91.9% of the incident energy. Only 8.1% of the incident light is absorbed in Au while the rest is transferred to the Si substrate via SPP-mediated tunneling.

It is worth noting that the power losses numerically obtained in Au and Si are consistent with those measured in Fig. 3(d); suppose that the total decay rate of the SPP mode is given by *γ*_{tot} = *γ*_{rad} + *γ*_{ohm} + *γ*_{LW}, where *γ*_{LW} represents the decay rates of the SPP to lossy waves in the substrate and *γ*_{nr} = *γ*_{ohm} + *γ*_{LW}. At critical coupling, *γ*_{nr} = *γ*_{tot}/2, which is measured by 7.023 × 10^{12} Hz from the FWHM of *R* in Fig. 3(d). Letting *γ*_{ohm} = 7.75 × 10^{11} Hz for the surface of semi-infinite thick Au, we obtain the ratios of *γ*_{ohm}/*γ*_{nr} = 11% and *γ*_{LW}/*γ*_{nr} = 89%. They are close to the numerical losses of 8.1% in Au and 91.9% in Si. The difference of about 3% may be due to the fact that *γ*_{ohm} on a thin Au film must be smaller than that on semi-infinite thick Au.

## 4. A coupled-mode interpretation of total absorption

Finally, we develop an intuitive way of understanding the phasor diagrams in Figs. 2(d)–2(f). As shown in Fig. 4(b), assume a resonator with ohmic damping (*γ*_{ohm}) coupling to one radiation port (*γ*_{rad}) and five lossy-wave ports (*γ*_{LW} = *γ*_{1} + … + *γ*_{5}). The temporal coupled-mode theory yields a complex reflection coefficient [1]

*γ*

_{nr}=

*γ*

_{ohm}+

*γ*

_{LW}, where

*ω*is angular frequency of incident light,

*ω*

_{0}is resonance frequency, and

*ϕ*is non-resonant (background) reflection phase. We assume the non-resonant reflection going back to the radiation port to be lossless; this is acceptable for noble metals like Ag, Au, and Al in the visible and infrared spectral ranges. Note that the complex form in Eq. (2) provides a physical interpretation of poles and zeros in the polological descriptions of light scattering [13]; the spectral positions of poles and zeros correspond to

*ω*=

*ω*

_{0}−

*i*(

*γ*

_{nr}+

*γ*

_{rad}) in the denominator and

*ω*=

*ω*

_{0}−

*i*(

*γ*

_{nr}−

*γ*

_{rad}) in the numerator, respectively. Therefore, the resonance condition

*ρ*(0)=0 corresponds to critical coupling,

*γ*

_{nr}=

*γ*

_{rad}.

The circle with radius *b* = *γ*_{rad}/*γ*_{tot} (blue) in Fig. 5(a)
shows a phasor representation of *ρ*(*ω*) on a complex plane [1,10]. We refer to this *ρ*(*ω*) trace as a resonance-response circle (RRC). As *ω* increases, *ρ*(*ω*) traces the RRC counter-clockwise from the open circular dot (*ω* << *ω*_{0}), passes through the closed circular dot at *ω* = *ω*_{0}, and again approaches the open dot for *ω* >> *ω*_{0}. Note that the RRC reproduces the measured phasor diagrams well; for example, the (black) circles of D9 in Figs. 2(d)–2(f), showing an increment in *b* as *γ*_{tot} depends on *t*_{Au}. The three traces of *ρ*(*ω*) in Fig. 5(b), red (*γ*_{rad} < *γ*_{nr}), green (*γ*_{rad} = *γ*_{nr}), and blue (*γ*_{rad} > *γ*_{nr}) RRC’s, respectively represent typical behaviors of under-coupled (b < 0.5), critically-coupled (b = 0.5), and over-coupled (b > 0.5) cases. Total absorption *ρ*(*ω*) = 0 at *ω* = *ω*_{0} occurs only at the critically coupled case (the red circular dot at origin); otherwise directly scattered and leakage radiations cannot completely cancel each other even at *ω* = *ω*_{0} (the black and blue dots on the same dashed line). Some of the phasor diagrams in Fig. 2 are depicted with the same colors corresponding to these three cases. These graphical constructions were originally provided by Haus [10].

Another interesting result obtained from the RRC description is the fact that a maximum spectral variation of Δ shown in Fig. 2 quantitatively relates to the minimum value of *R*. For example, on the black RRC with *b* < 0.5 in Fig. 5(b), the phase of *ρ*(*ω*) starts to increase from *ϕ* when *ω* << *ω*_{0} up to its maximum of *ϕ* + *ξ*/2 at lower vicinity of *ω*_{0} before it falls down again to *ϕ* for *ω* >> *ω*_{0} via its minimum of *ϕ* − *ξ*/2. The Δ variation is limited by *ξ* = 2sin^{−1}[*b*/(1–*b*)] and the minimum reflectance by *R*_{min} = (1–2*b*)^{2}. Therefore, the analytic relation between *ξ* and the peak absorbance *A*_{max} = 1– *R*_{min} can be written by

In Fig. 6 , the experimental data (squares), extracted from the measured spectra in Fig. 2, strongly supports our intuitive understanding of total resonant absorption using this simple resonator model based on the coupled-mode theory.

## 5. Conclusions

We experimentally demonstrate SPP-mediated non-ohmic total absorption of light in Au gratings on Si substrate. The non-ohmic total absorption at SPP resonance is explained by resonant light transfer to multiple diffraction orders of lossy waves that give rise to nearly 10-fold enhancement in the absorbing rate when compared to the ohmic damping rate associated with an SPP on a flat Au surface. The basic physics behind this effect is identical to that for a dissipative quasi-bound resonator. In particular, the simple analytic model for a dissipative quasi-bound resonator quantitatively describes the experimentally observed relation between the peak absorbance and the reflection phase change. This absorbance-phase relation can be used for precise optimization of absorbing resonances in practical device applications. For example, this narrow angle, narrow spectral absorption can be used in direction-sensitive optical detectors. Additionally, use of stationary plasmonic modes in metallic void or core-shell arrays may improve the resonant forward diffraction to be effective over wider angular extent [3].

## Acknowledgments

This work was supported in part by the UT System Texas Nanoelectronics Research Superiority Award funded by the State of Texas Emerging Technology Fund. Additional support was provided by the Texas Instruments Distinguished University Chair in Nanoelectronics endowment. Moreover, this work was supported in part by the National Research Foundation of Korea grant funded by the Korea Government (MEST) [2010-0000256] and the IT R&D program [2008-F-022-01] of the MKE/IITA, Korea.

## References and links

**1. **J. Yoon, K. H. Seol, S. H. Song, and R. Magnusson, “Critical coupling in dissipative surface-plasmon resonators with multiple ports,” Opt. Express **18**(25), 25702–25711 (2010). [CrossRef] [PubMed]

**2. **Y. P. Bliokh, J. Felsteiner, and Y. Z. Slutsker, “Total absorption of an electromagnetic wave by an overdense plasma,” Phys. Rev. Lett. **95**(16), 165003 (2005). [CrossRef] [PubMed]

**3. **T. V. Teperik, F. J. García de Abajo, A. G. Borisov, M. Abdelsalam, P. N. Bartlett, Y. Sugawara, and J. J. Baumberg, “Omnidirectional absorption in nanostructured metal surfaces,” Nat. Photonics **2**(5), 299–301 (2008). [CrossRef]

**4. **W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption,” Science **331**(6019), 889–892 (2011). [CrossRef] [PubMed]

**5. **A. N. Grigorenko, P. I. Nikitin, and A. V. Kabashin, “Phase jumps and interferometric surface plasmon resonance imaging,” Appl. Phys. Lett. **75**(25), 3917–3919 (1999). [CrossRef]

**6. **A. V. Kabashin and P. I. Nikitin, “Surface plasmon resonance interferometer for bio- and chemical-sensors,” Opt. Commun. **150**(1-6), 5–6 (1998). [CrossRef]

**7. **T. D. Heidel, J. K. Mapel, M. Singh, K. Celebi, and M. A. Baldo, “Surface plasmon polariton mediated energy transfer in organic photovoltaic devices,” Appl. Phys. Lett. **91**(9), 093506 (2007). [CrossRef]

**8. **K. Tvingstedt, N.-K. Persson, O. Inganas, A. Rahachou, and I. V. Zozoulenko, “Surface plasmon increase absorption in polymer photovoltaic cells,” Appl. Phys. Lett. **91**(11), 113514 (2007). [CrossRef]

**9. **C. Hägglund and S. P. Apell, “Resource efficient plasmon-based 2D-photovoltaics with reflective support,” Opt. Express **18**(S3Suppl 3), A343–A356 (2010). [CrossRef] [PubMed]

**10. **H. A. Haus, *Waves and Fields in Optoelectronics* (Prentice-Hall, Englewood Cliffs, NJ, 1984).

**11. **J. Chandezon, M. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. **72**(7), 839–846 (1982). [CrossRef]

**12. **E. D. Palik, *Handbook of Optical Constants of Solids II* (Academic Press, San Diego, 1998).

**13. **D. Maystre, M. Nevière, and P. Vincent, “On the general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” J. Mod. Opt. **25**(9), 905–915 (1978). [CrossRef]