# Kramers–Kronig Relation for Attenuated Total Reflection from a Metal–Dielectric Interface Where Surface Plasmon Polaritons Are Excited

## Abstract

**:**

## 1. Introduction

## 2. ATR in a Kretschmann–Raether Configuration

## 3. Phase Dispersion Calculation with Kramers–Kronig Relation

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Hecht, E. Optics, 5th ed.; Pearson: London, UK, 2016. [Google Scholar]
- Saleh, B.E.A.; Teich, M.C. Fundamentals of Photonics, 2nd ed.; Wiley: Hoboken, NJ, USA, 2007. [Google Scholar]
- Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; Wiley & Sons, Inc.: Hoboken, NJ, USA, 1983. [Google Scholar]
- Raether, H. Surface Plasmons on Smooth and Rough Surfaces and on Gratings; Springer: Berlin/Heidelberg, Germany, 1988. [Google Scholar]
- Maier, S.A. Plasmonics: Fundamentals and Applications; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Homola, J. Present and future of surface plasmon resonance biosensors. Anal. Bioanal. Chem.
**2003**, 377, 528–539. [Google Scholar] [CrossRef] [PubMed] - Hayashi, S.; Okamoto, T. Plasmonics: Visit the past to know the future. J. Phys. D Appl. Phys.
**2012**, 45, 433001. [Google Scholar] [CrossRef] - Sharma, A.K.; Jha, R.; Gupta, B.D. Fiber-optic sensors based on surface plasmon resonance: A comprehensive review. IEEE Sens. J.
**2007**, 7, 1118–1129. [Google Scholar] [CrossRef] - Tran, V.T.; Yoon, W.J.; Lee, J.H.; Ju, H. DNA sequence-induced modulation of bimetallic surface plasmons in optical fibers for sub-ppq (parts-per-quadrillion) detection of mercury ions in water. J. Mater. Chem. A
**2018**, 6, 23894–23902. [Google Scholar] [CrossRef] - Kim, J.; Son, C.; Choi, S.; Yoon, W.J.; Ju, H. A plasmonic fiber based glucometer and its temperature dependence. Micromachines
**2018**, 9, 506. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kim, J.; Kim, S.; Nguyen, T.T.; Lee, R.; Li, T.; Yun, C.; Ham, Y.; An, S.S.A.; Ju, H. Label-free quantitative immunoassay of fibrinogen in Alzheimer disease patient plasma using fiber optical surface plasmon resonance. J. Electron. Mater.
**2016**, 45, 2354–2360. [Google Scholar] [CrossRef] - Vukusic, P.S.; Bryan-Brown, G.P.; Sambles, J.R. Surface plasmon resonance on gratings as a novel means for gas sensing. Sens. Actuators B
**1992**, 8, 155–160. [Google Scholar] [CrossRef] - Byun, K.M.; Kim, S.J.; Kim, D. Grating-coupled transmission-type surface plasmon resonance sensors on dielectric and metallic gratings. Appl. Opt.
**2007**, 46, 5703–5708. [Google Scholar] [CrossRef] [PubMed] - Kotlarek, D.; Vorobii, M.; Ogieglo, W.; Knoll, W.; Rodriguez-Emmenegger, C.; Dostalek, J. Compact grating-coupled biosensor for the analysis of thrombin. ACS Sens.
**2019**, 4, 2109–2116. [Google Scholar] [CrossRef] [PubMed] - Luo, X.; Tsai, D.; Gu, M.; Hong, M. Extraordianry optical fields in nanostructures: From sub-diffraction-limited optics to sensing and energy conversion. Chem. Soc. Rev.
**2019**, 48, 2458–2494. [Google Scholar] [CrossRef] [PubMed] - Guo, Y.; Zhang, Z.; Pu, M.; Huang, Y.; Li, X.; Ma, X.; Xu, M.; Luo, X. Spoof plasmonic metasurfaces with catenary dispersion for two-dimensional wide-angle focusing and imaging. iScience
**2019**, 21, 145–156. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gerislioglu, B.; Dong, L.; Ahmadivand, A.; Hu, H.; Nordlander, P.; Halas, N.J. Monolithic metal dimer-on-film structure: New plasmonic properties introduced by the underlying metal. Nano Lett.
**2020**, 20, 2087–2093. [Google Scholar] [CrossRef] [PubMed] - Born, M.; Wolf, E. Principles of Optics, Electromagnetic Theory of Propagation, Interface and Diffraction of Light, 5th ed.; Pergamon Press: Oxford, UK, 1975. [Google Scholar]
- Pedrotti, F.L.; Pedrotti, L.M.; Pedrotti, L.S. Introduction to Optics, 3rd ed.; Pearson: London, UK, 2007. [Google Scholar]
- Shalabney, A.; Abdulhalim, I. Electromagnetic fields distribution in multilayer thin film structures and the origin of sensitivity enhancement in surface plasmon resonance sensors. Sens. Actuators A
**2010**, 159, 24–32. [Google Scholar] [CrossRef] - RefractiveIndex.INFO. Available online: https://refractiveindex.info/?shelf=main&book=Au&page=Yakubovsky-53nm (accessed on 21 October 2021).
- Yakubovsky, D.I.; Arsenin, A.V.; Stebunov, Y.V.; Fedyanin, D.Y.; Volkov, V.S. Optical constants and structural properties of thin gold films. Opt. Exp.
**2017**, 25, 25574–25587. [Google Scholar] [CrossRef] [PubMed][Green Version] - Haes, A.J.; Zou, S.; Zhao, J.; Schatz, G.C.; Van Duyne, R.P. Localized surface plasmon resonance spectroscopy near molecular resonances. J. Am. Chem. Soc.
**2006**, 128, 10905–10914. [Google Scholar] [CrossRef] [PubMed] - Zhao, J.; Jensen, L.; Sung, J.; Zou, S.; Schatz, G.C.; Van Duyne, R.P. Interaction of plasmon and molecular resonances for rhodamine 6G adsorbed on silver nanoparticles. J. Am. Chem. Soc.
**2007**, 129, 7647–7656. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematic for ATR in a Kretschmann–Raether configuration to excite SPP at the Au film–air interface using TM polarization of light. Given an incident angle of light (${\theta}_{in}$), reflectance reduces at a narrow spectral band as a result of SPP generation (wavelength interrogation method).

**Figure 2.**(

**A**) The reflectance $R\left(\lambda \right)\equiv {\left|r\left(\lambda \right)\right|}^{2}$ calculated by a field transfer matrix-based formula (Equation (1)), assuming the Kretschmann–Raether configuration where SPP is excited at the interface between the 50 nm-thick Au film and air. The incident angle (${\theta}_{in}$) is 43.85°. The reflection dip occurs at a wavelength of about 615 nm. (

**B**) The natural logarithm of 1/$R\left(\lambda \right)$ where $R\left(\lambda \right)$ is given by (

**A**).

**Figure 3.**The wavelength dependence of the phase shift of the reflected and transmitted electric fields calculated using a field transfer matrix formula Equations (1) and (2), respectively.

**Figure 4.**The wavelength-dependent phase calculated using the KK relation in comparison to the phase of $r\left(\lambda \right)$ calculated from a field transfer matrix formula (Equation (1)). The incident angle (${\theta}_{in}$) is 43.85°.

**Figure 5.**At other incident angles (${\theta}_{in}$) of light, the phase calculated using the KK relation (empty circles) in comparison to the phase directly obtained from $r\left(\lambda \right)$ calculated from a field transfer matrix formula Equation (1) (solid curves). The phase offsets used are ${\varphi}_{0}=-2.28,-2.20,-2.06,-2.00$ radians for (

**A**–

**D**), respectively.

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**MDPI and ACS Style**

Ju, H. Kramers–Kronig Relation for Attenuated Total Reflection from a Metal–Dielectric Interface Where Surface Plasmon Polaritons Are Excited. *Nanomaterials* **2021**, *11*, 3063.
https://doi.org/10.3390/nano11113063

**AMA Style**

Ju H. Kramers–Kronig Relation for Attenuated Total Reflection from a Metal–Dielectric Interface Where Surface Plasmon Polaritons Are Excited. *Nanomaterials*. 2021; 11(11):3063.
https://doi.org/10.3390/nano11113063

**Chicago/Turabian Style**

Ju, Heongkyu. 2021. "Kramers–Kronig Relation for Attenuated Total Reflection from a Metal–Dielectric Interface Where Surface Plasmon Polaritons Are Excited" *Nanomaterials* 11, no. 11: 3063.
https://doi.org/10.3390/nano11113063