Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Tianqi
Sai
^{ab},
Matthias
Saba
^{a},
Eric R.
Dufresne
^{b},
Ullrich
Steiner
*^{a} and
Bodo D.
Wilts
*^{a}
^{a}Adolphe Merkle Institute, University of Fribourg, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland. E-mail: ullrich.steiner@unifr.ch; bodo.wilts@unifr.ch
^{b}Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 5, CH-8093 Zürich, Switzerland

Received
26th February 2020
, Accepted 26th March 2020

First published on 22nd April 2020

For a number of optical applications, it is advantageous to precisely tune the refractive index of a liquid. Here, we harness a well-established concept in optics for this purpose. The Kramers–Kronig relation provides a physical connection between the spectral variation of the (real) refractive index and the absorption coefficient. In particular, a sharp spectral variation of the absorption coefficient gives rise to either an enhancement or reduction of the refractive index in the spectral vicinity of this variation. By using bright commodity dyes that fulfil this absorption requirement, we demonstrate the use of the Kramers–Kronig relation to predictively obtain refractive index values in water solutions that are otherwise only attained with toxic specialised liquids.

To this end, it is useful to consider the fundamental theories of the propagation of light. Based on the finite speed of the propagation of light, the consequences of causality for mathematical relationships of optical constants were considered nearly 100 years ago: for the scattering of light from an object, causality requires that “no scattered wave can exist before the incident wave has reached the scattering center”.^{9} Based on this consideration, Kramers^{10} showed that the refractive index of a medium can be calculated from its absorption spectrum. Combined with Kronig’s^{11} argument that a “dispersion relation is a sufficient and a necessary condition for strict causality to hold”, this establishes the well-known Kramers–Kronig relations, which constitute the connection between the in- and out-of-phase responses of a system to sinusoidal perturbations. Mathematically, the Kramers–Kronig relations connect the real and imaginary parts of any complex function that is analytic in the upper half-plane and vanishes sufficiently fast for large arguments.^{9,12} Physically, such a function corresponds to an analytical linear response function of finite width in the time domain.

For an optical system with a complex refractive index n = n + iκ, with n(ω) the ‘real’ refractive index and κ(ω) the extinction coefficient, both of which vary with the angular frequency of the electromagnetic field ω, the Kramers–Kronig relation can be written as^{9}

(1) |

When used in spectroscopy, it is useful to write eqn (1) in terms of the optical wavelength λ = 2πc/ω, where c is the speed of light. A substitution of the integration variable yields

(2) |

An additional requirement of the Kramers–Kronig relation is the superconvergence theorem,^{9} leading to

(3) |

(4) |

The use of the Kramers–Kronig relation in optics is extremely well established.^{9,13–15} It is often used to determine the spectral variation n(λ) of an analyte from a measured absorption spectrum, and vice versa.

While identical in terms of the underlying physics, this work explores an alternative approach to creating new high refractive index liquids with a designed optical response, by exploring the Kramers–Kronig relation. This concept was illustrated by the refractive index dispersion of pigments in biological samples,^{14,15} the oxygen saturation levels of human blood,^{13} and the high refractive index materials in the wing scales of pierid butterflies.^{16}

n_{C} = A + B/λ^{2}, | (5) |

Fig. 1 Refractive index spectra, calculated by numerically solving eqn (2). (a) Hypothetical extinction spectrum confined to the 100–400 nm band. (b) Refractive index change Δn corresponding to (a). (c) Alternative hypothetical extinction spectrum and (d) corresponding Δn spectrum. Similar plots with extinction spectra given by single Lorentzian peaks are shown in Fig. S3.† |

While perhaps somewhat unrealistic, Fig. 1a and b illustrate the scope of our approach. Importantly, the enhancement of Δn extends to wavelengths where the absorption coefficient is practically zero. The strong enhancement close to, but above, the absorption band makes this idea particularly attractive in applications where only narrow-band illumination is required, e.g. telecommunication^{18} or lenses for immersion lithography.^{19} By carefully selecting a short-wavelength absorber, a high refractive index can be obtained in a controlled fashion.

Fig. 1c and d show a prediction which is closer to physical parameters that can be achieved for liquids: an extinction coefficient κ = 0.52 which corresponds to α ≈ 10.3 μm^{−1} at λ = 630 nm for a 0.6 M Brilliant Blue solution in water (see below). In this simulation, the absorption edge was smeared out over ca. 35 nm. Here, an overall refractive index enhancement of ≈1 is expected at 400 nm compared to the dye-free solution and a value of Δn ≈ 0.8 is expected at 430 nm, where the dye absorption has decayed to just a few percent of its maximal value. While perhaps not as impressive as the hypothetical case in Fig. 1b, this concept has the potential to enhance the refractive index of common liquids into the range commonly reserved for specialised toxic liquids.^{3}

To start our investigation, UV-Vis transmission spectra of all four dyes were acquired. Water solutions of the four dyes were prepared with a series of concentrations c_{d} up to the solubility limit of the dyes (between 300 mM and 600 mM). The spectral absorption coefficients were then determined according to the Beer–Lambert law, as shown in Fig. 2. The absorption spectrum of Allura Red shows rather broad spectral peaks with a broad decay toward long wavelengths and an absorption peak at about 500 nm. In contrast, Brilliant Blue, Quinoline Yellow, and Pyranine show sharper spectral decays towards long wavelengths at 630 nm, 412 nm, and 428 nm, respectively, which combine the prerequisite for a strong Δn enhancement at these wavelengths according to eqn (2), with the possibility to achieve significant Δn enhancement in the spectral domain where light absorption is insignificant (see Fig. 1). In particular, Brilliant Blue is an excellent candidate dye with a molar absorbance of 17.25 ± 0.04 cm^{−1} at 630 nm.

Spectroscopic ellipsometry was employed to measure the spectral variation n(λ) of the various dye solutions, according to the protocol of Synowicki et al.^{21} In this method, dye solution drops are placed onto a rough glass substrate. The rough glass substrate enables the spreading of the water–dye solutions to provide a planar liquid surface and it suppresses the specular back reflection of the incident light beam from the liquid–substrate interface. Since only the phase relationship of the light reflected from the liquid–air surface is analysed, the ellipsometry data analysis is not very sensitive to the imaginary part of the refractive index κ. Therefore, only the real part of the refractive index n(λ) was extracted from the ellipsometric data. To test the reliability of this approach, the variation of the refractive index of water was determined in the 400–1000 nm wavelength range and an agreement with literature data of better than 0.5% was found (Fig. S2 in the ESI†). As a further test of the measurement protocol, a 6-wavelength (455–655 nm) Abbe refractometer was also used, with a measurement accuracy for refractive index of 4 × 10^{−4}.

The results of the ellipsometry and refractometry measurements for the highest concentrations of the four dyes in water are shown in Fig. 2. A comparison with the spectral variation of the absorption curves bears out the qualitative variation of Δn in Fig. 1, i.e. all curves exhibit the signature predicted by the Kramers–Kronig relations.

To calculate the variation of n(λ) from the absorption curves in Fig. 2, the limitations of the integration of eqn (2) have to be discussed. The prediction of the Kramers–Kronig relation is only precise if the variation of the extinction coefficient κ is known for the entire spectral wavelength range (0 → ∞). Since this is generally impossible, we exploit the limiting properties of eqn (2) given by the Cauchy equation (eqn (5)).

Considering, for example, an absorption spectrum with two distinct spectral features (Fig. S3 in the ESI†), the integral in eqn (2) can be decomposed into two separate integrations, the results of which are then added to yield the overall spectral response n(λ).^{9} In the range of the long-wavelength absorption peak, the Δn contribution of the short-wavelength absorption peak can be approximated by eqn (5), which can be added to the result of eqn (2).

In the absence of spectral information below λ = 250 nm, which strongly affects the long-wavelength variation of Δn, the following approach was therefore adopted.

(1) Calculation of κ(λ) from the absorption curves in Fig. 2 using eqn (4), setting κ = 0 for λ < 250 nm and λ > 1000 nm.

(2) Integration of eqn (2) to obtain Δn(λ).

(3) Approximation of the refractive index variation of water n_{Cw} (Fig. S2 in the ESI†) by eqn (5).

(4) Calculation of n_{KK}(λ) = Δn(λ) + n_{Cw}(λ) + n_{Cd}(λ), with another Cauchy-type refractive index contribution of the dye, n_{Cd}(λ), as given by eqn (5). The coefficients A and B were fitted so that n_{KK} overlaps with the ellipsometry results in the short and long wavelength ranges far from the oscillatory feature in Δn (Table S1 in the ESI†).

The motivation for the fitting procedure resulting in n_{Cd} stems from the fact that κ(λ) of the dye is unknown for λ < 250 nm, the presence of which will give rise to a Cauchy-type decay for λ > 250 nm. Using this approach, an excellent agreement between calculated and measured refractive index spectra in the visible wavelength range was found (Fig. 2b, d, f and h).

Fig. 3 Refractive index variation of Brilliant Blue solutions in water as a function of dye concentration. (a) Ellipsometry measurements (solid lines) are compared to n_{KK}(λ) calculated using the Kramers–Kronig relation (eqn (2), dashed lines). Note the increase in the refractive index at short wavelengths, which is indicative of additional dye absorption peaks below 250 nm (see text).^{22} (b) A plot of the absorption coefficient vs. Δn clearly reveals a substantial Δn enhancement in spectral regions where the absorption coefficient is practically zero. |

The comparison of the calculated n_{KK} with the measured ellipsometry spectra is very good, given the fact that spectral information outside the 250–1000 nm wavelength window is missing and Cauchy approximations were employed instead. Note also that while the fit determining n_{Cd}(λ) improves the Kramers–Kronig description of the experimental data, the predictive quality of n_{KK} even in the absence of any fitting (i.e. setting n_{Cd} = 0) is very good, showing the strength of this approach (Fig. S4 in the ESI†).

A further way to control the refractive index of a liquid at a given wavelength is the combination of the “traditional” method, the addition of a transparent additive, with the present Kramers–Kronig approach. Fig. 4 shows the measured refractive index spectra of 0.1 M Brilliant Blue dissolved in two sugar solutions. The combination of raising the Cauchy background (eqn (5)) with the Kramers–Kronig response of the dye provides a versatile way to control the refractive index of the solution across large parts of the optical spectrum.

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## Footnote |

† Electronic supplementary information (ESI) available: Materials and methods, Table S1 and Fig. S1–S4. See DOI: 10.1039/d0fd00027b |

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