User:Antoine Moreau/Proposed/Kramers-Kronig relations

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Kramers-Kronig relations are primarily used in optical spectroscopy to find out the complex refractive index of a medium from the measured absorption, transmission or reflection spectrum. They provide fundamental constraints assessing the self-consistency of measured data or of the output of numerical models. The derivation of the Kramers-König relations is based on one of the fundamental properties of physics, i.e. causality. Other applications include the analysis of acoustic data and, more recently, the investigation of the response of general dynamical systems to small perturbations.


Derivation of Kramers-Kronig relations

Kramers-Kronig relations provide fundamental constrains for the linear frequency-dependent response of a system to an external perturbation and, as discussed by Kronig (1926), Kramers (1927), Nussenzveig (1972), Peiponen et al. (1999), and Lucarini et al. (2005), are basically equivalent to the physical property of causality of a system. In the case of interaction of an electromagnetic wave with a medium, causality implies that the medium will respond to the external perturbation, namely the incident wave, only after its incidence. In the spectral region of visible light, the response of the medium can be observed as the polarization of the electrons. If the electrons were polarized before the disturbance, the process would violate causality due to the discrepancy in the time ordering of the cause and the response. The polarization of electrons may have a resonance, i.e. they may feature a greatly amplified response, at some specific wavelength of the incident light. A special characteristic of resonances is that the incident light undergoes pronounced absorption, whose intensity is weaker as the wavelength departs from the resonant value. Transmittance spectroscopy often allows for measuring experimentally the absorption of a material. When measuring light transmittance across a medium with a spectrometer, an absorption band of the medium causes the intensity of the transmitted light to become weaker. From the transmittance, which is the ratio of the transmitted and incoming light, it is possible to get the absorption coefficient (\(\alpha\)) of the medium. Not only light is absorbed as a result of the interaction: also the velocity of the light depends on the wavelength of the incident light wave. Such a phenomenon is known in optics as light dispersion, which means that the velocity (\(v\)) of the electromagnetic wave in the medium is equal to \(v = c/n(\lambda)\ ,\) where \(c\) is the light velocity in vacuum, \(n\) is the refractive index of the medium and \(\lambda\) is the wavelength of light. Kramers and Kronig realized that the refractive index and so-called extinction coefficient (\(k\)), \(k = \lambda \alpha/4\pi\ ,\) of the medium are connected by integral relations. If the extinction coefficient of the medium is known then the refractive index can be calculated and vice versa. This implies that knowing over the full spectral range either the real or the imaginary part of the complex refractive index \(n + ik\ ,\) which is fundamental physical property of the medium, it is possible to reconstruct the full refractive index. The knowledge of the complex refractive index can be exploited in identification of chemical species, density, purity and strength of light interaction with a medium. The derivation of the Kramers-Kronig relations is based on assumptions imposed on the complex refractive index. These are:

  1. the complex physical quantity n + ik is an analytic function in the upper half of complex circular frequency plane,
  2. the function has a sufficiently fast fall-off at high frequencies and
  3. the real part of the complex quantity is an even and the imaginary part is an odd function of the variable circular frequency (\(\omega\)), \(\omega = 2\pi c/\lambda\ .\)

Under such assumptions and using complex contour integration the Kramers-Kronig relations can be derived.

Conventional and singly subtractive Kramers-Kronig relations

A pair of conventional Kramers-Kronig relations that have been used for data inversion in transmission spectroscopy read as follows:

\[\tag{1} n(\omega') -1 = \frac{2}{\pi}\, \mathcal{P} \int_0^\infty \frac{\omega\,k(\omega)}{\omega^2 - \omega'^2} \, d\omega\]


\[\tag{2} k(\omega') = -\frac{2\omega'}{\pi}\, \mathcal{P} \int_0^\infty \frac{n(\omega)-1}{\omega^2-\omega'^2} \, d\omega\]

where \(\mathcal{P}\) stands for Cauchy principal value which is a limiting process to approach the singularity namely \(\omega'\) symmetrically from the left and the right. Transmission measurement incorporated with the Kramers-Kronig relation (#kk1) is powerful in spectral studies of solid, liquid, and gaseous phases if the medium is optically not very dense or thick, to find out light dispersion. The present computer technology makes it possible to implement algorithms into the PC of a spectrophotometer so that the information of refractive index change of the medium can be calculated immediately after the measurements. The dispersion relation (#kk2) has been rarely used however, it is also essential because it assists in testing the success of the data inversion carried out with the dispersion relation (#kk1). It is important to underline that the pair of Kramers-Kronig formulas provide a powerful tool to test the self-consistency of the measured data. In Fig. (1) we present an example of investigation of the optical properties of color centers through Kramers-Kronig analysis.

Figure 1: Measured extinction curves of F and M color centers in a KBr crystal at different temperatures (a); the corresponding changes of the refractive index are computed via Kramers-Kronig relation (b). The curves correspond to 1–250 K, 2–170 K, 3–100 K, and 4–35 K, with peaks becoming sharper at low temperature. Adapted from Peiponen et al. (1984).

For instance metals and semiconductors are opaque, hence measurement of reflectance suits better for resolving the frequency (wavelength)-dependent complex refractive index of the medium. In typical Kramers-Kronig analysis the measurement is carried at normal incidence. In the case of oblique angle of incidence utilization of s-polarized light is preferred and the relevant reflectance obtained from Fresnel’s theory for light reflection. The choice of p-polarized light can be sometimes problematic especially when there are singularities of the complex reflection coefficient in the upper-half of complex frequency plane. In such a case, as proven by Toll (1956), Nussenzveig (1972), and Lucarini et al. (2005), so-called Blaschke product modifies the complex reflection coefficient of the medium, and causes phase jumps that conventional Kramers-Kronig relations do not take into account. Below are Kramers-Kronig dispersion relations for the case of normal reflection of light:

\[\tag{3} \phi(\omega') = \frac{2}{\pi} \mathcal{P} \int_0^\infty \frac{\ln r(\omega}{\omega^2 - \omega'^2} d\omega\]


\[\tag{4} \ln \left| r(\omega') \right| - \ln \left| r(\omega'') \right| = \frac{2}{\pi} \mathcal{P} \int_0^\infty\, \omega \,\phi(\omega) \,\left( \frac{1}{\omega^2-\omega'^2} - \frac{1}{\omega^2-\omega''^2}\right) d\omega\]

where \(r\) is the complex reflection coefficient \(r = |r|\,e^{i\phi}\ .\) Note that the dispersion relations involve the natural logarithm of the reflection coefficient, i.e., \(\ln r =\ln |r| +i \phi\ .\) Relation (#phi) is the one that is most frequently utilized in reflection studies, but also (#r) is useful in testing the consistency of measured data and success of the phase retrieval by (#phi). In the case of (#r) the reflectance has to be a priori known at an "anchor point" \(\omega ''\ .\)

The disadvantage of the conventional Kramers-Kronig relations is the practical restriction of data available at a finite spectral range hence introducing an error for the extracted optical properties of the medium. Ahrenkiel (1971) introduced the idea of improve the Kramers-Kronig analysis by adopting dispersion integrals, which have better convergence than the conventional Kramers-Kronig relations. These dispersion integrals are known as singly subtractive Kramers-Kronig relations. They are based on an assumption that a priori information on the complex refractive index or complex reflection coefficient is available at one anchor point (i.e. frequency) in the measured spectral range. In many cases the pair of singly subtractive Kramers-Kronig relations is sufficient for obtaining a good estimate for the wanted complex optical function. For a function \(f\) that may be the complex refractive index, reflection coefficient, nonlinear susceptibility or other function in physics and fulfilling the three assumptions stated above, one can write

\[\tag{5} u(x')-u(x_1) = \frac{2\,(x'^2-x_1^2)}{\pi}\,\mathcal{P} \int_0^\infty \frac{x\,\nu(x)}{(x^2-x'^2)(x^2-x_1^2)}\,dx\]

\[\tag{6} \nu(x_1)-\nu(x') = \frac{2\,(x'-x_1)}{\pi} \,\mathcal{P} \int_0^\infty \frac{x^2+x'\,x_1}{(x^2-x'^2)(x^2-x_1^2)} \,dx\]

where \(x\) denotes circular frequency or energy and \(x_1\) is the anchor point. The integrals (#integ1-#integ2) have stronger convergence than those of (#kk1) - (#r). Palmer et al. (1998) showed that it is also possible to derive multiply subtractive Kramers-Kronig relations which have more complicated mathematical expressions than equations (1) - (4). See also Lucarini et al. (2005) for a detailed description of these techniques.

As proven by Altarelli and Smith (1974), Kramers-Kronig relations have been used to get dispersion relations for the integer powers of optical functions and also their moments, and these relations provide a forum both for testing the success of data inversion and information on the physical properties of the medium via derived sum rules.

It is worth mentioning that, using the tools of quasi-equilibrium statistical mechanics, it is possible to derive Kramers-Kronig relations connecting the real and imaginary part of generalized susceptibilities, which describe the frequency-dependent response of a thermalised classical or quantum statistical mechanical system to external perturbations. As discussed in, e.g., Landau and Lifshitz (1980), by exploiting a formal link between the fluctuation-dissipation theorem and the Kramers-Kronig relations, one can derive the full properties of the linear response of a system to a given perturbation from the observation of the thermal fluctuations of a suitably defined observable.

Beyond the classical Kramers-Kronig theory

The classical Kramers-Kronig theory has mostly been developed with the aim of studying the linear response (Kubo 1957) of quasi-equilibrium statistical mechanical systems, both quantum and classical. In recent years, the theory has been mainly extended in two directions, namely by addressing the properties of the output signal resulting from nonlinear processes, and by studying the response of fully non-equilibrium (forced and dissipative) statistical mechanical systems.

It has been proved and experimentally verified on various media that specific classes of nonlinear phenomena – most notably general harmonic generation and non-degenerate Kerr processes – are described by susceptibilities which obey general Kramers-Kronig type dispersion relations and sum rules. See Fig. (2) for an example of application of conventional and singly subtractive Kramers-Kronig relations for the analysis of third harmonic generation in a polymer. On the contrary, the analysis of outputs of general nonlinear processes cannot be treated using the Kramers-Kronig formalism, while statistical approaches such as those falling in the class of maximum entropy methods can be successful. See Scandolo and Bassani (1991), Hutchings et al (1992), Peiponen et al (1999), and Lucarini et al. (2003, 2005) for a detailed discussion of this problem.

Figure 2: Efficacy of SSKK vs. K-K relations in retrieving (a) \(Re\{\chi^{(3)}(3\omega)\}\) and (b) \(Im\{\chi^{(3)}(3\omega)\}\) on polysilane (Dot in the graph=anchor point). Adapted from Lucarini et al. (2003).

Very recently, a great deal of attention has been paid to developing a rigorous theory for the investigation of the response of non-equilibrium systems to external perturbations. This is of great mathematical interest per se, especially because, as opposed to the quasi-equilibrium case, for non-equilibrium systems the fluctuation-dissipation theorem cannot be straightforwardly applied (Ruelle 1998). The Ruelle (1998,2009) response theory bears a great potential for studying the impact of perturbations on the statistical properties of forced and dissipative complex systems, such as the climate system. Also in this case, it has been proved that Kramers-Kronig theory, both in its linear and nonlinear versions (Lucarini 2008), can be used to study the frequency dependent response to perturbations and numerical evidences suggest that the practical applicability of the theory is solid. In [[#F3|Fig. (3)] we present the computed linear susceptibility for a global observable in a simplified model of the atmosphere and its reconstruction through Kramers-Kronig analysis.

Figure 3: Real (a) and Imaginary (b) part of the susceptibility of the observable total momentum of the Lorenz 96 model of the atmosphere against zonal forcing. The black lines are the measured quantities, the red lines are the extrapolations obtained using linear response theory, the blue (magenta) lines are the result of Kramers-Kronig transforms of the measured (extrapolated) data. Adapted from Lucarini and Sarno (2011).

Applications of Kramers-Kronig relations

Looking at the table books on optical constants one gets quickly an impression that Kramers-Kronig relations have been important to find out the complex refractive index especially of metals and semiconductors in the context of reflection spectroscopy, and, in general, have proven to be invaluable tool for studying the optical properties of condensed matter Bassani and Altarelli 1983). During the past decades other applications involve for instance data inversion of third harmonic nonlinear susceptibility in the field of nonlinear optics as proven by Lucarini et al (2003), detection of sample misplacement in terahertz (THz) time-domain reflection spectroscopy and correction of the THz spectra as proven by Peiponen and Saarinen (2009). Other interesting applications are in the field of acoustics (Mobley 2010), metamaterials (Szabó et al 2010), and negative refractive index media (Hickey et al 2010). Finally, recent applications of the theory by Reick (2002) and Lucarini (2009) include the analysis of the linear and nonlinear response of chaotic systems to perturbations. Proposed practical applications foresee the rigorous analysis of climate variability and climate change, as discussed in Lucarini and Sarno (2011).


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