# References and Further Reading *A curated bibliography across the domains touched by this guide — original papers, standard textbooks, and online references. Entries are grouped by topic rather than ordered alphabetically; within each group, classic sources appear first and modern references last.* ## Original papers - R. de L. Kronig, "On the theory of dispersion of X-rays", *Journal of the Optical Society of America*, vol. 12, no. 6, pp. 547–557, 1926. The first known dispersion relation, derived from a Sellmeier- style atomic model with no complex variables. Gives only the `n − 1`-from-absorption relation. - H. A. Kramers, "La diffusion de la lumière par les atomes", *Atti del Congresso Internazionale dei Fisici, Como*, vol. 2, pp. 545–557, 1927. Hilbert-pair statement on the *atomic polarizability*; the corresponding statement on the bulk susceptibility is left as a conjecture. - R. de L. Kronig, "Algemeene theorie der diëlectrische en magnetische verliezen" ("General theory of dielectric and magnetic losses"), *Nederlands Tijdschrift voor Natuurkunde*, vol. 9, pp. 402–409, 1942. The first model-independent derivation of the full K–K pair, using the Fourier transform of a causal polarization response function. - E. C. Titchmarsh, *Introduction to the Theory of Fourier Integrals*, Oxford University Press, 1937 (2nd ed. 1948). Contains the equivalence theorem now called Titchmarsh's theorem. - R. E. A. C. Paley and N. Wiener, *Fourier Transforms in the Complex Domain*, American Mathematical Society Colloquium Publications, vol. 19, 1934. - C. F. Bohren, "What did Kramers and Kronig do and how did they do it?", *European Journal of Physics*, vol. 31, no. 3, pp. 573–577, 2010. Historical correction: traces the original 1926, 1927, and 1942 papers and shows that the modern attribution of contour integration, analytic continuation, and signal-causality arguments to the original authors is largely posthumous. - J. Bechhoefer, "Kramers–Kronig, Bode, and the meaning of zero", *American Journal of Physics*, vol. 79, no. 10, pp. 1053–1059, 2011. Sharpens the distinction between K–K (an equality) and Bode (an inequality), with elementary examples — flexible objects, optical reflectance, bicycles — of how upper-half-plane zeros add phase beyond the K–K-predicted minimum. ## Textbooks — electromagnetism and optics - J. D. Jackson, *Classical Electrodynamics*, 3rd ed., Wiley, 1999. §7.10 covers K-K for the dielectric constant and derives the sum rules. - L. D. Landau and E. M. Lifshitz, *Electrodynamics of Continuous Media* (Course of Theoretical Physics, Vol. 8), 2nd ed., Pergamon, 1984. §82 presents K-K from first principles of causality. - H. M. Nussenzveig, *Causality and Dispersion Relations*, Academic Press, 1972. The deepest single-volume treatment of the mathematical foundations; UHP analyticity, subtractions, sum rules, and connections to scattering theory. - V. Lucarini, J. J. Saarinen, K.-E. Peiponen, E. M. Vartiainen, *Kramers–Kronig Relations in Optical Materials Research*, Springer Series in Optical Sciences, vol. 110, 2005. Comprehensive on applications and nonlinear generalisations. - F. Wooten, *Optical Properties of Solids*, Academic Press, 1972. Classic applied treatment of K-K in reflectance and ellipsometry. - M. Born and E. Wolf, *Principles of Optics*, 7th expanded ed., Cambridge University Press, 1999. ## Dielectric spectroscopy - A. K. Jonscher, *Dielectric Relaxation in Solids*, Chelsea Dielectrics Press, 1983. The standard on dispersion mechanisms in disordered materials. - F. Kremer and A. Schönhals (eds.), *Broadband Dielectric Spectroscopy*, Springer, 2003. Modern reference covering instrumentation, fitting, and K-K validation. - J. R. Macdonald, *Impedance Spectroscopy: Theory, Experiment, and Applications*, 2nd ed., Wiley, 2005. ## DSP and signal processing - A. V. Oppenheim and R. W. Schafer, *Discrete-Time Signal Processing*, 3rd ed., Pearson, 2010. Chapter 12 covers the discrete Hilbert transform; chapters 5 and 13 cover minimum phase and the cepstrum. - A. Papoulis, *The Fourier Integral and Its Applications*, McGraw-Hill, 1962. The classic derivation of the Hilbert transform as an analytic-signal operator. - R. N. Bracewell, *The Fourier Transform and Its Applications*, 3rd ed., McGraw-Hill, 2000. - F. W. King, *Hilbert Transforms*, vols 1–2, Cambridge University Press, 2009. The most thorough reference on the mathematical theory and numerical implementation of Hilbert transforms. - S. L. Marple, "Computing the Discrete-Time Analytic Signal via FFT", *IEEE Transactions on Signal Processing*, vol. 47, no. 9, pp. 2600–2603, 1999. ## Communications - J. G. Proakis and M. Salehi, *Digital Communications*, 5th ed., McGraw-Hill, 2008. - S. Haykin, *Communication Systems*, 5th ed., Wiley, 2009. - D. B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum", *Proceedings of the IEEE*, vol. 54, no. 2, pp. 329–330, 1966. - A. Mecozzi, C. Antonelli, M. Shtaif, "Kramers–Kronig coherent receiver", *Optica*, vol. 3, no. 11, pp. 1220–1227, 2016. ## Control theory - H. W. Bode, *Network Analysis and Feedback Amplifier Design*, Van Nostrand, 1945. The original gain–phase theorem; still worth reading for the physical intuition. - K. J. Åström and R. M. Murray, *Feedback Systems: An Introduction for Scientists and Engineers*, 2nd ed., Princeton University Press, 2020. ## Microwave and interconnect - P. Triverio, S. Grivet-Talocia, M. S. Nakhla, F. G. Canavero, R. Achar, "Stability, Causality, and Passivity in Electrical Interconnect Models", *IEEE Transactions on Advanced Packaging*, vol. 30, no. 4, pp. 795–808, 2007. - D. M. Pozar, *Microwave Engineering*, 4th ed., Wiley, 2011. - S. Grivet-Talocia and B. Gustavsen, *Passive Macromodeling: Theory and Applications*, Wiley, 2015. ## Computational electromagnetics (FDTD) - A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method*, 3rd ed., Artech House, 2005. - D. M. Sullivan, *Electromagnetic Simulation Using the FDTD Method*, 2nd ed., Wiley-IEEE Press, 2013. ## Acoustics and audio - H. Kuttruff, *Room Acoustics*, 6th ed., CRC Press, 2016. - A. D. Pierce, *Acoustics: An Introduction to Its Physical Principles and Applications*, 3rd ed., Springer, 2019. - J. Blauert and P. Laws, "Group delay distortions in electroacoustical systems", *Journal of the Acoustical Society of America*, vol. 63, pp. 1478–1483, 1978. ## Electrochemistry and impedance - B. A. Boukamp, "A linear Kronig–Kramers transform test for immittance data validation", *Journal of the Electrochemical Society*, vol. 142, pp. 1885–1894, 1995. ## Scattering theory and particle physics - M. L. Goldberger and K. M. Watson, *Collision Theory*, Wiley, 1964. The original application of dispersion relations to forward- scattering amplitudes. - R. Newton, *Scattering Theory of Waves and Particles*, 2nd ed., Dover, 2002. ## Nonlinear extensions - S. Scandolo and F. Bassani, "Nonlinear sum rules: The three-level and the anharmonic-oscillator models", *Physical Review B*, vol. 51, pp. 6925–6934, 1995. - V. Chernyak and S. Mukamel, "Generalized sum rules for optical nonlinearities of many-electron systems", *Journal of Chemical Physics*, vol. 103, pp. 7640–7644, 1995. ## Online references Wikipedia provides reliable overview articles that are a good entry point and that link to primary literature: - [en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations](https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations) - [en.wikipedia.org/wiki/Hilbert_transform](https://en.wikipedia.org/wiki/Hilbert_transform) - [en.wikipedia.org/wiki/Minimum_phase](https://en.wikipedia.org/wiki/Minimum_phase) - [en.wikipedia.org/wiki/Dispersion_relation](https://en.wikipedia.org/wiki/Dispersion_relation) - [en.wikipedia.org/wiki/Titchmarsh_theorem](https://en.wikipedia.org/wiki/Titchmarsh_theorem) - [en.wikipedia.org/wiki/Analytic_signal](https://en.wikipedia.org/wiki/Analytic_signal) Discussion on Physics Stack Exchange and Signal Processing Stack Exchange often clarifies sign conventions and edge cases; use the site's own search rather than relying on remembered URLs. Publisher landing pages useful for recent research: - Optica (formerly OSA) at [www.optica.org](https://www.optica.org) - IEEE Xplore at [ieeexplore.ieee.org](https://ieeexplore.ieee.org) - APS journals at [journals.aps.org](https://journals.aps.org) ## Historical notes The name "Kramers–Kronig" reflects the near-simultaneous but independent 1926–1927 derivations by Kronig (working at Columbia on X-ray dispersion) and Kramers (at Copenhagen, working on optical dispersion in the context of the Bohr–Kramers–Slater theory). Both derivations, however, started from a Sellmeier-style atomic model with the `1/(Ω² − ω²)` denominator built in; *neither* paper made the modern argument from linearity, causality and analyticity, and neither mentioned signal-speed at all. Kronig 1926 contains no complex variables and gives only the `n − 1` half of the eponymous pair; the χ′-from-χ″ relation was left as a conjecture by Kramers in 1927 and first derived from causality alone by Kronig in 1942 (in a Dutch- language journal article that is rarely cited). The mathematical content — a Hilbert transform on the real line arising from UHP analyticity — was already known to complex analysts (Sokhotski 1873; Plemelj 1908; Hilbert 1912). Titchmarsh (1937) is the first clean mathematical statement of the equivalence between causality and the Hilbert pair. Bohren's 2010 article (cited above) is the standard reference for this corrected historical narrative. ## See also - [01_introduction.md](01_introduction.md) - [02_historical_context.md](02_historical_context.md) - [25_vna_measurements.md](25_vna_measurements.md) - [30_pitfalls_limitations.md](30_pitfalls_limitations.md)