On the Kramers-Kronig Relation for the Phase Spectrum
Abstract
The accepted result for the Kramers-Kronig relation for the phase spectrum ψ(ω) of the complex reflection coefficient at normal incidence r(ω) is shown to contain an error. Previously, phase shifts computed from reflectance data using Kramers-Kronig analysis have in some cases been shifted by ± π with no basic justification other than a claim that this provides agreement with accepted values and consistency with the incomplete Stokes treatment. The theoretical basis for this shift is provided in this short note. We derive the correct formula, and show that well-known model dielectrics are consistent with our result and not the accepted result.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The phase of the reflection coefficient can be extracted from its amplitude and, conversely, the amplitude can be extracted from its phase via the Kramers-Kronig relations.
“These two equations state that the phase of the reflection coefficient can be extracted from its amplitude and, conversely, the amplitude of the reflection coefficient can be extracted from its phase.”
From Robust phase correction techniques for terahertz time-domain reflection spectroscopy (2024) · cites this paper as [34]
The Kramers-Kronig relations are bidirectional complex relations based on causality that connect the real and imaginary parts of a response function, typically the optical permittivity.
“The Kramers-Kronig relations are bidirectional complex relations that work on the principle of causality and are commonly used in optics to connect the real and imaginary parts of a response function, typically the optical permittivity.”
From Robust phase correction techniques for terahertz time-domain reflection spectroscopy (2024) · cites this paper as [34]
Cited by
Sorted by citations
Biosensing applications of all-dielectric SiO2-PDMS meta-stadium grating nanocombs
Ghasemi, M., Roostaei, N., Sohrabi, F., Hamidi, S. M., & Choudhury, P. K. (2020) · Optical Materials Express · doi:10.1364/ome.389361 · Cited by 26 · see citing works
Polymer dispersed liquid crystal films for modulating infra-red radiation
McKenna, L., Miller, L. S., & Peterson, I. R. (2004) · doi:10.1016/j.polymer.2004.07.033 · Cited by 22
II: Dispersion Relations and Phase Retrieval in Optical Spectroscopy
Peiponen, K.-E., Vartiainen, E. M., & Asakura, T. (1997) · Progress in Optics · doi:10.1016/s0079-6638(08)70337-8 · Cited by 21 · see citing works
Phase retrieval of reflection and transmission coefficients from Kramers-Kronig relations.
Gralak, B., Lequime, M., Zerrad, M., & Amra, C. (2014) · Journal of The Optical Society of America A-optics Image Science and Vision · doi:10.1364/josaa.32.000456 · Cited by 21
Saarinen, J., Vartiainen, E., & Peiponen, K. (2009) · Sensors and Actuators B Chemical · doi:10.1016/j.snb.2009.02.033 · Cited by 12 · see citing works
On the Kramers–Kronig transform with logarithmic kernel for the reflection phase in the Drude model
André, J., Guen, K. L., Jonnard, P., Mahne, N., Giglia, A., & Nannarone, S. (2010) · doi:10.1080/09500340.2010.506015 · Cited by 6
Sinha, M., Modi, M. H., Ghosh, H., Yadav, P., & Gupta, R. (2018) · Journal of Synchrotron Radiation · doi:10.1107/s1600577518002771 · Cited by 4
“The phase spectrum ’(E) obtained by KK analysis of the complex reflection coefficient contains an error of (Nash et al., 1995) which is included in our analysis.”
Effect of reflection and refraction on NEXAFS spectra measured in TEY mode
Filatova, E., & Sokolov, A. (2018) · Journal of Synchrotron Radiation · doi:10.1107/s1600577517016253 · Cited by 4
“It is known (Toll, 1956; Stern, 1963; Young, 1977; Plaskett & Schatz, 1963; Vinogradov et al., 1988; Sheik-Bahae, 2005; Lucarini et al., 2005; Nash et al., 1995; Smith, 1977; Lee & Sindoni, 1997; André et al., 2010) that generally an integral in the form (8) ambiguously defines the dependence…”
BackgroundSaarinen¶, J. J., Vartiainen∥, E. M., & Peiponen∥, K.-E. (2006) · Journal of Modern Optics · doi:10.1080/09500340500473349 · Cited by 4 · see citing works
Solving certain principal value integrals by reduction to the dilogarithm
Lee, M. (1996) · doi:10.1016/s0378-4371(96)00297-x · Cited by 4
Kiranjot, & Modi, M. H. (2021) · Journal of Applied Physics · doi:10.1063/5.0059736 · Cited by 2
Sinha, M., Singh, A., Gupta, R., & Modi, M. H. (2019) · doi:10.1063/1.5084605 · Cited by 2
Modi, M. H., Gupta, R., Yadav, P. K., Gupta, S., Mukherjee, C., & Idir, M. (2023) · Journal of Applied Physics · doi:10.1063/5.0145828 · Cited by 1
Kiranjot, Modi, M. H., Gupta, R., Sinha, M., & Yadav, P. K. (2020) · Journal of Synchrotron Radiation · doi:10.1107/s1600577520011960 · Cited by 1 · see citing works
Robust phase correction techniques for terahertz time-domain reflection spectroscopy
Jatkar, K. D., Yeh, T.-T., Pancaldi, M., & Bonetti, S. (2024) · arXiv · doi:10.48550/arxiv.2412.18662
“[31, 34, 35].”
Baux, D., Hermet, P., Campidelli, S., Bantignies, J., Rousseau, E., & Izard, N. (2023) · Journal of Physical Chemistry C · doi:10.1021/acs.jpcc.3c02962
Continuous optical constants (δ & β) spectra of aluminium oxide near Al L and O K-edge region
Sinha, M., Gupta, R., Dasilva, P., Mercère, P., & Modi, M. H. (2019) · DAE SOLID STATE PHYSICS SYMPOSIUM 2018 · doi:10.1063/1.5113046
Sharma, S., Gupta, R., Sinha, M., Yadav, P. S., Singh, A., & Modi, M. H. (2016) · doi:10.1063/1.4947879
- Authors
- Nash, P. L., Bell, R. J., & Alexander, R.
- Affiliations
- The University of Texas at San Antonio (US)
- Missouri University of Science and Technology (US)
- Venue
- Journal of Modern Optics
- Publisher
- Informa UK Limited
- Published
- 1995-09
- Volume
- 42
- Issue
- 9
- Pages
- 1837-1842
- ISSN
- 0950-0340 (ISSN-L)
- 1362-3044
- OpenAlex
- W2073510700
- Cited by
- 21
- References
- 0
- Citations by year
- 2023: 3 · 2021: 1 · 2020: 2 · 2019: 2 · 2018: 1 · 2017: 1 · 2016: 1 · 2015: 1
- FWCI
- 0.73
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Modern Optics (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Engineering › Biomedical Engineering › Optical Polarization and Ellipsometry
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Electrical and Electromagnetic Research
- Keywords
- Kramers-Kronig relations, phase spectrum, complex reflection coefficient, reflectivity, normal incidence, Stokes parameters
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
