J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025) 211
For less intuitive cases where the transmitted wave is inhomogeneous, we provide visual illustrations to highlight the physical significance of phase delays between coexisting electric fields. Additionally, our approach offers practical methods for determining the exponential decay rate, wavelength, and propagation direction, as outlined in equations( A. 15)–( A. 17) of Appendix. This includes the special case of evanescent waves.
In summary, the procedure presented here for determining the electric fields at the interface is more practical and efficient than previously established formal methods [ 9, 10 ]. Moreover, it highlights the underlying physics of the fields at the interface while addressing critical properties of light in the second medium, such as its propagation direction and penetration depth.
Funding
This work was supported by projects TED2021-129639B-I00, CNS2022-136051 and PID2022-138699OB-I00 from the Ministerio de Ciencia e Innovación of Spain. Work also supported by TED2021-129639B-I00 / AEI / 10.13039 / 501100011033 / Unión Europea NextGenerationEU / PRTR
Conflicts of interest The authors do not have any kind of conflict of interest.
Data availability statement This article has no associated data generated and / or analyzed.
Author contribution statement
The first author is responsible for formulating the original idea, while the second author contributed essential comments that are reflected inthefinal text.
References
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Appendix
We summarize the properties of inhomogeneous plane waves that are relevant for the present paper.
Inhomogeneous versus homogeneous plane waves
What are monochromatic inhomogeneous plane waves? What is their physical significance? First, remember that monochromaticity for an electromagnetic field means that the space ð ~ rÞ and time( t) variations can be factorized in the form h i h i physical f ield ¼ Re Eð ~ ~ r; tÞ ¼ Re Eð ~ ~ rÞ expð�ixtÞ ðA: 1Þ where x is the angular frequency and Eð ~ ~ rÞ is a 3-D vector with( in the most general case) complex components Eð ~ ~ rÞ ¼ ðE x ð ~ rÞ; E y ð ~ rÞ; E z ð ~ rÞÞ.
Linear operations on the fields( addition, derivation...) will not mix real and imaginary parts of the complex expressions and, therefore, complex form will always be used in our work for convenience. Specifically, we will only focus on Eð ~ ~ rÞ as it contains all the significant information.
In 3-D space, a particular case of( A. 1) above is the monochromatic homogeneous plane wave inside a transparent medium with refractive index n. It may be simply written in terms of the scalar product between a standard wavevector ~ k ¼ ðk x; k y; k z Þ and the position vector ~ r. Thevector ~ k has three real-valued components, points in the direction of the wave propagation and is related to the wavelength by k ¼ 2p = j ~ kj. Thus, the monochromatic homogeneous plane wave is written
E ~ ð~ r; tÞ ¼ E~ 0 exp i ~ k ~ r � xt; ðA: 2Þ
where E~ 0 is a constant vector with three complex components. Since all the relevant information on the wave is contained in the spatial part
E ~ ðÞ¼ ~ r E~ 0 exp i ~ k ~ r; ðA: 3Þ
we conclude that the couple of vectors ~ E 0 and ~ k fully define our homogeneous monochromatic plane wave.
Now consider not just one but two standard vectors, namely ~ k R; ~ k I, each with three real components( thus each one indicating
one direction in 3-D space), and consider the following mathematical expression
~ k ~ kR þ i ~ k I; ðA: 4Þ
as the definition of the complex wavevector ~ k and, for the electric field, consider the expression
E ~ ðÞ¼ ~ r E~ 0 exp i ~ k ~ r ¼ E~ 0 exp i kR ~ þ i k ~
I ~ r
¼ exp � k ~
I ~ r E ~ 0 exp i k ~
R ~ r: ðA: 5Þ