ABSTRACT Reflectance, transmittance, and absorption of materials are also known as materials’ Fresnel problem. It is widely accepted that Interface model can be utilized to solve Fresnel

problem of two dimensional materials. Here, we question the validity of Interface model. Theoretical and experimental results of two dimensional materials are analyzed, and theoretical

optical response of two dimensional materials is derived based on thin film model. A new simple, approximate formula of 4_πnkd/λ_ is proposed for calculation of absorption of two dimensional

materials. It is found that, in essence, Interface model is a kind of approximate style of thin film model, the main difference between two models is term of (_n_2 − _k_2) at normal

better agreement with experimental reflectance results than Interface model. Unexpectedly, on contrary to other remarkable, intriguing properties, two dimensional materials exhibit an

ordinary Fresnel optical response, which is same with thin film. SIMILAR CONTENT BEING VIEWED BY OTHERS OPTICAL SECOND HARMONIC GENERATION IN ANISOTROPIC MULTILAYERS WITH COMPLETE

MULTIREFLECTION OF LINEAR AND NONLINEAR WAVES USING ♯SHAARP._ML_ PACKAGE Article Open access 29 March 2024 TERAHERTZ MICROSCOPY THROUGH COMPLEX MEDIA Article Open access 05 April 2025

REVISITING THE RYTOV APPROXIMATION IN DIFFUSE OPTICS AND ITS APPLICATIONS FOR THE INVERSE AND FORWARD PROBLEMS Article Open access 28 December 2024 INTRODUCTION Since the first successful

fabrication of graphene in 2004, its electrical, optical and magnetic properties have been widely studied. Reflectance, transmittance, and absorption of materials are also known as

materials’ Fresnel problem. Two models are utilized in solving Fresnel problem of graphene. One is that graphene is treated as an infinitesimally thin sheet, which is Interface (IF)

model1,2,3,4,5; the other is the thin film (TF) model6,7,8,9,10, and it is an open question that which one is correct. One of unique features in optical properties of graphene is that its

universal absorption in visible band is solely defined by fine structure constant or optical conductivity1,2,3,4,5. Derivation for absorption of standing-free graphene (_A_ = _πa_ = 2.3%,

where _A_ is absorption, and _a_ is fine structure constant) from Stauber was based on IF model. Experimental results agreed well with theoretical simulation in 450–800 nm1. However, there

was a deviation in other wavelength bands1, and the possible reason for this deviation was qualitatively attributed to surface contamination (<450 nm). This discrepancy still cannot be

elucidated quantitatively by present available theories. From then on, several theoretical jobs were successfully done to reproduce this special absorption value (_A_ = _πa_ = 2.3%) from

different routes11,12,13,14,15,16,17,18. Some jobs were done to study Fresnel optics of two dimensional material (TDM) on the substrate14,19, they exhibited different properties as compared

to standing-free samples, and they attribute these changes to the substrate effect. It was found that stepwise absorption of two dimensional InAs on the substrate is 1.6% rather than 2.3%,

and Fang proposed a formula of (2/(1 + _n_s))2π_α_ (_n_s is refractive index of the substrate) to describe this changes19. Although some formulas were proposed based on idea of substrate

effect, underlying physical mechanism remains unclear. Until now, formulas of Fresnel Optics for TDM with14,19 and without a substrate11,12,13,14,15,16,17,18 have been derived, their

derivation routes involved IF model3,17,18, quantum mechanics1,11,14,15,16,19, and dielectric constant12,13. These calculations were complex, scattered, and they need to be summarized into a

general theory. In this paper, we treat TDMs as a thin film with a thickness of several Angstrom. In order to solve three questions mentioned above, we attempt to solve Fresnel problem of

TDM by TF model based on basic optical coating theory. The origin of IF model is also reexamined. In addition, derived formulas are also utilized to interpret previous experimental results.

THEORY FREE-STANDING SAMPLE NORMAL INCIDENCE In modern optical coating theory and software, a thin film with a thickness of several Angstrom like graphene also can be treated as a thin film

rather than an interface. Here, first we employ basic optical coating theory based on TF model to derive the equation of absorption for TDM at normal incidence. We just give a brief

derivation, the details can be found in classical optical books20,21,22. Figure 1 shows TF model of TDM. _E_a, _H_a, _N_0, are electric field vector, magnetic field vector, and complex

refractive index of air, respectively. _N_ (_n_ − i_k_, _n_ is refractive index, and _k_ is extinction coefficient), _d_, are the complex refractive index, and thickness of thin film,

respectively. _E_b, _H_b, _N_m, are electric field vector, magnetic field vector, and complex refractive index of the substrate, respectively. According to boundary condition, electric and

magnetic vectors between boundary a and b can be connected in matrix notation, which is shown in Eq. (1), and the first item on the right side of Eq. (1) is characteristic matrix of thin

film. $$[\begin{array}{c}{E}_{a}\\ {H}_{a}\end{array}]=[\begin{array}{cc}\cos \,\delta & (i\,\sin \,\delta )/y\\ iy\,\sin \,\delta & \cos \,\delta

\end{array}]\,[\begin{array}{c}{E}_{b}\\ {H}_{b}\end{array}]$$ (1) where _δ_ is phase thickness of thin film, which is defined by Eq. (2), _d_ is thickness of thin film, _λ_ is wavelength,

and _y_ is optical admittance of TDM, which is defined by Eq. (3), and _Y_ is optical admittance of free space. $$\delta =2\pi Nd/\lambda $$ (2) $$y=H/E=NY$$ (3) Equation (1) can be

normalized by dividing by _E__b_, Eq. (4) is obtained, where _y__m_ is optical admittance of the substrate, B and C are normalized electric and magnetic vectors.

$$[\begin{array}{c}{E}_{a}/{E}_{b}\\ {H}_{a}/{E}_{b}\end{array}]=[\begin{array}{c}B\\ C\end{array}]=[\begin{array}{cc}\cos \,\delta & (i\,\sin \,\delta )/y\\ iy\,\sin \,\delta & \cos

\,\delta \end{array}]\,[\begin{array}{c}1\\ {y}_{m}\end{array}]$$ (4) _δ_ is very small because _d_/_λ_ is roughly less than 0.001, so cos_δ_ ≈ _1_, _sinδ_ ≈ _δ_. Thus, characteristic

matrix can be simplified to be Eq. (5). $$[\begin{array}{cc}1 & (i\delta )/y\\ iy\delta & 1\end{array}]$$ (5) Then, as shown in ref. 20, transmittance, reflectance, absorption of TDM

can be calculated by Eqs. (6–8), respectively. $$T=\frac{4{y}_{0}\mathrm{Re}({y}_{s})}{({y}_{0}B+C){({y}_{0}B+C)}^{\ast }}$$ (6)

$$R=(\frac{{y}_{0}B-C}{{y}_{0}B+C}){(\frac{{y}_{0}B-C}{{y}_{0}B+C})}^{\ast }$$ (7) $$A=\frac{4{y}_{0}\mathrm{Re}(B{C}^{\ast }-{y}_{s})}{({y}_{0}B+C){({y}_{0}B+C)}^{\ast }}$$ (8) If we set

_N_0 = _N_m = 1, combine Eqs. (2–5) into (6–8), Eqs. (9–11) are obtained. $${T}_{TF}=\frac{4}{{(2+4\pi nkd/\lambda )}^{2}+4{\pi }^{2}{d}^{2}{({n}^{2}-{k}^{2}+1)}^{2}/{\lambda }^{2}}$$ (9)

$${R}_{TF}=\frac{{(4\pi nkd/\lambda )}^{2}+4{\pi }^{2}{d}^{2}{(1-{n}^{2}+{k}^{2})}^{2}/{\lambda }^{2}}{{(2+4\pi nkd/\lambda )}^{2}+4{\pi }^{2}{d}^{2}{({n}^{2}-{k}^{2}+1)}^{2}/{\lambda

}^{2}}$$ (10) $${A}_{TF}=\frac{4\pi nkd/\lambda +4{\pi }^{2}{d}^{2}({n}^{2}-{k}^{2})/{\lambda }^{2}}{{(1+2\pi nkd/\lambda )}^{2}+{\pi }^{2}{d}^{2}({n}^{2}-{k}^{2}+1)/{\lambda }^{2}}$$ (11)

For comparison, formulas of transmittance, reflectance, absorption of TDM derived based on IF model are given in Eqs. (12–14)3. It should be noted that the term of _4πnkd/λ_ is equal to

_G/cε_0 in equations (50–53) of ref. 3, which will be discussed in Section 2.1.3. As shown in Eqs. (9–11) and Eqs. (12–14), the main difference between TF model and IF model is the term

including of (n2 − k2) involved in Eqs. (9–11). There are slight differences in transmittance and absorption between two models, but significant discrepancy in reflectance because the term

including of (n2 − k2) has the same magnitude with (4πnkd/λ)2. In other words, eliminating the term including of (n2 − k2), formulas of TF model reduced to be the ones of IF model. When d/λ

is extremely small, this treatment is reasonable for transmittance and absorption, not for reflectance. $${T}_{Interface}=\frac{4}{{(2+4\pi nkd/\lambda )}^{2}}\approx 1-\frac{4\pi

nkd}{\lambda }$$ (12) $${R}_{Interface}=\frac{{(4\pi nkd/\lambda )}^{2}}{{(2+4\pi nkd/\lambda )}^{2}}$$ (13) $${A}_{Interface}=\frac{4\pi nkd/\lambda }{{(1+2\pi nkd/\lambda )}^{2}}\approx

\frac{4\pi nkd}{\lambda }$$ (14) We utilize Eq. (9) of TF model and Eq. (12) of IF model to calculate transmittance of graphene and bilayer graphene, and compare with experimental,

theoretical results of Nair1. The optical constants of graphene are cited from ref. 9 (annealing sample). Figure 2a demonstrates comparison between our calculation and experimental,

theoretical results of Nair1. Nair’s theoretical calculations were based on IF model, but he ignored wavelength dependence of optical conductance. Here wavelength dependence of optical

constant is considered in IF model simulation. TF model and IF model simulation all agree well with experimental results, and even better than Nair’s theoretical prediction. There is no

significant difference between TF model and IF model simulation because second term is very small compared with first term in denominator of Eq. (9). Also, there is no significant difference

in 600–740 nm among Nair’s theoretical calculation, TF model and IF model simulation because 4πnkd/λ is equivalent to _πα_, which will be discussed in Section 2.1.3. Thus, dispersion of

grapheme at <450 nm can answer the first question raised in Introduction section. From Eq. (14), one can see that there is a thickness dependence on the absorption, which can be used to

interpret the layer effect of the absorption for the graphene in Fig. 2a. As for deviation from _πα_ in 400–450 nm for experiment results in ref. 1, the reason is that optical conductance

_G_* is not strictly universal in visible region, and it is wavelength dependence. Several jobs were done to discuss this problem, and optical conductance was described by Eq. (15)23,24:

$${G}^{\ast }/{G}_{0}=1+C{\alpha }^{\ast }+O({\alpha }^{\ast 2})$$ (15) where _α__*_ is a renormalized, dimensionless coupling constant24, and _C_ is a constant, but there are several values

for this debated constant23,24. If optical responses of TDM are measured, optical conductance can be accurately determined based on TF model, which will contribute most to precisely study

optical conductivity of TDM, and unveil mechanism of unique physical properties of TDM. We also use Eq. (10) of TF model and Eq. (13) of IF model to calculate reflectance of graphene, the

results are shown in Fig. 2b. The reflectance calculated is less than 0.1%, which is consistent with experimental results1. The calculated reflectance based on TF model is up to three times

of the one based on IF model, and this can be explained by the fact that the term including of (_n_2 − _k_2) has the same magnitude with (4_πnkd/λ_)2 in Eq. (10). OBLIQUE INCIDENCE Fresnel

problem of TDM at oblique incidence is discussed based on TF model in ref. 25. One can see that refractive angle of TDM is taken into account in our calculation, which significantly differs

from theoretical results of Stauber3,26. Figure 3 shows theoretical p- and s-polarized transmittance (a) and reflectance (b) of monolayer MoS2 based on TF model and IF model at 45°

incidence. Compared with normal incidence, besides reflectance, transmittance also shows obvious difference between two models. There are more pronounced differences of s-polarization than

p-polarization in transmittance and reflectance. Also, there are significant polarization-independent peak shifts between the two models. For example, in 600–650 nm region, the peaks of both

p- and s-polarization curves for IF model are at 616 nm, while the peaks of the curves for TF model are at 625 nm. However, there is no shift in the transmittance curves. The reason can be

found by comparison equations (S9, S10, S13, S14) in ref. 25 with equations (51, 53, B3) in ref. 3. For transmittance, compared with IF model, in TF model, the term, 1/cosθ at

p-polarization, or cos3θ/cosθ0 at s-polarization is added to multiply term of πα/2, and this term just modifies the amplitude, not shifts the peaks. For reflectance, besides the term added

multiplying πα/2, another positive term including of (n2 − k2) is added plus term of π2α2 in the numerator. Thus, peak reflectance in TF model is red-shifted. In addition, as shown in

equations (S13, S14), s-polarized reflectance and transmittance based on TF model is dependent of incidence angle, which is not same with IF model3. However, our formulas for the case of

oblique incidence still need further verification of experimental results. 4ΠNKD/Λ The term of 4πnkd/λ we proposed is readily to be understood and used. Because 4πk/λ is absorption

coefficient, we yields another style of approximate absorption of TDM (17, 18): $$A=\alpha dn$$ (16) where _α_ is absorption coefficient. Holovský (17, 18) also obtained this result, but he

did not give the origin of αdn. It is contradictory that he admitted that the graphene was a thin film with _n_, _k_, and _d_, at the same time he added the value of αdn to formulas derived

from IF model. It should be noted that Lee (20) also did initial, qualitatively derivation in discussion of absorption for very thin film, but he did not give final result. Table 1

summarizes four kinds of approximate formulas for calculation of absorption of TDM11,12,13,14,15,16,17,18 in visible and infrared range, and _γ_ is layer number. The first formula is

equivalent to other three ones only for some special condition (like graphene), the last three formulas are equivalent. Now we derive other three formulas from the term of 4πnkd/λ we

proposed. Complex refractive index is defined by Eqs. (17) and (18), where _ε__r_ and _μ__r_ are the relative dielectric permittivity and permeability, _σ_ is optical conductivity, _ω_ is

angular frequency, and _ε_’_r_ is relative complex dielectric permittivity18,20,21. $${N}^{2}={(n-ik)}^{2}={\varepsilon }_{r}{\mu }_{r}-i\frac{{\mu }_{r}\sigma }{\omega {\varepsilon }_{0}}$$

(17) $${N}^{2}={(n-ik)}^{2}={\mu }_{r}{\varepsilon }_{r}\text{'}={\mu }_{r}({\varepsilon }_{r}-i\frac{\sigma }{\omega {\varepsilon }_{0}})$$ (18) Derived from Eqs. (17) and (18),

optical conductivity and imaginary part of relative complex dielectric permittivity are related to _nk_ as shown in Eq. (19), where _ε_0 is dielectric permittivity of vacuum, and μr = 1.

$$nk=\frac{{\mu }_{r}\sigma }{2\omega {\varepsilon }_{0}}=\frac{\text{Im}({\varepsilon ^{\prime} }_{r})}{2}$$ (19) Then17,18 $$A=\frac{4\pi nkd}{\lambda }=\frac{\sigma d}{{\varepsilon

}_{0}c}=\frac{G}{{\varepsilon }_{0}c}$$ (20) where _c_ is the speed of light in free space, and _G_ is optical conductance. For graphene in 600–740 nm: $$G(\omega )\approx {G}_{0}=\frac{\pi

{e}^{2}}{2h}=\frac{{e}^{2}}{4\hslash }$$ (21) where _e_ is the electron charge, _h_ is Plank’s constant, and _ħ_ is reduced Plank’s constant. Then $$A=\frac{4\pi nkd}{\lambda }=\frac{\sigma

d}{{\varepsilon }_{0}c}=\frac{G}{{\varepsilon }_{0}c}=\frac{{e}^{2}}{4{\varepsilon }_{0}c\hslash }$$ (22) Because the fine structure constant $$\alpha =\frac{{e}^{2}}{4\pi {\varepsilon

}_{0}c\hslash }$$ (23) Then $$A=\frac{4\pi nkd}{\lambda }=\frac{\sigma d}{{\varepsilon }_{0}c}=\frac{G}{{\varepsilon }_{0}c}=\frac{{e}^{2}}{4{\varepsilon }_{0}c\hslash }=\pi \alpha $$ (24)

Thus, we recover the result of ref. 1, this is the reason that absorption in 600–740 nm is same in value between our calculation and Nair’s theoretical prediction. In addition, combining Eq.

(19) into Eq. (14), we yield absorption _A_: $$A=\frac{4\pi nkd}{\lambda }=\frac{\omega d}{c}\text{Im}({\varepsilon ^{\prime} }_{r})$$ (25) We recover the results of refs. 10,11.

ON-SUBSTRATE SAMPLE (NORMAL INCIDENCE) Generally, TDM needs to be transferred onto a substrate. Several jobs were done to discuss this issue14,17,18,19, TDM exhibited different properties as

compared to standing-free samples, and they attributed these changes to the substrate effect. It was found that stepwise absorption of two dimensional InAs on the substrate is 1.6% rather

than 2.3%, and Fang proposed a formula of (2/(1 + _n_s))2π_α_ (_n_s is refractive index of the substrate) to describe this changes19. Here we also calculate Fresnel optics of on-substrate

TDM. If we set _N_0 = 1, combine Eqs. (2–5) into (6–8), respectively, Eqs. (26–28) are obtained, where _n__s_ is refractive index of the substrate. $$A=\frac{4(4\pi nkd/\lambda +4{\pi

}^{2}{d}^{2}({n}^{2}-{k}^{2}){n}_{s}/{\lambda }^{2})}{{(1+{n}_{s}+4\pi nkd/\lambda )}^{2}+(4{\pi }^{2}{d}^{2}{({n}^{2}-{k}^{2}+{n}_{s})}^{2}/{\lambda }^{2})}$$ (26)

$$T=\frac{4{n}_{s}}{{(1+{n}_{s}+4\pi nkd/\lambda )}^{2}+(4{\pi }^{2}{d}^{2}{({n}^{2}-{k}^{2}+{n}_{s})}^{2}/{\lambda }^{2})}$$ (27) $$R=\frac{{(1-{n}_{s}-4\pi nkd/\lambda )}^{2}+4{\pi

}^{2}{d}^{2}{({n}_{s}-{n}^{2}+{k}^{2})}^{2}/{\lambda }^{2}}{{(1+{n}_{s}+4\pi nkd/\lambda )}^{2}+(4{\pi }^{2}{d}^{2}{({n}^{2}-{k}^{2}+{n}_{s})}^{2}/{\lambda }^{2})}$$ (28) For comparison,

formulas of transmittance, reflectance, absorption of TDM derived based on IF model are given in Eqs. (29–31)3. Similar to standing-free case, as shown in Eqs. (26–28) and Eqs. (29–31), the

main difference between TF model and IF model is the term including of (n2 − k2) involved in Eqs. (26–28). There are slight differences in transmittance and absorption between two models,

but significant discrepancy in reflectance because the term including of (n2 − k2) has the same magnitude with (4πnkd/λ)2. In other words, eliminating the term including of (n2 − k2),

formulas of TF model reduce to be the ones of IF model. When d/λ is extremely small, this treatment is reasonable for transmittance and absorption, not for reflectance.

$$A={(\frac{2}{(1+{n}_{s}+4\pi nkd/\lambda )})}^{2}\frac{4\pi nkd}{\lambda }\approx {(\frac{2}{(1+{n}_{s})})}^{2}\frac{4\pi nkd}{\lambda }$$ (29) $$T=\frac{4{n}_{s}}{{(1+{n}_{s}+4\pi

nkd/\lambda )}^{2}}$$ (30) $$R=\frac{{(1-{n}_{s}-4\pi nkd/\lambda )}^{2}}{{(1+{n}_{s}+4\pi nkd/\lambda )}^{2}}$$ (31) We utilize TF model and IF model to calculate reflectance of MoS2 and

MoSe2 on fused silica substrate, and optical constants are cited from ref. 27. The thicknesses of MoS2 and MoSe2 are 0.615 nm and 0.646 nm, respectively27. Figure 4 shows theoretical and

experimental27 reflectance results of MoS2 (a) and MoSe2 (b) on fused silica substrate. TF model simulations exhibit better agreement with experimental results than IF model. ORIGIN OF IF

MODEL refs. 3,5,28 provided derivations for calculation of optical response of TDM based on IF model. In equation (49) of ref. 3, _σ_ is different from the one in this job. The unit is

_Ω_−1, and it should be referred to optical conductance, not optical conductivity (the unit of _Ω_−1._m_−1) in this job. While in ref. 5 and Eq. (8) of ref. 28, _σ_ is same with the one in

this job. The relationship between _σ__IF_ in IF model and _σ__TF_ is described in Eq. (32)26. We can see that in refs. 3,5,28, they all took thickness of TDM into account of their

calculation of electric current density _j_, which is contradicted with the assumption that TDM is an interface. Thus, in essence, IF model is also an approximate style of TF model. A larger

error is introduced into reflectance calculation when IF model is utilized. Thus, TF model rather than IF model is an accurate way to solve Fresnel problem of TDM. $${\sigma

}_{{\rm{IF}}}={\sigma }_{{\rm{TF}}}d$$ (32) CONCLUSION It should be noted that when _d/λ_ is not extremely small, in other word, the wavelength extend to ultraviolet and X ray range, Eq. (5)

is not correct, an accurate form of first term on the right side of Eq. (1) must be used. Our job does not deny the fact that there are many fantastic optical properties existing in

multilayer TDM, such as interaction between bilayer graphene. These new properties will give some small corrections to optical conductivity, which can be macroscopically characterized by

optical constants (n and k). There are no contradiction between our job and new optical properties. Alternatively, it is found that the product of refractive index _n_, absorption

coefficient _a_, thickness _d_, plays a key role in the approximate calculation of absorption of TDM at normal incidence in visible and infrared range. Previous calculation of Fresnel optics

for TDM (standing-free and on-substrate samples) all can be recovered from our derived formulas by eliminating the term of (_n_2 − _k_2) at normal incidence, and they are just approximate

results of our calculation based on TF model. IF model, in essence, is a kind of approximate TF model, and the main difference between two models is the term of (_n_2 − _k_2) involved in TF

model at normal incidence. A significant error is introduced into reflectance calculation of two dimensional materials when IF model is utilized. The differences between two models are

complicated at oblique incidence, which needs further experimental verification. IF model is not suitable for solve Fresnel problem of two dimensional materials. Our job reveals that TDM is

a thin film rather than an interface or a boundary. It indicates that basic optical coating theory and traditional optical software apply to TDM, and our job will pave the way for widely

application of TDM in optical coating field. Especially, our job will contribute most to design novel filters based on TDM multilayer that was referred to as van der Waals heterostructures

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Article  CAS  Google Scholar  Download references ACKNOWLEDGEMENTS This work is supported by the Joint Research Fund in Astronomy (U1731114) under cooperative agreement between the National

Natural Science Foundation of China (NSFC) and Chinese Academy of Science (CAS), and partially supported by the Strategic Priority Research Program of Chinese Academy of Science (CAS), Grant

No. XDA15320103. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of

Sciences, Changchun, 130033, China Xiaodong Wang & Bo Chen Authors * Xiaodong Wang View author publications You can also search for this author inPubMed Google Scholar * Bo Chen View

author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS X.D.W. designed research and wrote the paper. B.C. reviewed the manuscript. CORRESPONDING AUTHOR

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dimensional materials. _Sci Rep_ 9, 17825 (2019). https://doi.org/10.1038/s41598-019-54338-0 Download citation * Received: 15 September 2019 * Accepted: 07 November 2019 * Published: 28

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