- Email: [email protected]
On the refractive index and photon mass
Accepted Manuscript Title: On the refractive index and photon mass Author: A.I. Arbab PII: DOI: Reference:
S0030-4026(16)30317-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.04.040 IJLEO 57523
To appear in: Received date: Accepted date:
15-2-2016 13-4-2016
Please cite this article as: A.I. Arbab, On the refractive index and photon mass, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.04.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
On the refractive index and photon mass A. I. Arbab
(a)
PACS PACS
41.20.Jb – Electromagnetic wave propagation 78.20.Ci – Refractive index 42.25.Bs – Wave propagation
cr
PACS
ip t
Department of Physics, College of Science, Qassim University, P.O. Box 6644, 51452 Buraidah, KSA
us
Abstract. - A new theoretical law for the refractive index is proposed. The law provides a limit to the photon mass inside optical materials. Experimental data reveal that the rest-mass energy of the photon inside a semiconductor or an insulator is found to be a few eV (m0 ∼ 10−6 me ). This energy could be related to the band gap energy of the material. The electric (optical) conductivity of the material is found to be related√to the photon mass. The 2
energy is given by, m0 c = √ εµ m0
√
n n2 −1
(
n−
n2 −1 n
)
, and its rest-mass
, where λ is the wavelength of the incident light. The skin depth inside the √
at very high frequency, and δ− =
2h ¯ m0 c
at very low frequency.
M
medium is δ+ =
h ¯
hc λ
hc λ
an
photon kinetic energy inside the material, with a refractive index n, is, EK =
The refractive index is a quantity that tells how a material deflects light when
ed
Introduction. –
passes into. It is defined as the ratio between the velocity of light in vacuum to the phase velocity (vp ) of light in a medium, viz., n = c/vp . When light enters a medium its wavelength (λ) changes. The
pt
refractive index of materials varies with the wavelength. Recall that when white light is incident on a
Ac ce
prism the refracted light has several colors. That means each color (wavelength) has a different refractive index, and hence refracted differently. There are two scenarios defining the wavelength in the medium by λ ′ = nλ, which is due to Abraham, and the second one defining λ ′ = λ/n , is due to Minkowski [1, 2]. It is recently shown that the two scenarios are correct where the former is related to the kinetic momentum, while the latter is related to canonical momentum of light in the medium [3]. The refractive index for visible light is generally greater than unity. Transparent media for visible light have refractive indices in the range between 1 and 3. It is generally understood that the refractive index cannot be lower than 1. However, in the x-ray regime the refractive indices are lower than but very close to unity. This may appear problematic with the theory of special relativity. For an electron transiting between the maximum of the valence band and the minimum of the conduction band, or vice versa, the conservation (a) [email protected]
p-1
Page 1 of 9
A. I. Arbab1
of momentum, however, can’t be fulfilled with the absorption or emission of a photon alone in an indirect semiconductor that because the magnitude of the momentum of a photon is several orders of magnitude smaller than that of an electron in a semiconductor. This conservation can be reconciled if the emitted phon had mass. We all understand that the photon behaves like a particle in the photoelectric and Compton effects, but no mass is introduced to it.
ip t
Thus, the particle nature of the photon is only apparent if the photon exits inside a medium and not in vacuum (free space) where it is massless. The rest-mass of the photon depend on the properties of the
cr
medium (e.g., refractive index) in which it exits. It is thus influenced by gravitational interactions while traveling inside the medium. Therefore, if we don’t live in a complete vacuum, photons would contribute
us
much to the total mass of the universe. With this minute mass, massive photons could resolve the dark matter and dark energy problems inflicting our present theories of gravitation.
an
In 1931 Yakov Frenkel proposed the idea of an excitonthat is a bound state of anelectronand a holethat are attracted to each other by the electrostatic Coulomb force [4]. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and in some liquids. It is considered as an elementary excitation
M
that can transport energy but not electric charge. It is formed whenever a photon is absorbed by a semiconductor or an insulator. It thus excites an electron from the valence band into the conduction
ed
band. It travels in a particle-like manner through the lattice. Thus, the idea of massive photon could replace that of an exicton where the binding energy of the exciton is equal to the rest-mass of the photon. The dynamics of the massive photon is shown in [5]
pt
and the references there in. One can define an effective classical radius of massive photon as r0 =
ke2 m0 c2 ,
where k is the Coulomb constant, which for a typical photon rest-mass energy is of the order of some few
Ac ce
nanometers.
We derive in this paper a dispersion law relating the refractive index to the wavelength of the incident light and the rest-mass energy of the photon. This law relies on the hypothesis that the photon has a non-zero effective mass inside the medium. The band gap of the material can be related to the photon rest-mass. Other optical properties resulting from this law are investigated. We make comparison with the predicted values and the experimental data. We also outlined some competing laws (models) relating the refractive index with the band gap energy.
Cauchy and Sellmeier equations. –
Optical materials can be properly designed by knowing its
refractive index and the parameters related to it. Cauchy had introduced an empirical formula for the p-2
Page 2 of 9
On the refractive index and photon mass
refraction index of a medium, and its relation to wavelength of the incident light, as [6] B C + 4 + ··· , 2 λ λ
n=A+
(1)
where A, B and C are constants to be determined from the experiment. This formula works quite well for visible light. More generally, for a transparent optical material a better relation applicable only to
n2 = 1 +
ip t
the wavelength region, where the absorption is negligible, is described by Sellmeier equation [6, 7] B1 λ2 B1 λ2 B3 λ2 + + , λ2 − C1 λ2 − C2 λ2 − C3
(2)
cr
where B1 , C1 , B2 , C2 , B3 , C3 are constants. The refractive index of a material changes also with temper-
ature, pressure, etc. Note that Sellmeier data are very useful for evaluating the chromatic dispersion of a
us
material. The Sellmeier model is an empirical model, which is basically used to describe the dependence of the refractive index in the transparent region. The model assumes κ to be zero so that [6, 7] Aλ2 , λ2 − B
(3)
an
n2 = A + where A and B are constants.
The wavelength dependence of the refractive index in optical wavelengths for various materials is often
n=1+
M
given by Feynman as [8]
N e2 , 2ε0 me (ω02 − ω 2 )
(4)
ed
where ω0 is a resonant frequency of an electron bound in an atom, and N is the charge number density. Light is absorbed by the medium if its frequency is resonant with the transition frequencies of the atoms
pt
in the medium. If the light with intensity I0 incident on the sample with thickness d, the intensity that is transmitted is expressed by the Lambert-Beer-Bouguer Law as [9]
Ac ce
α=
1 ln(1/T ) , , d
(5)
where T is the transmittance that is related to the reflectance by the relation T = 1 − R. However, the transmission coefficient is found experimentally by the relation [10–12] ( ) 1 (1 − R)2 α = ln . d T
(6)
The dielectric constant of the material can be written as ε = ε1 ± iε2 = (n ± iκ)2 ,
ε1 = n2 − κ2 ,
ε2 = 2nκ ,
(7)
where κ is called the extinction coefficient. Moreover, the electric conductivity of the material is related to ε2 by the relation, σ = ε2 ε0 ω. This can be written as σ=
4πε0 nκc . λ
(8)
p-3
Page 3 of 9
A. I. Arbab2
However, above the transmission edge the transmission coefficient it is described by [13, 14] α E = A(E − Eg )n ,
(9)
where E is the incident photon energy, Eg is the band gap energy of the material, A and n are constants. It is found that n = 1/2 and 2 for direct and indirect allowed transitions of the material, respectively.
ip t
The plot of (αE)1/n versus the photon energy, E is known as the Tauc’s plot [13]. Light with energy greater than the band gap energy of the material will be absorbed, while that with less energy than the band gap will be transmitted. The absorption coefficient is related to the extinction 4πκ . λ
Equations (8) and (10) yield the relation σ=
4π nκ . µ0 cλ
(10)
(11)
an
σ = nε0 cα ,
us
α=
cr
coefficient by the relation [15, 16]
This is an interesting relation connecting the electric conductivity to the refractive index and absorption coefficient. An optical conductivity is normally defined as σO = ncα/4π which is of special significance
M
[17, 18]. This increases with the frequency of the incident light.
Many theoretical and experimental investigations on the optical behaviour of thin films are based on reflection, transmission and absorption properties and their relation to the optical constants. The optical
ed
parameters of semimetal are obtained from transmittance as well as absorbance and the film thickness measurements.
pt
For metals and other opaque media the normal incidence reflectance R is given by
Ac ce
R=
Optical properties of material. –
(n − 1)2 + κ2 . (n + 1)2 + κ2
(12)
The basic optical properties of materials are obtained from
ellipsometry by studying the change in the polarization of light. This is done in terms of amplitude ratio (Ψ) and the phase difference (∆). Consequently, depending on the thickness of the material under study, the optical constants can be determined. The electric field of the incident light contains parallel and vertical components to the plane of incident. The ratio of the two amplitudes is related to the phase difference by the relation [19]
rp = ei∆ tan(Ψ) , rs
(13)
where rp and rs are the Fresnel reflection coefficients for the p- and s- polarized light, respectively. Various particular algorithms are used to find the refractive index n, the extinction constant κ, and other parameters. For this to work finely different wavelengths are employed. p-4
Page 4 of 9
On the refractive index and photon mass
In particular, when a sample structure is simple, the amplitude ratio Ψ is characterized by the refractive index n, while ∆ represents light absorption described by the extinction coefficient κ. In this case, the two values (n , κ) can be determined directly from the two ellipsometry parameters (Ψ , ∆) obtained from a measurement by applying the Fresnel equations. This is the basic principle of ellipsometry measurement. Note that Ψ represents the angle determined from the amplitude ratio between reflected
ip t
p- and s-polarizations, while ∆ expresses the phase difference between reflected p- and s- polarizations [19].
cr
There have been different empirical laws and formulae relating the refractive index to the energy band gap. Of these models are the Moss equation [21]
us
n4 Eg = 95 eV , the Ravindra equation
and the Herve-Vandamme
( 2
(15)
)2
,
(16)
M
n =1+
A B + Eg
an
n = 4.084 − 0.62 Eg ,
(14)
where A = 13.6 eV and B = 3.47 eV are constants. Moreover, we have, respectively, the Reddy and
ed
Kumar and Singh equations [21]
(17)
n = KEgC ,
(18)
pt
and
n4 (Eg − 0.365) = 154 ,
Ac ce
where K = 3.3668 and C = −0.32234.
Massive photon refractive index. –
Let us assume here that light has an effective mass (m0 )
inside the medium. In this case, one can write the relativistic energy equation for the photon as 1 γ=√ . 1 − v 2 /c2
E ′ = mc2 = m0 c2 γ ,
(19)
Furthermore, inside the medium, one has n=
c , v
n γ=√ . 2 n −1
(20)
Substitute eq.(20) in eq.(19) to obtain E ′ = m0 c2 √
n n2
−1
.
(21)
p-5
Page 5 of 9
A. I. Arbab3
Using eq.(21), the photon momentum inside the material is given by p′ = mv = m0 γc/n .
(22)
Using eq.(20), eq.(22) yields m0 c . n2 − 1
(23)
ip t
p′ = √ The relativistic relation
p c2 , v
(24)
cr
E=
E′ =
p ′ c2 = p ′c n , v
us
can be assumed to be valid inside the medium too, i.e.,
(25)
upon using eq.(20). The relativistic equation in eq.(24) is valid for all relativistic particles whether they
an
are massive or massless. Two scenarios for the photon momentum inside the medium are expounded; the first one is due to Minkowski where p ′ = np and the second is due to Abraham where p ′ = p/n.
M
Applying these cases in eq.(25) yields the two equations:
E M = n2 E .
EA = E ,
(26)
ed
Equation (23) now yields the mass and refractive index due to Abraham model, viz., λ2 n = 2 c 2, λc − λ
mA 0
h = λc
√
n2 − 1 , n
pt
2
λc =
h . m0 c
(27)
The Minkowski model yields
√
Ac ce
1 n2 = + 2
1 λ2 + , 4 λ2c
mM 0 =
h √ 2 n n − 1. λc
(28)
It is thus apparent that Minkowski model predicts a bigger mass and total energy for the photon in the medium. Abraham model gives the same energy of the photon in the medium though its has a non-zero mass provided the relativistic Einstein relation, eq.(24), is valid. The data of the dispersive refractive index may be analyzed using the single-effective-oscillator model that take the form [20] n2 = 1 +
E0 Ed . E02 − E 2
(29)
where Ed is the oscillator strength or dispersion energy and E0 is the single-oscillator energy. Moreover, it is observed that in eq.(24) E0 ≈ 2Eg . p-6
Page 6 of 9
On the refractive index and photon mass
Equation (27) and (28) can be used to find the rest-mass energy of the photon inside the material due to Abraham and Minkowski as hc = λ
and
√ n2 − 1 , n
1.243 E0 (eV ) = λ(µm)
hc √ 2 n n − 1, λ
E0M =
E0 (eV ) =
√
n2 − 1 , n
1.243 √ 2 n n − 1, λ(µm)
respectively.
(30)
(31)
ip t
E0A
cr
Comparison of the experimental data with the formula in eqs.(27) and (28) show that E0 can be
related to the band-gap energy of the material. It should be understood that when the photon is emitted
us
inside the material, as a result of the excitation of the electron from the valence band to the conduction band, the emitted photon is not massless but has mass. And since the photon transmits electric field, then it will given rise to conduction.
an
Thus, the total energy of the photon inside the material is obtained from eqs.(19) and (21) as (32)
M
E = m0 c2 + EK ,
ed
upon using eqs.(21) and (29). The kinetic energy of the massive photon due to Abraham is ( ) √ hc n − n2 − 1 A EK = , λ n and that due to Minkowski is
Hence, one obtains the relation
√ hc n(n − n2 − 1) . λ
pt
M EK =
Ac ce
M A EK = n2 EK .
The analogy between matter and electromagnetic has give rise to a photonic conductivity of the type [5]
σ0 =
2m0 , µ0 ¯h
(33)
that can be associated with the massive photons. It is evident that σ0 has a quantum character. If this conductivity is equal to that in eq.(11), then one finds m0 = n κ
h , cλ
(34)
associated with the massive photon. Equation (33) can be written as σ0A,M =
2E0 ε0 , h ¯
σ0A,M = 2.6843 × 104 E0A,M Ω−1 m−1 ,
(35)
p-7
Page 7 of 9
A. I. Arbab4
where E0 is measured in eV . Since the conductivity of the material should somehow depend on the energy band gap of the material, then E0 has to be related to Eg . Thus, it is apparent from eq.(35) that increasing the rest-mass energy would tend to make the material more conducting. The real part of the optical conductivity describes the dissipation of electromagnetic energy in the medium, while the imaginary part describes screening of
ip t
the applied field. Therefore, σ0 should represent the imaginary part of the electric conductivity.
Electric field due to massive photon. – The electric field due to massive photon is given by [5]
cr
⃗ ⃗ ( mc )2 1 ∂2E 2m0 ∂ E 2⃗ ⃗ = 0. − ∇ E + + E c2 ∂t2 ¯h ∂t ¯h
us
Assuming a plane wave solution in the medium of the form ⃗ =E ⃗ 0 exp i(⃗k · ⃗r − ωt) , E
k 2 = ω 2 εµ −
(37)
an
yields the dispersion relation
(36)
( m c )2 2m ω 0 0 + i. ¯h ¯h
(38)
M
Now expressing k in a complex for, as k = α + iβ, yields
ed
[ ( ] ( m c )2 )2 4m2 ω 2 ( m c )2 1/2 1 0 0 0 2 1/2 2 β = √ { ω εµ − + . } − ω εµ + h ¯ ¯h ¯h2 2 m0 c2 2¯ h ),
pt
In the very high frequency range (ω >>
(39)
the skin depth (δ = β −1 ) is given by √ ¯h εµ , m0
δ+ =
(40)
Ac ce
while the skin depth in the very low frequency rage is given by δ− =
√ 2 ¯h . m0 c
(41)
Using eq.(39), the skin depth in general becomes δ
−1
]1/2 )1/2 1 2π [( 4 , − n2 + α 2 =√ n − 2n2 α2 + α4 + 4α2 2 λ
Concluding remarks. –
√ α=
n2 . −1
n2
(42)
We have presented in this paper a new dispersion law relating the re-
fractive index to the wavelength of the incident light and the photon mass inside the material. We also outlined several other competing empirical models hitherto known. The derived law relies on the idea that photons inside the material have an effective mass. The rest-mass energy associated with photons can be related to the energy band gap of the material. The model indicates that the mass of the photon p-8
Page 8 of 9
On the refractive index and photon mass
inside the material is of the order of a few eV . The relativistic relation E =
c2 p v ,
where v is the speed of
the massive photon, is found to be valid for photons inside and outside the material. The electric (optical) conductivity is found to depend on the wavelength of the incident light and the photon rest-mass energy inside the material. The penetration depth of the electromagnetic wave depends on the refractive index.
ip t
REFERENCES
Ac ce
pt
ed
M
an
us
cr
[1] Abraham, M., Zur Elektrodynamik bewegter Krper Rendiconti del Circolo Matematico di Palermo, 1 (1909) 28. [2] Minkowski, H., Die Grundgleichungen fr die elektromagnetischen Vorgnge in bewegten Krpern, Math. Ann., 68 (1910) 472. [3] Barnett, S. M. and Loudon, R., The enigma of optical momentum in a medium, Phil. Trans. R. Soc. A, 368 (2010) 927. [4] Barnett, S. M. and Loudon, R., On the Transformation of light into Heat in Solids. I, Phys. Rev., 37 (1931) 17. [5] Arbab, A. I., The analogy between matter and electromagnetic waves, EPL, 94 (2011) 50005. [6] Sellmeier, W., Spectroscopic Ellipsometry: Principles and Applications, Ann. Phys. Chem., 143 (1871) 271. [7] Born, M. and Wolf, E., Principles of Optics, (Pergamon, Oxford), 1980. [8] Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman lectures on physics, 6th ed., (AddisonWesley Publishing Company), 1964. [9] Hummel, R. E., Electronic properties of materials 4th edition (Springer, New Yourk, 2011), p. 215. [10] Chopra, N., Mansingh, A. and Chadha, G.K., J. NonCryst. Solids, 126 (1999) 194. [11] Sharma, P., Sharma, V. and Katyal, S.C.,Chalcogenide Lett., 3 (2006) 73. [12] Pankove, J. I., Optical processes in semiconductors, (New Jersey: Prentice-Hall, 1971) p. 93. [13] Tauc, J., Optical properties of non-crystalline solids, in Optical Properties of Solid: F. Abeles, Ed., (NorthHolland, Amsterdam, The Netherlands), 1972, p. 277. [14] Davis, E.A. and Mott, N.F., Philos. Mag., 22 (1970) 903. [15] Forouhi, A. R. and Bloomer, I., Optical properties of crystalline semiconductors and dielectrics, Phys. Rev. B, 38 (1988) 1865. [16] Gittleman, J. I. , Sichel, E.K. and Arie, Y. Sol. Energy Mater, 1 (1979) 93. [17] Pankove, J. I., Optical processes in semiconductors, (Dover Publications, Inc. NewYork, 1975) p. 91. [18] Sharma, P. and Katyal, S. C. J. Phys. D: Appl.Phys., 40 (2007) 2115. [19] Fujiwara, H., Spectroscopic Ellipsometry: Principles and Applications, (John Wiley) 2007 [20] Wemple, S. H. and Didomenico, M., Phys. Rev. B, 3 (1971) 1338: S. H. Wemple, Phys. Rev. B, 7 (1973) 3767. [21] Tripathy, S. K., Refractive Indices of Semiconductors from Energy gaps, (arXiv:1508.03511). [22] Veselago, V. G., The electrodynamics of substances with simultaneously negative values of ε and µ, Soviet Physics Uspekhi, 10 (1968) 509.
p-9
Page 9 of 9
S0030-4026(16)30317-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.04.040 IJLEO 57523
To appear in: Received date: Accepted date:
15-2-2016 13-4-2016
Please cite this article as: A.I. Arbab, On the refractive index and photon mass, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.04.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
On the refractive index and photon mass A. I. Arbab
(a)
PACS PACS
41.20.Jb – Electromagnetic wave propagation 78.20.Ci – Refractive index 42.25.Bs – Wave propagation
cr
PACS
ip t
Department of Physics, College of Science, Qassim University, P.O. Box 6644, 51452 Buraidah, KSA
us
Abstract. - A new theoretical law for the refractive index is proposed. The law provides a limit to the photon mass inside optical materials. Experimental data reveal that the rest-mass energy of the photon inside a semiconductor or an insulator is found to be a few eV (m0 ∼ 10−6 me ). This energy could be related to the band gap energy of the material. The electric (optical) conductivity of the material is found to be related√to the photon mass. The 2
energy is given by, m0 c = √ εµ m0
√
n n2 −1
(
n−
n2 −1 n
)
, and its rest-mass
, where λ is the wavelength of the incident light. The skin depth inside the √
at very high frequency, and δ− =
2h ¯ m0 c
at very low frequency.
M
medium is δ+ =
h ¯
hc λ
hc λ
an
photon kinetic energy inside the material, with a refractive index n, is, EK =
The refractive index is a quantity that tells how a material deflects light when
ed
Introduction. –
passes into. It is defined as the ratio between the velocity of light in vacuum to the phase velocity (vp ) of light in a medium, viz., n = c/vp . When light enters a medium its wavelength (λ) changes. The
pt
refractive index of materials varies with the wavelength. Recall that when white light is incident on a
Ac ce
prism the refracted light has several colors. That means each color (wavelength) has a different refractive index, and hence refracted differently. There are two scenarios defining the wavelength in the medium by λ ′ = nλ, which is due to Abraham, and the second one defining λ ′ = λ/n , is due to Minkowski [1, 2]. It is recently shown that the two scenarios are correct where the former is related to the kinetic momentum, while the latter is related to canonical momentum of light in the medium [3]. The refractive index for visible light is generally greater than unity. Transparent media for visible light have refractive indices in the range between 1 and 3. It is generally understood that the refractive index cannot be lower than 1. However, in the x-ray regime the refractive indices are lower than but very close to unity. This may appear problematic with the theory of special relativity. For an electron transiting between the maximum of the valence band and the minimum of the conduction band, or vice versa, the conservation (a) [email protected]
p-1
Page 1 of 9
A. I. Arbab1
of momentum, however, can’t be fulfilled with the absorption or emission of a photon alone in an indirect semiconductor that because the magnitude of the momentum of a photon is several orders of magnitude smaller than that of an electron in a semiconductor. This conservation can be reconciled if the emitted phon had mass. We all understand that the photon behaves like a particle in the photoelectric and Compton effects, but no mass is introduced to it.
ip t
Thus, the particle nature of the photon is only apparent if the photon exits inside a medium and not in vacuum (free space) where it is massless. The rest-mass of the photon depend on the properties of the
cr
medium (e.g., refractive index) in which it exits. It is thus influenced by gravitational interactions while traveling inside the medium. Therefore, if we don’t live in a complete vacuum, photons would contribute
us
much to the total mass of the universe. With this minute mass, massive photons could resolve the dark matter and dark energy problems inflicting our present theories of gravitation.
an
In 1931 Yakov Frenkel proposed the idea of an excitonthat is a bound state of anelectronand a holethat are attracted to each other by the electrostatic Coulomb force [4]. It is an electrically neutral quasiparticle that exists in insulators, semiconductors and in some liquids. It is considered as an elementary excitation
M
that can transport energy but not electric charge. It is formed whenever a photon is absorbed by a semiconductor or an insulator. It thus excites an electron from the valence band into the conduction
ed
band. It travels in a particle-like manner through the lattice. Thus, the idea of massive photon could replace that of an exicton where the binding energy of the exciton is equal to the rest-mass of the photon. The dynamics of the massive photon is shown in [5]
pt
and the references there in. One can define an effective classical radius of massive photon as r0 =
ke2 m0 c2 ,
where k is the Coulomb constant, which for a typical photon rest-mass energy is of the order of some few
Ac ce
nanometers.
We derive in this paper a dispersion law relating the refractive index to the wavelength of the incident light and the rest-mass energy of the photon. This law relies on the hypothesis that the photon has a non-zero effective mass inside the medium. The band gap of the material can be related to the photon rest-mass. Other optical properties resulting from this law are investigated. We make comparison with the predicted values and the experimental data. We also outlined some competing laws (models) relating the refractive index with the band gap energy.
Cauchy and Sellmeier equations. –
Optical materials can be properly designed by knowing its
refractive index and the parameters related to it. Cauchy had introduced an empirical formula for the p-2
Page 2 of 9
On the refractive index and photon mass
refraction index of a medium, and its relation to wavelength of the incident light, as [6] B C + 4 + ··· , 2 λ λ
n=A+
(1)
where A, B and C are constants to be determined from the experiment. This formula works quite well for visible light. More generally, for a transparent optical material a better relation applicable only to
n2 = 1 +
ip t
the wavelength region, where the absorption is negligible, is described by Sellmeier equation [6, 7] B1 λ2 B1 λ2 B3 λ2 + + , λ2 − C1 λ2 − C2 λ2 − C3
(2)
cr
where B1 , C1 , B2 , C2 , B3 , C3 are constants. The refractive index of a material changes also with temper-
ature, pressure, etc. Note that Sellmeier data are very useful for evaluating the chromatic dispersion of a
us
material. The Sellmeier model is an empirical model, which is basically used to describe the dependence of the refractive index in the transparent region. The model assumes κ to be zero so that [6, 7] Aλ2 , λ2 − B
(3)
an
n2 = A + where A and B are constants.
The wavelength dependence of the refractive index in optical wavelengths for various materials is often
n=1+
M
given by Feynman as [8]
N e2 , 2ε0 me (ω02 − ω 2 )
(4)
ed
where ω0 is a resonant frequency of an electron bound in an atom, and N is the charge number density. Light is absorbed by the medium if its frequency is resonant with the transition frequencies of the atoms
pt
in the medium. If the light with intensity I0 incident on the sample with thickness d, the intensity that is transmitted is expressed by the Lambert-Beer-Bouguer Law as [9]
Ac ce
α=
1 ln(1/T ) , , d
(5)
where T is the transmittance that is related to the reflectance by the relation T = 1 − R. However, the transmission coefficient is found experimentally by the relation [10–12] ( ) 1 (1 − R)2 α = ln . d T
(6)
The dielectric constant of the material can be written as ε = ε1 ± iε2 = (n ± iκ)2 ,
ε1 = n2 − κ2 ,
ε2 = 2nκ ,
(7)
where κ is called the extinction coefficient. Moreover, the electric conductivity of the material is related to ε2 by the relation, σ = ε2 ε0 ω. This can be written as σ=
4πε0 nκc . λ
(8)
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Page 3 of 9
A. I. Arbab2
However, above the transmission edge the transmission coefficient it is described by [13, 14] α E = A(E − Eg )n ,
(9)
where E is the incident photon energy, Eg is the band gap energy of the material, A and n are constants. It is found that n = 1/2 and 2 for direct and indirect allowed transitions of the material, respectively.
ip t
The plot of (αE)1/n versus the photon energy, E is known as the Tauc’s plot [13]. Light with energy greater than the band gap energy of the material will be absorbed, while that with less energy than the band gap will be transmitted. The absorption coefficient is related to the extinction 4πκ . λ
Equations (8) and (10) yield the relation σ=
4π nκ . µ0 cλ
(10)
(11)
an
σ = nε0 cα ,
us
α=
cr
coefficient by the relation [15, 16]
This is an interesting relation connecting the electric conductivity to the refractive index and absorption coefficient. An optical conductivity is normally defined as σO = ncα/4π which is of special significance
M
[17, 18]. This increases with the frequency of the incident light.
Many theoretical and experimental investigations on the optical behaviour of thin films are based on reflection, transmission and absorption properties and their relation to the optical constants. The optical
ed
parameters of semimetal are obtained from transmittance as well as absorbance and the film thickness measurements.
pt
For metals and other opaque media the normal incidence reflectance R is given by
Ac ce
R=
Optical properties of material. –
(n − 1)2 + κ2 . (n + 1)2 + κ2
(12)
The basic optical properties of materials are obtained from
ellipsometry by studying the change in the polarization of light. This is done in terms of amplitude ratio (Ψ) and the phase difference (∆). Consequently, depending on the thickness of the material under study, the optical constants can be determined. The electric field of the incident light contains parallel and vertical components to the plane of incident. The ratio of the two amplitudes is related to the phase difference by the relation [19]
rp = ei∆ tan(Ψ) , rs
(13)
where rp and rs are the Fresnel reflection coefficients for the p- and s- polarized light, respectively. Various particular algorithms are used to find the refractive index n, the extinction constant κ, and other parameters. For this to work finely different wavelengths are employed. p-4
Page 4 of 9
On the refractive index and photon mass
In particular, when a sample structure is simple, the amplitude ratio Ψ is characterized by the refractive index n, while ∆ represents light absorption described by the extinction coefficient κ. In this case, the two values (n , κ) can be determined directly from the two ellipsometry parameters (Ψ , ∆) obtained from a measurement by applying the Fresnel equations. This is the basic principle of ellipsometry measurement. Note that Ψ represents the angle determined from the amplitude ratio between reflected
ip t
p- and s-polarizations, while ∆ expresses the phase difference between reflected p- and s- polarizations [19].
cr
There have been different empirical laws and formulae relating the refractive index to the energy band gap. Of these models are the Moss equation [21]
us
n4 Eg = 95 eV , the Ravindra equation
and the Herve-Vandamme
( 2
(15)
)2
,
(16)
M
n =1+
A B + Eg
an
n = 4.084 − 0.62 Eg ,
(14)
where A = 13.6 eV and B = 3.47 eV are constants. Moreover, we have, respectively, the Reddy and
ed
Kumar and Singh equations [21]
(17)
n = KEgC ,
(18)
pt
and
n4 (Eg − 0.365) = 154 ,
Ac ce
where K = 3.3668 and C = −0.32234.
Massive photon refractive index. –
Let us assume here that light has an effective mass (m0 )
inside the medium. In this case, one can write the relativistic energy equation for the photon as 1 γ=√ . 1 − v 2 /c2
E ′ = mc2 = m0 c2 γ ,
(19)
Furthermore, inside the medium, one has n=
c , v
n γ=√ . 2 n −1
(20)
Substitute eq.(20) in eq.(19) to obtain E ′ = m0 c2 √
n n2
−1
.
(21)
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A. I. Arbab3
Using eq.(21), the photon momentum inside the material is given by p′ = mv = m0 γc/n .
(22)
Using eq.(20), eq.(22) yields m0 c . n2 − 1
(23)
ip t
p′ = √ The relativistic relation
p c2 , v
(24)
cr
E=
E′ =
p ′ c2 = p ′c n , v
us
can be assumed to be valid inside the medium too, i.e.,
(25)
upon using eq.(20). The relativistic equation in eq.(24) is valid for all relativistic particles whether they
an
are massive or massless. Two scenarios for the photon momentum inside the medium are expounded; the first one is due to Minkowski where p ′ = np and the second is due to Abraham where p ′ = p/n.
M
Applying these cases in eq.(25) yields the two equations:
E M = n2 E .
EA = E ,
(26)
ed
Equation (23) now yields the mass and refractive index due to Abraham model, viz., λ2 n = 2 c 2, λc − λ
mA 0
h = λc
√
n2 − 1 , n
pt
2
λc =
h . m0 c
(27)
The Minkowski model yields
√
Ac ce
1 n2 = + 2
1 λ2 + , 4 λ2c
mM 0 =
h √ 2 n n − 1. λc
(28)
It is thus apparent that Minkowski model predicts a bigger mass and total energy for the photon in the medium. Abraham model gives the same energy of the photon in the medium though its has a non-zero mass provided the relativistic Einstein relation, eq.(24), is valid. The data of the dispersive refractive index may be analyzed using the single-effective-oscillator model that take the form [20] n2 = 1 +
E0 Ed . E02 − E 2
(29)
where Ed is the oscillator strength or dispersion energy and E0 is the single-oscillator energy. Moreover, it is observed that in eq.(24) E0 ≈ 2Eg . p-6
Page 6 of 9
On the refractive index and photon mass
Equation (27) and (28) can be used to find the rest-mass energy of the photon inside the material due to Abraham and Minkowski as hc = λ
and
√ n2 − 1 , n
1.243 E0 (eV ) = λ(µm)
hc √ 2 n n − 1, λ
E0M =
E0 (eV ) =
√
n2 − 1 , n
1.243 √ 2 n n − 1, λ(µm)
respectively.
(30)
(31)
ip t
E0A
cr
Comparison of the experimental data with the formula in eqs.(27) and (28) show that E0 can be
related to the band-gap energy of the material. It should be understood that when the photon is emitted
us
inside the material, as a result of the excitation of the electron from the valence band to the conduction band, the emitted photon is not massless but has mass. And since the photon transmits electric field, then it will given rise to conduction.
an
Thus, the total energy of the photon inside the material is obtained from eqs.(19) and (21) as (32)
M
E = m0 c2 + EK ,
ed
upon using eqs.(21) and (29). The kinetic energy of the massive photon due to Abraham is ( ) √ hc n − n2 − 1 A EK = , λ n and that due to Minkowski is
Hence, one obtains the relation
√ hc n(n − n2 − 1) . λ
pt
M EK =
Ac ce
M A EK = n2 EK .
The analogy between matter and electromagnetic has give rise to a photonic conductivity of the type [5]
σ0 =
2m0 , µ0 ¯h
(33)
that can be associated with the massive photons. It is evident that σ0 has a quantum character. If this conductivity is equal to that in eq.(11), then one finds m0 = n κ
h , cλ
(34)
associated with the massive photon. Equation (33) can be written as σ0A,M =
2E0 ε0 , h ¯
σ0A,M = 2.6843 × 104 E0A,M Ω−1 m−1 ,
(35)
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Page 7 of 9
A. I. Arbab4
where E0 is measured in eV . Since the conductivity of the material should somehow depend on the energy band gap of the material, then E0 has to be related to Eg . Thus, it is apparent from eq.(35) that increasing the rest-mass energy would tend to make the material more conducting. The real part of the optical conductivity describes the dissipation of electromagnetic energy in the medium, while the imaginary part describes screening of
ip t
the applied field. Therefore, σ0 should represent the imaginary part of the electric conductivity.
Electric field due to massive photon. – The electric field due to massive photon is given by [5]
cr
⃗ ⃗ ( mc )2 1 ∂2E 2m0 ∂ E 2⃗ ⃗ = 0. − ∇ E + + E c2 ∂t2 ¯h ∂t ¯h
us
Assuming a plane wave solution in the medium of the form ⃗ =E ⃗ 0 exp i(⃗k · ⃗r − ωt) , E
k 2 = ω 2 εµ −
(37)
an
yields the dispersion relation
(36)
( m c )2 2m ω 0 0 + i. ¯h ¯h
(38)
M
Now expressing k in a complex for, as k = α + iβ, yields
ed
[ ( ] ( m c )2 )2 4m2 ω 2 ( m c )2 1/2 1 0 0 0 2 1/2 2 β = √ { ω εµ − + . } − ω εµ + h ¯ ¯h ¯h2 2 m0 c2 2¯ h ),
pt
In the very high frequency range (ω >>
(39)
the skin depth (δ = β −1 ) is given by √ ¯h εµ , m0
δ+ =
(40)
Ac ce
while the skin depth in the very low frequency rage is given by δ− =
√ 2 ¯h . m0 c
(41)
Using eq.(39), the skin depth in general becomes δ
−1
]1/2 )1/2 1 2π [( 4 , − n2 + α 2 =√ n − 2n2 α2 + α4 + 4α2 2 λ
Concluding remarks. –
√ α=
n2 . −1
n2
(42)
We have presented in this paper a new dispersion law relating the re-
fractive index to the wavelength of the incident light and the photon mass inside the material. We also outlined several other competing empirical models hitherto known. The derived law relies on the idea that photons inside the material have an effective mass. The rest-mass energy associated with photons can be related to the energy band gap of the material. The model indicates that the mass of the photon p-8
Page 8 of 9
On the refractive index and photon mass
inside the material is of the order of a few eV . The relativistic relation E =
c2 p v ,
where v is the speed of
the massive photon, is found to be valid for photons inside and outside the material. The electric (optical) conductivity is found to depend on the wavelength of the incident light and the photon rest-mass energy inside the material. The penetration depth of the electromagnetic wave depends on the refractive index.
ip t
REFERENCES
Ac ce
pt
ed
M
an
us
cr
[1] Abraham, M., Zur Elektrodynamik bewegter Krper Rendiconti del Circolo Matematico di Palermo, 1 (1909) 28. [2] Minkowski, H., Die Grundgleichungen fr die elektromagnetischen Vorgnge in bewegten Krpern, Math. Ann., 68 (1910) 472. [3] Barnett, S. M. and Loudon, R., The enigma of optical momentum in a medium, Phil. Trans. R. Soc. A, 368 (2010) 927. [4] Barnett, S. M. and Loudon, R., On the Transformation of light into Heat in Solids. I, Phys. Rev., 37 (1931) 17. [5] Arbab, A. I., The analogy between matter and electromagnetic waves, EPL, 94 (2011) 50005. [6] Sellmeier, W., Spectroscopic Ellipsometry: Principles and Applications, Ann. Phys. Chem., 143 (1871) 271. [7] Born, M. and Wolf, E., Principles of Optics, (Pergamon, Oxford), 1980. [8] Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman lectures on physics, 6th ed., (AddisonWesley Publishing Company), 1964. [9] Hummel, R. E., Electronic properties of materials 4th edition (Springer, New Yourk, 2011), p. 215. [10] Chopra, N., Mansingh, A. and Chadha, G.K., J. NonCryst. Solids, 126 (1999) 194. [11] Sharma, P., Sharma, V. and Katyal, S.C.,Chalcogenide Lett., 3 (2006) 73. [12] Pankove, J. I., Optical processes in semiconductors, (New Jersey: Prentice-Hall, 1971) p. 93. [13] Tauc, J., Optical properties of non-crystalline solids, in Optical Properties of Solid: F. Abeles, Ed., (NorthHolland, Amsterdam, The Netherlands), 1972, p. 277. [14] Davis, E.A. and Mott, N.F., Philos. Mag., 22 (1970) 903. [15] Forouhi, A. R. and Bloomer, I., Optical properties of crystalline semiconductors and dielectrics, Phys. Rev. B, 38 (1988) 1865. [16] Gittleman, J. I. , Sichel, E.K. and Arie, Y. Sol. Energy Mater, 1 (1979) 93. [17] Pankove, J. I., Optical processes in semiconductors, (Dover Publications, Inc. NewYork, 1975) p. 91. [18] Sharma, P. and Katyal, S. C. J. Phys. D: Appl.Phys., 40 (2007) 2115. [19] Fujiwara, H., Spectroscopic Ellipsometry: Principles and Applications, (John Wiley) 2007 [20] Wemple, S. H. and Didomenico, M., Phys. Rev. B, 3 (1971) 1338: S. H. Wemple, Phys. Rev. B, 7 (1973) 3767. [21] Tripathy, S. K., Refractive Indices of Semiconductors from Energy gaps, (arXiv:1508.03511). [22] Veselago, V. G., The electrodynamics of substances with simultaneously negative values of ε and µ, Soviet Physics Uspekhi, 10 (1968) 509.
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