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LAL 07-
July 2007
A particle mechanism for the index of refraction
Marcel URBAN
LAL, Univ Paris-Sud, IN2P3/CNRS, Orsay, France
Abstract. We propose to go away from the electromagnetic wave description of light and to explain through a purely corpuscular and neutral approach, the phenomenon of the slowing down of light in a transparent medium. The quantum predictions and ours are compared against experimental refractive indices. In our framework the optical Kerr phenomenon finds a very natural interpretation and its numerical values are obtained easily.
1. Introduction Leon Foucault showed experimentally [1] that light slows down in water with respect to vacuum. More generally visible light velocity is reduced in transparent media. In a homogeneous material this is described by a single constant: the index of refraction n, which is the ratio of the velocity of light in vacuum to its velocity in the medium. Molecules have energy states quantized and in a gas these states are very well separated as opposed to solids where they can merge and form continuous bands. A photon propagating through a gas is absorbed by a molecule only if its energy corresponds to an allowed difference of energies in this molecule. For instance if the first excited state is 10 eV above the ground state, visible photons of a few eV should do nothing to these molecules. Why then do we have an index of refraction for a continuum of photon energies? Or put differently, how come a photon whose energy does not fit an atomic line can be influenced by the medium? We begin to recall the wave approach to the origin of the index of refraction. Then we propose another mechanism for the slowing down of the photons. The idea is to push the particle model of light as far as possible without using wave’s properties. We describe our model of the interaction between a photon and a molecule in matter and we undertake a comparison between the two approaches. In this corpuscular framework, we predict that there will be a dispersion of the arrival times for mono energetic photons traversing a sheet of matter. This could have been already seen in 1987. Another benefit of the corpuscular approach is that we understand the optical Kerr phenomenon easily and that we derive its magnitude in gases and in condensed matter. 2. The usual explanation for the index of refraction To the best of our knowledge there is only one explanation for the origin of the index of refraction and a clear description is made in [2] and in [3]. The first step considers an electromagnetic monochromatic plane wave falling on a sphere of dielectric whose diameter is very much smaller than
the wave length λ. The electric field of the incident wave induces a vibrating electric dipole in the
sphere which emits a spherical wave of the same frequency as the one of the incoming wave. This is the incoherent Rayleigh scattering on a single scattering centre. The downstream amplitude of the incident plane wave is slightly reduced but there is no change in its velocity. The second step considers a continuous slab of dielectric having a small depth and a large extension perpendicular to the axis of propagation of the plane wave. The slab of dielectric is decomposed into
small volumes, as compared to , where the incident plane wave induces electric dipoles. These
elementary vibrating dipoles emit secondary spherical waves according to the Rayleigh incoherent cross section. The reason to consider a very thin slab of matter is that the sum of these secondary waves on one particular elementary dipole is negligible compared to the incident light amplitude. The driving of the elementary dipoles is then proportional only to the amplitude of the incident light. All these secondary waves add up coherently downstream to the incoming light. The coherent integration
λ^3
of these secondary waves is phase shifted by π / 2 with respect to the incident wave. Since the thickness of the slab is small, the amplitude iA of this integration has a smaller modulus than the amplitude of the incident plane wave taken as 1. The sum, downstream, of the incident light and of the secondary waves takes the form 1+iA which is approximated as eiA. Thus instead of being 1 the incident wave is transformed into eiA^ which displays a phase shift interpreted as a delay in time due to a smaller speed in the slab of dielectric. This at last gives the index of refraction. The strength of the induced electric dipoles is given by their amount of polarisation under the influence of the incident electric field. This approach was envisaged for a continuous distribution of matter, and when the atomic nature of matter was established it became possible to have a microscopic model for the incoherent polarisation. At the same time we have the difficulty to go from a discrete to a continuum. The continuum is necessary in order to use the framework of the coherent integration producing the phase shift and thus explaining the index of refraction. The small dielectric sphere of the incoherent Rayleigh scattering is replaced by the molecule. The electric field of the incident wave sets the electrons in the molecule into a dipolar periodic motion and these electrons are supposed to reemit light spherically and at the same frequency as the one of the incoming wave. Any approach to the prediction of the index of refraction is based upon the calculation of the average electric polarisation of the molecule, produced by the incident electric field of the plane wave. Then, as soon as possible, go to the macroscopic polarisation. Once in the continuum theory framework the
dielectric constant ε is obtained as a function of frequency and finally the index of refraction is
predicted through the Maxwell formula:. We will call this line of thought the Maxwell- quantum framework.
n^2 = ε
3. What is found in the literature? The semi classical and quantum predictions of the index of refraction can be found in textbooks [4]. The semi classical formula for the index of refraction of a photon of energy admits that the
electrons in the molecule, behave like oscillators with discrete energies
E γ W i
∑ −
i (^) i
i e W E
N
n^2 14 r ( c )^222 γ
π h (^) elec i
∑ N^ i = N (3.1)
Nelec is the number of valence electrons per unit volume and re = 2.83 10-15^ m, is the classical radius of
the electron. h c = 197. 3 MeVfm. The hydrogen atom has W 0 = -13.6 eV and W 1 = -3.5 eV.
When the polarisation of the molecule is calculated with the quantum mechanics, we get what we call the Maxwell-quantum formula
∑ −
i (^) i
i e elec E E
f n^2 14 rN ( c )^222 γ
π h (^) ∑ = 1 i
f (^) i (3.2)
If the ground state energy of the atom is W 0 then: E i = Wi − W 0.
The hydrogen atom has E 1 = W 1 -W 0 = 10.2 eV. Therefore the Maxwell-quantum formula differs mainly in the characteristic energies Ei which are smaller than the Wi of the semi classical formula. In a condensed medium, Lorentz and Lorenz try to take into account the difference between the incident and the local electric field and suggest the following formula:
∑ −
i (^) i
i e elec E E
f rN c n
n 2 2
2 2
2 ( ) 3
γ
h (3.3)
So we understand that the mediums where we have a chance to predict correctly the index of refraction are the gases. However a surprise awaits us. All textbooks concerning the index of refraction theory spend a fair amount of pages and mathematics to first derive the semi classical formula (3.1) then the Maxwell-quantum formula (3.2) and yet no comparison with real data is made! They are in fact copying each other about this subject. A. Sommerfeld is an exception because he does compare, quickly, the classical formula with the experimental data on gaseous hydrogen [5] and he recognises that ‘the theory is still very crude…’ Because of this copying we may as well go to one of the first textbook like H. A. Lorentz. Arriving more than half way through the book he states: “We could now enter upon a comparison of our dispersion formula with the measurements of the indices of refraction, but I shall omit this, because we must not attach too much importance to the particular form which we have found for the equations”.
geometric cross section of the molecule seen from the photon. The average volume that a molecule occupies is obtained from the mass density of the solid or the liquid, the mass of a mole and the Avogadro number.
The mean free path for a photon is then: Λ =( ασ ⊥ Nmol ) −^1. Nmol is the number of molecules per unit
volume. Finally our prediction for the refractivity is:
[^ n^ −^1 ]corpuscular =ασ^ ⊥ N^ mol h c^
fi
∑ i Ei − E γ
We simplify this formula by defining an average molecular excitation energy EAV such that
i (^) i AV
i
E E E E
f
γ γ
Then we get:
[^ n^ −^1 ]corpuscular ≈^ N^ mol^ ασ^ ⊥
h c
E AV − E γ
4.2 Propagation through a condensed medium The formula for the index of refraction in a condensed medium is going to be simplified with respect to what we have in a gas. In a crystal we have the molecules packed and touching each other. The numerical density of these molecules can be expressed in terms of the spacing between them.
If we have three axes Ox, Oy and Oz and the corresponding spacings: δ x , δ y and δ z
then: N mol =(δ x δ y δ z. If the photon is propagating along Ox we have:
− 1
σ ⊥ = δ y δ z =>
x
Nmol δ
E E
c
E E
c
n N
AV x AV
x −^ = ⊥ mol − = −
h h
This shows that having a cross section in our index formula leads to sensitivity to the spacing in the direction of propagation. When the molecule is very asymmetrical and fixed in position like in a crystal, there will be a varying index with the direction of the light in the crystal. In the Maxwell- quantum formula this does not happen because the number of valence electrons in a molecule does not depend upon the direction of propagation. This could be an important aspect of the physics behind the phenomenon of birefringence.
5. Questions to the Maxwell-quantum model We list now five questions which are in fact criticisms to the Maxwell-quantum framework and which motivated the development of our corpuscular approach.
1- What happens to light going through a gas with, for instance, less than one molecule per^ λ^3?
This question concerns the limits in the use of the Maxwell theory of electromagnetism in macroscopic media. In our corpuscular approach the photon being a neutral object has no electric field and thus there is no problem since we do not use Maxwell theory of light. 2- What do we do for condensed mediums? This is linked to the local field not being the incident field. The light is considered to be a wave and has therefore a spatial extent. This question is coming about because of this extended aspect of light. In our approach there is no problem of local fields versus incident fields since we do not have fields. 3- If instead of an everlasting plane wave we envisage a more realistic burst of light it seems clear that a large part of the secondary waves will not have time to catch up with the incident light and will not therefore change the incident burst. Does the velocity appear to change as we move downstream? Again this is linked to the extended aspect of light in the classical theory and it disappears in our local photon theory.
4 - If instead of steady we consider moving molecules like in the Fizeau experiment with running water, then the molecules will reemit Doppler shifted waves with a frequency which will depend upon the angle of emission. Therefore the summation with the incident plane wave will be composed of different frequencies. Then the resulting apparent phase shift depends upon time! Is it reasonable to have an index of refraction which depends upon time? This is a mix between the extended aspect and the summing of the secondary wavelets to the main incident wave. We do not have the extended problem and our photons do not scatter, they keep going straight. Further it is well known that the Lorentz transformation applied to the photons in water gives back the results of Fizeau. 5- Is it reasonable to imagine that the atomic electrons reemit light of the same frequency as the incoming wave when we know that atoms in a gas emit lines and not a continuum? This is a very serious concern for the Maxwell-quantum framework. The way we see it in our approach is that since a photon does not possess an electric field it does not set the atomic electrons into a periodic motion. In its way through matter, a photon is either absorbed momentarily by the molecule or it ignores the molecule. In any case the molecules do not emit photons on top of the propagating one.
6. Comparisons between the Maxwell-quantum and the corpuscular expressions. We will compare the two formulas (3.2) and (4.3) in gases. As we did for our corpuscular model we will approximate (3.2) by an average excitation energy EAV.
[^ n^ −^1 ]Maxwell ≈^ N^ elec^2 π^ re
(h c ) 2
E AV^2 − E γ^2
[^ n^ −^1 ]corpuscular ≈^ N^ mol^ ασ^ ⊥
h c
E AV − E γ
The number density Nelec of dispersion electrons is not the number of electrons in a molecule (14 for N 2 ) times the number density of molecules Nmol. It is in fact the number of valence electrons: nvalence times Nmol. nvalence is the number of electrons missing to get to 8, a complete shell. For instance in N 2 this is nvalence = 2x3 = 6. Since N (^) elec = Nmolnvalence we have:
[^ n^ −^1 ]Maxwell ≈^ N^ mol n^ valence^2 π^ re
(h c ) 2
E AV^2 − E γ^2
The two formulas (6.2) and (6.3) differ both in the constants and in the energy dependences.
6.1 Close to an atomic line We can imagine that if we get close to an atomic line, E 0 , the data will show which form is right. In alkali vapours it is possible to get close to a line with visible light. For instance, within a few percent of the yellow doublet of Na, the data [8] are best fit with
[^ n^ −^1 ]data ∝
λ 0 = 0. 58910 −^6 m ,
0
0
E
πh c
γ
E
2 h c
[^ n^ −^1 ]data ∝
λ 0 E γ
2 π h c −λ 0 E γ
[^ n^ −^1 ]data ∝ −^1 +^
2 π h c
2 π h c −λ 0 E γ
E 0
E 0 − E γ
In [8] Eγ is within a few percent of E 0 ,. We can therefore drop (-1) and we get
[^ n^ −^1 ]data ∝^
E 0
E 0 − E γ
This is exactly our corpuscular formula.
The transverse area is thus estimated as π / 4 (VO2)2/3^ = 10.2 10-20^ m^2.
We calculate then the weighted average transverse area for Air :< σ ⊥> = 11.4 10-20^ m^2.
⎡⎣ n g − 1 ⎤⎦corpuscular =
10 25 11.4 10−^20
197 10−^9 eVm
E AV − E γ
= 10 −^3
E AV − E γ
In th e M ax w el l - q u an t u m mo d el we need the number density of the valence electrons. For N 2 , nvalence = 2x3 and for O 2 , nvalence = 2x2. For Air this will average to 0.86+0.24 = 5.6 valence electrons per air molecule. Then under our Air conditions we get: Nelec = 14.28 10^25 valence electrons/m^3. As for the corpuscular formula we try a one term approach with an average energy. The phase refractivity is then approximately
[^ n^ P −^1 ]Maxwell =^
98.5 10−^3
E AV^2 − E γ^2
The energies EAV and Eγ are in eV. Then we make use of (7.1) to get the group index of the Maxwell-quantum approach.
Figure 1. (ng-1)*10^3 for Air, as a function of the photon energy in eV.
The figure 1, shows the Maxwell-quantum prediction with EAV = 16.7 eV. It is clear that it needs to be reduced. When EAV = 19 eV, the group index values are fine but the Maxwell-quantum curve is then too flat. We conclude that the two predictions for Air are not very different except that the Maxwell- quantum formula demands the average excitation energy to be larger than in the corpuscular formula.
7.2 Fused silica The data for synthetic fused silica, SiO 2 , were found in [12]. The weigh of 6 10^23 molecules is 60.1 g and the density at 25°C is 2200 kg/m^3. Thus a molecule occupies a volume: 60.1/(6 10^23 22 10^5 ) = 45.4 10-30^ m^3. In other words, there are 2.2 10^28 molecules per m^3.
The spacing is: δ SiO 2 = (45.4 10 −^30 ) , and the corpuscular prediction is
1/ 3
m = 3.57 10−^10 m
⎡⎣ n gSiO 2 − 1 ⎤⎦corpuscular =
α δ SiO 2
h c
E AV − E γ
4.08 eV
E AV − E γ
In the Maxwell-quantum framework we use the Lorenz-Lorentz formula (3.3). The number of valence electrons for SiO 2 is 4+2x2 = 8 The number density is: 2.2 10^28 *8 = 17.6 10^28 valence electrons/m^3.
A n E E E E
n
P AV AV
P =
2 2 2 2
15 28 14 2
γ γ
⎡⎣ n p − 1 ⎤⎦ LL =
1 + 2 A
1 − A
Lorentz explains in his book that formula (3.3) is not always working fine and that the old Laplace formula (3.2) is to be considered too because experimental data sometimes stand in between. So we will also draw the refractivity obtained with Laplace formula:
⎡⎣ n p − 1 ⎤⎦ Laplace = 1 + 3 A − 1
Then we calculate ng from np with formula (7.1).
Figure 2. The fused silica group refractivity: ng-1, as a function of the photon energy in eV.
The energy bandwidth is given by:
γ
γ E
E
, Δ E γ < 0. 018 eV and ε< 0. 009 eV
Experimentally [14]: 1. 3102 ( )^1
( 2 ) 1. 5 and = − −
≈ = eV dE
dn E eV n (^) KDP E eV KDP
γ γ
FWHM ( t (^) traversal )= 8610 −^12 L ε
This is linear with the thickness of the crystal and with the energy bandwidth of the photons.
FWHM ( t (^) traversal )< 730 L fs
The length of the KDP crystal in this experiment is 8cm and the pair production is a rare
process, thus the average thickness is = 4cm. We end up with
FWHM ( t (^) traversal )< 30 fs
8.3 effect of the statistics of the number of stops In our corpuscular model, the average number of stops follows a Poisson law.
We will assume that (^) δ (^) KDP ≈ 3. 610 −^10 m and E (^) AV ≈ 12 eV
The average number of stops is:
L
L L
N
KDP KDP
stop
And the average stop-time:
s E E
t AV
stop
≈ 60 10 −^18
γ
h
FWHM ( t traversal , 1 γ)= 2. 35 < tstop > Nstop = 2. 356010 −^15 4. 5 L = 630 L fs
This is proportional to the square root of the thickness.
Again with an average = 4cm we have
FWHM ( t (^) traversal , 1 γ)≈ 120 fs
We have to fold the two Gaussian distributions since we have two photons and this is
FWHM ( t traversal , 2 γ)≈ 2120 fs = 170 fs
8.4 Comparison with the experimental result [13]
The observed FWHM is 56fs but it has to be doubled because when the beam splitter moves
so as to reduce one path length it increases at the same time, the other path. Further if the
beam splitter is attacked with a 45° incident angle there is a further multiplicative factor
1/cos(45°). The real time FWHM turns out to be 5621.414 = 158 fs. The effect of the
bandwidth is predicted to be larger than 350 fs. The effect of the dispersion is less than 30 fs.
The only effect which corresponds to the measurements is our statistical stop time prediction.
We conclude that light does not seem to be a wave and that it may very well be that our
prediction of the statistical distribution of traversal times has been already seen in 1987!
9. Understanding the optical Kerr effect An intense beam of light produces, in a medium, an increase of its refractive index [15], [16], [17], [18], [19], [20]. This phenomenon has been observed in solids, liquids and gases and is known by the names of AC Kerr, optical Kerr or self focusing effect. This is usually expressed empirically with the
formula: n = n 0 + n 2 I. n 0 is the usual index of refraction at low light intensity I. The constant n 2 is
comprised between 10-20^ m^2 /W for dense media and 10-23^ m^2 /W for gases.
9.1 The physics of the Kerr effect in our corpuscular approach
A photon crossing a molecule has a probability α to stop. If the photon stops and if we have, at the
same time, 1 / α photons on top of the first, we will have on average two photons stopping in the molecule. The stop time for a photon is proportional to 1 /Δ E where Δ E is the energy borrowed to
get from E γ to (^) E 1. These two photons need only borrow 2
E 1 − 2 E γ each. When , this is
roughly half the value for a single photon. Therefore they will stop for a time twice as long and since the refractivity: (n
E 1 >> E γ
0 -1), is proportional to the stop time, we expect a doubling of the refractivity. Thus when the density of photons is 1 / α in a molecular volume, the refractivity is roughly multiplied by a factor two. Let us consider visible photons of energy 1 eV propagating through water, and estimate the energy flux necessary to get such a density. A water molecule occupies approximately, a volume of 30 10-30^ m^3.
30 3 301030 4.^610 /
photons m photons (^) V (^) molecule
photons m s n
c photons photons 1. 33 10 / /
8
Φ =ρ =^30 ≈
This can be translated into an energy flux
I ( W / m^2 ) E 10391. 61019 1. 61020 W / m^2 = Φ photon = = − γ So we understand three things: 1- the index of refraction increases when the flux of photons increases. n = n 0 + n 2 I
2- n 2 ≈ 10 -20^ m^2 /W in water. 3- the refractivity: n – 1, is proportional to the number density of incoming photons and is also roughly proportional to the density of the medium. We understand therefore why n 2 is about 1000 times smaller for gases than for condensed mediums.
9.2 The prediction for n 2 and the comparison to the data Consider an energy flux I (W/m^2 ) of photons of energy: , falling on a slab of matter of thickness L.
The number density of photons in the slab is
E γ
ρ photons , and we have: (4.1).
( )−^1
Λ = ασ⊥ Nmol
When ρ photons Vmolecule << 1 / α, we consider two possibilities only:
- a single photon stops in a molecule and stays for a time Δ t 1
- two photons stop in a molecule and stay for a time (^) Δ t 2. If L <<Λthe average number of photons which stop per m^2 and per second is:
γ γ
E
I
N L
E
L I
N = ⊥ mol Λ
they will stop for a time
E E γ
t
AV −
h
1
A fraction of these: N 2 = N 1 αρ photons Vmolecule , will be absorbed together with a companion and this
corresponds to a stop time of
E E γ E E γ
t
AV AV^2
h h
The stop time Δ t 2 affects two photons.
This leads to an average stop time which is given by:
2 [ ( 2 )] / /^211221
2 1
(^1 2) N t N t t I
E
t I E
N
t I E
N N
t Δ + Δ = Δ + Δ −Δ
<Δ >= γ γ γ
γ
γ γ
αρ
E E
E E
V
E E
L
t
AV
AV photon molecule
AV^2
h
The total time spent crossing the distance L is: L / c + <Δ t >.
Acknowledgements I wish to thank my colleagues: Barrand G, Haïssinski J and Zomer F, for numerous and helpful discussions.
References [1] Foucault L 1850 CR Acad. Sci. Paris 30 551 [2] Feynman R P, Leighton R B, and Sands M L 1965 The Feynman Lectures on Physics vol 1 (Addison-Wesley, New York) [3] van de Hulst H C 1981 Light Scattering by Small Particles (Dover publications, New York) sections 4.3 and 4. [4] Lorentz H A 1952 The Theory of Electrons , second edition, (Dover publications, New York) Item 132 p 152 Sommerfeld A 1954 Optics , Lectures on Theoretical Physics, vol 4 (Academic Press Inc) pp 91 and 92 Rosenfeld L 1951 Theory of Electrons (North Holland Publishing Company, Amsterdam) Jackson J D 1975, Classical Electrodynamics , second edition, (John Wiley & sons) Panofsky W K H and Phillips M 1955 Classical Electricity and Magnetism (Addison-Wesley) Born M and Wolf E 1999 Principles of Optics , seventh edition, (Cambridge University Press) [5] Koch J 1909 Nova Acta Upsal. 2 [6] Breit G 1932 Quantum theory of dispersion Reviews of Modern Physics 4 504 Breit G 1933 Quantum theory of dispersion Reviews of Modern Physics 5 91 [7] Vinti J P 1932 Phys. Rev. 42 632 Wheeler J A 1933 Phys. Rev. 43 258 [8] Korff S A 1929 Phys. Rev. 34 457 [9] Brillouin L 1960 Wave Propagation and Group Velocity (Academic Press Inc. New York) [10] Steinberg A M, Kwiat P G, and Chiao R Y 1992 Phys. Rev. Let. 68 2421 [11] Ciddor P E and Hill R J 1999 Applied Optics 38 1663 [12] Malitson I H 1965 J. Opt. Soc. Am. 55 1205 [13] Hong C K, Ou Z Y and Mandel L 1987 Phys. Rev. Let. 59 2044 [14] Zernike F 1964 J. Opt. Soc. Am. 54 1215 [15] Maker P D, Terhune R W and Savage C M 1964 Phys. Rev. Let. 12 507 [16] Ho P P and Alfano R R 1979 Phys. Rev. A 20 2170 [17] Gong Q H, Li J L, Zhang T Q and Yang H 1998 Chin. Phys. Let. 15 30 [18] Kolesic M, Wright E M and Moloney J V 2004 Phys. Rev. Let. 92 253901 [19] Hellwarth R W, Pennington D M and Henesian M A 1990 Phys. Rev. A 41 2766 [20] Nibbering E T J, Grillon G, Franco M A, Prade B S and Mysyrowicz A 1997 J. Opt. Soc. Am. B 14 650
