Abstract

We demonstrate the fulfilment of the conservation laws with respect to fluxes of momentum, energy, spin, and photons when a plane circularly polarized electromagnetic wave reflects from a receding mirror at normal incidence. The given calculations show that spin occurs to be the same natural property of a plane electromagnetic wave, as energy and momentum. Recognizing the existence of photons with momentum, energy and spin in a plane electromagnetic wave (1.3), it is strange to deny the existence of spin in such a wave, as is done in modern electrodynamics.
Reflection of light from a moving mirror
Radi I. Khrapko
Moscow Aviation Institute - Volokolamskoe shosse 4, 125993 Moscow, Russia
We demonstrate the fulfilment of the conservation laws with respect to fluxes of
momentum, energy, spin, and photons when a plane circularly polarized
electromagnetic wave reflects from a receding mirror at normal incidence.
Key Words: spin, circular polarization, spin tensor
PACS 75.10.Hk
1. Introduction
The reflection of light from a moving mirror is essentially exhaustively investigated in a famous
paper [1]. Nevertheless, it seems interesting to demonstrate the implementation of the conservation
laws with respect to fluxes of momentum, energy, spin, and number of photons within such a
reflection. The Maxwell tensor in Minkowski space [2],
µλ
µλ
αβλβ
µλ
αµαβ
+= FFgFFgT
, (1.1)
is used for the calculation of momentum and energy, and the canonical spin tensor [3,4],
νµλλµν
=
Υ
][
2
F
A
, (1.2)
is used for the calculation of spin. Here
µλ
F is the electromagnetic field tensor, and
λ
A
is the
magnetic vector potential. The number of photons is calculated by means of a division of energy by
ω
h
or by means of a division of spin by
h
. We confine ourselves by the normal incidence of light
on a mirror and, for concreteness, we consider the case of a receding mirror. Due to the spin count,
we consider the incident plane wave of circular polarization
)exp()(
1111
tizikiE ω+= yxE
[V/m],
101
EH ci
ε=
[A/m],
11
ω=ck
(1.3)
and, respectively, the reflected wave
)exp()(
2222
tizikiE ω+= yxE
,
202
EH ci
ε=
.
22
ω=ck
(1.4)
As is well known [1], the frequency ratio of the reflected and incident waves coincides with
the ratio of the amplitudes of these waves and is given by the formula
β+
β
==
ω
ω
1
1
1
2
1
2
E
E
, (1.5)
where cv /
=
β
, and
v
is the speed of the mirror.
2. Momentum flux density, i.e. pressure
The wave, which impinges on the moving mirror, has the frequency related to the mirror, according
to the Doppler effect [5, § 48],
β+
β
ω=ω 1
1
10
(2.1)
and, respectively, has the amplitude
β+
β
=1
1
10
EE . (2.2)
We consider the mirror to be superconducting, thus the magnetic field doubles on the mirror under a
zero electric field
)exp()(2
0000
tiicE ω+ε= yxH . (2.3)
Therefore the pressure on the mirror is defined by the formula =
0
P
2/
2
0
><µ>=< HT
zz
and
turns out to be equal to
β+
β
ε=ε=+µ>==< 1
1
224/}{
2
10
2
00000
EEHHHHT
yyxx
zz
P
[N/m
2
]. (2.4)
In addition to the momentum flux, which gives pressure on the mirror, there is a filling of
the space vacated by the moving mirror by momentum. The volume density of the filling,
>+=<
ztztz
TTG
21
, consists of two parts, belonging to the incident and to the reflected waves:
)(
2222111121
yt
zy
xt
zx
yt
zy
xt
zx
zzztzt
FFFFFFFFgTT +++=+
(2.5)
2/)(
2222111121
yt
zy
xt
zx
yt
zy
xt
zx
ztztz
DBDBDBDBTTG −ℜ>=+=<
2
2
10
2
1
2
2
2
10
2
2
2
1
0
)1(
4
)1()( β+
βε
=
ε
=
ε
=c
E
E
E
c
E
EE
c
[Ns/m
3
]. (2.6)
This filling requires the momentum flux density vG
z
, which we will call
~
:
2
22
10
)1(
4
~
β+
βε
== E
vG
z
P
[N/m
2
]. (2.7)
The total flux density is equal to:
2
2
2
10
2
22
10
2
100
)1(
1
2
)1(
2
1
1
2
~
β+
β+
ε=
β+
βε
+
β+
β
ε=+= E
E
E
PPP
(2.8)
This total flux density is provided by the oncoming flux density >+=<
zzzz
TT
21
P
. Really, in
accordance with the formula (1.1), we have expressions such as
2/)(
yt
yt
yx
yx
xt
xt
yz
zy
xz
zx
tz
zt
zzzz
FFFFFFFFFFFFgT +++++=
2/)(
y
y
x
x
yz
zy
xz
zx
DEDEHBHB +=
, (2.9)
2
0
22
0
22
0
4/)(4/)( EEEHHT
xyxy
zz
ε=+ε++µ>=<
(2.10)
for the incident or reflected waves. Thus the total momentum flux density,
2
2
2
10
2
2
2
10
2
1
2
2
2
10
2
2
2
1021
)1(
1
2
)1(
)1(
1)1()( β+
β+
ε=
β+
β
+ε=+ε=+ε>=+=< EE
E
E
EEETT
zzzz
P
, (2.11)
coincides with expression (2.8).
3. Law of Conservation of Energy
The pressure on the mirror
0
P
(2.4) produces a work because of the movement of the mirror. The
corresponding mass-energy flux density is equal to:
β
β+
β
ε
==Π 1
1
2
2
10
2
0
0
c
E
c
v
P
sm
kg
2
(3.1)
In addition, there is a filling of the space vacated by the moving mirror by mass-energy. The
volume density of this filling, >+=<
tttt
TTu 21
, consists of two parts, belonging to the incident and
to the reflected waves. Taking into account formula (1.1), we have expressions such as
2/)(
yz
yz
xz
xz
xy
xy
zt
tz
yt
ty
xt
tx
tttt
FFFFFFFFFFFFgT +++++=
)2/()( 2
cHBHBDEDE
yz
yz
xz
xz
y
y
x
x
+++=
, (3.2)
22
0
222
0
222
0/)4/()()4/()( cEcHHcEET
xyyx
tt
ε=+µ++ε>=<
[kg/m
3
]. (3.3)
for the incident or reflected waves. Thus the total volume mass-energy equals:
2
2
2
2
10
2
2
2
2
10
2
1
2
2
2
2
10
22
2
2
1021 )1(
1
2
)1(
)1(
1)1(/)( β+
β+
ε
=
β+
β
+
ε
=+
ε
=+ε>=+=< c
E
c
E
E
E
c
E
cEETTu
tttt
. (3.4)
This filling requires the mass-energy flux density, which we call
uv=Π
~
,
β
β+
β+
ε
==Π
2
2
2
10
)1(
1
2
~
c
E
uv
. (3.5)
The total mass-energy flux density,
2
2
10
2
2
2
10
0)1(
4
)1(
1
1
1
2
~
β+
βε
=β
β+
β+
+
β+
β
ε
=Π+Π c
E
c
E
sm
kg
2
(3.6)
is provided by the Poynting vector >+=<Π
tztz
TT 21
. Really,
)( 2222111121
xt
zx
xt
zx
yt
zy
xt
zx
zztztz
FFFFFFFFgTT +++=+ )( 12121111
y
zy
x
zx
y
zy
x
zx
DBDBDBDB =
,
)(
2222111100
yxxyyxxy
EHEHEHEH
+εµ= , (3.7)
2
2
10
2
2
2
10
2
1
2
2
2
10
2
2
2
1
0
21 )1(
4
)1(
)1(
1)1()( β+
βε
=
β+
β
ε
=
ε
=
ε
>=+=<Π c
E
с
E
E
E
с
E
EE
с
TT
tztz
. (3.8)
The value (3.8) coincides with (3.6).
4. Conservation of the number of photons
The volume density of photons,
n
, in the space, vacated by the moving mirror, is obtained by
dividing the portions of the energy density (3.4) by the energy of a single photon, i.e. by
1
ω
h
or by
2
ω
h
)1(
2
1
1
1)1()(
1
2
10
1
2
10
1
2
1
2
10
2
2
2
1
2
1
0
β+ω
ε
=
β+
β
+
ω
ε
=
ω
ω
+
ω
ε
=
ω
+
ω
ε=
hhhhh
EEEEE
n
[1/m
3
]. (4.1)
Due to the motion of the mirror the number of the photons increases. This requires the photon
number flux density
)1(
2
1
2
10
β+ω
ε
=
h
vE
nv
[1/m
2
s]. (4.2)
This flux density is provided by the difference of Poynting vectors from formula (3.8)
)1(
2
1
1
1)1()(
1
2
10
1
2
10
1
2
1
2
10
2
2
2
1
2
1
0
2
2
1
1
1
β+ω
ε
=
β+
β
ω
ε
=
ω
ω
ω
ε
=
ω
ω
ε=>
ω
+
ω
<
hhhhhhh
vEcEcEEE
cc
TT
tztz
. (4.3)
Photon number flux density (4.3) coincides with flux density (4.2).
5. Conservation of spin
The number of photons can be calculated not only on the basis of wave energy, but also on the basis
of wave spin. The volume density of wave spin is given by the component of the canonical spin
tensor (1.2)
xyyx
tyxxyt
DADAFA +==Υ ][
2
[Js/m
3
], (5.1)
and the spin flux density is given by the component
yyxx
zyxxyz
HAHAFA +==Υ ][
2
[J/m
2
]. (5.2)
Note that the lowering of the spatial index of the vector potential is related to the change of the sign
in the view of the metric signature )(
+
.
Since for a monochromatic field
ω== /
kkk
iEdtEA , densities (5.1), (5.2) can be
expressed through the electromagnetic field:
ω=Υω=Υ /)(,/)(
yyxx
xyz
xyyx
xyt
HiEHiEDiEDiE
. (5.3)
In our case of reflection from a moving mirror (1.3), (1.4) volume density of the spin is equal to:
2/}/)(/){( 2222211111 ω+ω>=Υ<
xyyxxyyx
xyt
DiEDiEDiEDiE
)1(
2
1
1
1)1()(
1
2
10
1
2
10
1
2
1
2
10
2
2
2
1
2
1
0
β+ω
ε
=
β+
β
+
ω
ε
=
ω
ω
+
ω
ε
=
ω
+
ω
ε= EEEEE , (5.4)
and the photon density is given through the division by
h
and coincides with the value (4.1).
The spin flux density is equal to:
2/}/)(/){( 2222211111 ω+ω>=Υ<
yyxxyyxx
xyz
HiEHiEHiEHiE
)1(
2
1
1
1)1()(
1
2
10
1
2
10
1
2
1
2
10
2
2
2
1
2
1
0
β+ω
ε
=
β+
β
ω
ε
=
ω
ω
ω
ε
=
ω
ω
ε= vEcEcEEE
c, (5.5)
and the photon flux density is given through the division by
h
and coincides with the value (4.3).
Naturally, the increase in the amount of spin is provided by the spin flux:
>Υ=<>Υ<
xyzxyt
v (5.6)
6. Conclusion
The given calculations show that spin occurs to be the same natural property of a plane
electromagnetic wave, as energy and momentum. Recognizing the existence of photons with
momentum, energy and spin in a plane electromagnetic wave (1.3), it is strange to deny the
existence of spin in such a wave, as is done in modern electrodynamics.
I am eternally grateful to Professor Robert Romer, having courageously published my question:
"Does a plane wave really not carry spin?” [6], and to the reviewer of JMO, who evaluated my
paper [7] positively.
See also "Note" in http://khrapkori.wmsite.ru/ftpgetfile.php?id=9&module=files
References
1. Bolotovskii B. M, Stolyarov S, N “Reflection of light from a moving mirror and related
problems” Sov. Phys. Usp.
32
813–827 (1989) http://ufn.ru/en/articles/1989/9/e/
2. Jackson J. D., Classical Electrodynamics, (John Wiley, 1999), p. 609.
3. Corson E M Introduction to tensors, spinors, and reativistic wave-equation NY, Hafner, 1953
p.71
4. Soper D. E., Classical Field Theory (N.Y.: Dover, 2008), p. 114
5. Landau L. D., Lifshitz E. M. The Classical Theory of Fields (Pergamon, N. Y. 1975)
6. Khrapko R I "Does plane wave not carry a spin?" Amer. J. Phys. 69, 405 (2001)
http://khrapkori.wmsite.ru/ftpgetfile.php?id=10&module=files
7. Khrapko R.I. Mechanical stresses produced by a light beam J. Modern Optics, 55, 1487-1500
(2008) http://khrapkori.wmsite.ru/ftpgetfile.php?id=9&module=files
Addition
History of rejections
AJP rejected the paper
-- The content of this manuscript consists only of calculations, with no attempt at providing any
physical insight or discussion as to why such calculations might be useful or interesting to physics
educators.
David P. Jacksonand Daniel V. Schroeder
-- My Appeal:
This calculation proves reality and activity of the canonical spin tensor, which is presented in
textbooks.
The given calculations show that spin occurs to be the same natural property of a plane
electromagnetic wave as energy and momentum. Recognizing the existence of photons with
momentum, energy and spin in a plane electromagnetic wave it is strange to deny the existence of
spin in such a wave, as is done in the modern electrodynamics.
OC rejected the paper
-- I feel your manuscript is not appropriate for this journal's readership.
Martin Booth Editor Optics Communications
-- My answer:
Dear Martin Booth, thank you for your kind answer. However I think OC's readers are interested to
know that a current statement, "A plane wave has no angular momentum", is not true.
Really:
Heitler W ["The Quantum Theory of Radiation" (Oxford: Clarendon, 1954) p. 401]:
"A plane wave travelling in z-directionand with infinite extension in the xy-directions can have no
angular momentum about the z-axis, because S is in the z-direction and (r x S)_z = 0".
Allen L., M. J. Padgett, [“Response to Question #79. Does a plane wave carry spin angular
momentum?” Am. J. Phys. 70, 567 (2002)]:
"For a plane wave there is no (radial intensity) gradient and the spin density is zero"
Simmonds J. W., M. J. Guttmann, States, Waves and Photons [Addison-Wesley, Reading, MA,
1970]:
“The skin region (of a beam) is the only in which the z-component of angular momentum does not
vanish”
However,
We demonstrate the fulfilment of the conservation laws with respect to fluxes of momentum,
energy, spin, and photons when a plane circularly polarized electromagnetic wave reflects from a
receding mirror at normal incidence
The given calculations show that spin occurs to be the same natural property of a plane
electromagnetic wave, as energy and momentum. Recognizing the existence of photons with
momentum, energy and spin in a plane electromagnetic wave (1.3), it is strange to deny the
existence of spin in such a wave, as is done in the modern electrodynamics.
... The presented calculation of the work over the absorber, like other calculations [12][13][14][15], does not depend on existence of a boundary of the electromagnetic wave. Such calculations are important because according to a nowadays theory of electrodynamics spin, "a plane wave travelling in z-direction and with infinite extension in the xy-directions can have no angular momentum about the z-axis" [16]. ...
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A simple calculation shows that when an electromagnetic wave of circular polarization is absorbed, the rotation of the absorber changes the flow of electromagnetic energy to the absorber by the amount of mechanical work, positive or negative, performed by the spin of this wave. The result obtained confirms the existence of the classical spin of a plane electromagnetic wave of circular polarization.
... This corresponds to the power density 2 0 0 0 / E ε µ . Therefore, the canonical tensor could be successfully used in [9,[17][18][19][20][21][22][23]. However, this tensor gives the wrong result for directions perpendicular to the direction of propagation. ...
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... J. H. Poynting). The spin tensor is used in the publications [9][10][11][12][13][14][15][16][17][18][19][20]. However, the spin tensor is ignored in works expressing the common point of view, e.g. ...
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... This proves that the spin of electromagnetic radiation is actually described by the spin tensor. The spin tensor has been successfully used to describe also plane waves and beams [11][12][13][14][15][16][17]. ...
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Jefimenko's generalization of the Coulomb and Bio-Savard laws is used to calculate the reaction of the radiation on the rotating electric dipole. It is found that the energy taken from the dipole is equal to the recognized value of the radiated energy. At the same time, it is confirmed that the angular momentum flux exceeds the generally accepted value by the spin radiation not seen before.
... The canonical spin tensor (1.4) was successfully used in [9] in order to confirm the fulfillment of the conservation laws with respect to spin when a plane circularly polarized electromagnetic wave reflects from a moving mirror, and in [10] where the absorption of spin of a plane wave is described. These calculations prove the functionality of the spin tensor and show that spin is the same natural property of a perfect plane electromagnetic wave, as energy and momentum; and spin density is proportional to energy density. ...
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We demonstrate the transfer of momentum, energy, and spin from a plane circularly polarized electromagnetic wave into an absorber. Lorentz transformations are used for the flux densities because a moving absorber is considered. The given calculations show that spin is the same natural property of a plane electromagnetic wave, as energy and momentum.
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It is shown that the electromagnetic field in the well-known Beth’s experiment contains no linear momentum. This means that the angular momentum of the field in Beth’s experiment is zero, since the angular momentum, by definition, is the moment of the linear momentum. Nevertheless, the half-wave plate in Beth’s experiment receives an angular momentum from the field, which, by definition, does not have this angular momentum. This means that the definition of the angular momentum of an electromagnetic field should be changed to explain Beth’s experiment. The angular momentum of an electromagnetic field contains a spin term that does not depend on the linear momentum.
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The use of the electrodynamics spin tensor in parallel with the energy-momentum tensor proves absorption of spin and energy of a plane circularly polarized electromagnetic wave and so confirms the absorption, which was calculated in a previous paper by the dynamical way, i.e. by the use of the concept of mechanical torque.
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When the canonical spin tensor is used, a spin is detected along the axis of rotation of a rotating electric dipole. Early this spin radiation was obtained by Feynman in the frame of quantum mechanics. The magnitude of the spin flux is half the flux of the angular momentum that is emitted by a rotating dipole, according to modern electrodynamics, and this angular momentum flux is recognized here as an orbital angular momentum. Thus, the total angular momentum flux exceeds 1.5 times the value now recognized. It is shown that the torque experienced by a rotating dipole from the field is equal in magnitude to this total angular momentum flux.
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A circularly polarized electromagnetic beam is considered, which is absorbed by a plane, and the mechanical stress produced in the plane by the beam is calculated. It is shown that the central part of the beam produces a torque at the central region of the plane due to the spin of the beam, and the wall of the beam produces an additional torque due to orbital angular momentum of the beam. The total torque acting on the plane equals twice the power of the beam divided by the frequency. This fact contradicts the standard electrodynamics, which predicts the torque equals power of the beam divided by frequency, and means the electrodynamics, as well as the whole classical field theory, is incomplete. Introducing the spin tensor corrects the electrodynamics.
Book
Continuum mechanics, electrodynamics and the mechanics of electrically polarized media, and gravity are formulated by means of the principle of least action.
  • J D Jackson
Jackson J. D., Classical Electrodynamics, (John Wiley, 1999), p. 609.