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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

P

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=+ℜµ>==< 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

P

~

:

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−ℜ>=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−−ℜ>=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.