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Topic 9: The Spin of the FieldFirst Draft

Dr. Bill PezzagliaCSUEB Physics

Updated Nov 20, 2010

1Lecture Series: The Spin of the Matter, Physics 4 250, Fall 2010

Outline

A. Energy & Momentum in WavesB. The Spin in WavesC. Gravitational AnalogiesD. References

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A. Energy & Momentum in EM Waves

1. Energy in Field

2. Momentum in Waves

3. Minkowski vs Abraham Controversy

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Electromagnetic Wave1865 Maxwell shows his equations predict that electromagnetic waves can exist in vacuum.

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(a) Energy in Electric Fields• Attributed to Maxwell (article 631, of the Treatise)

• Energy in Electric Field:

• Electric Displacement Vector D might not be parallel to Electric field E. Related by the permittivity tensor εεεε

1. Energy in EM Fields 5

kjk

j EEEDu ε21

21 =•=

rr

kjkj ED ε=

• Attributed to Maxwell (article 636, of the Treatise)

• Energy in Magnetic Field:

• Magnetic Intensity Vector H might not be parallel to Magnetic field B. Related by the permeability tensor µµµµ

(b) Energy in Magnetic Fields 6

kjkj HB µ=

kjk

j HHHBu µ21

21 =•=

rr

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• 1884 Poynting describes energy flow in electromagnetic wave

• Given speed of wave is c, the flux of energy is given: S=cu[Sunlight is 1400 Watts/m2]

• Situation is less clear inside of media. The speed is slower, so we’d need the energy density to be higher to have the same flux.

(c) Energy Flux 7

HEcuSrrr

×==

John Henry Poynting(1852-1914)

(a) Light Pressure

• 1619 Kepler (1746 Euler) proposes comet tails are pushed by sun’s rays

• 1871 Maxwell deduces that light should exert a pressure, but this was from an “induction” argument, rather than from Lorentz force (as this was before the theory of electron conduction)

• 1876 Bartole predicts light pressure from 2nd law of thermodynamics

• 1884 Poynting describes energy flow in electromagnetic wave, and I believe shows that light pressure would be given by Poynting vector divided by c (hence very small)

2. Momentum 8

HEuSP cc

rrr×=== 11

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b. Experiments

• P.N. Lebedev, Ann. Phys. (Leipzig) 6, 433 (1901)First experiment to demonstrate light pressure. However, it is later felt his experiment was somewhat flawed.

• 1901 Nichols & Hull construct a radiometer which unambiguously measures light pressure.E.F. Nichols and G.F. Hull, Phys. Rev. 13, 307 (1901)

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Ernest Fox Nichols1869-1924

Gordon Ferrie Hull 1870-1956

• There is disagreement over the correct form of Poynting “momentum”vector inside of media. They have been arguing over it for 100 years.

• 1908 Minkowski:

Hence momentum is proportional to index of refraction (i.e. flux moves with phase velocity). Sometimes called “pseudomomentum”

• 1910 Abraham:

Hence momentum is inversely proportional to index of refraction (i.e. flux moves with group velocity). Favored by recent experiments.

3. Minkowski-Abraham Controversy10

BDcSrrr

×=

HES c

rrr×= 1

Minkowski, H., Nachr. Ges. Wiss. Gottingen, 53 (1908).Abraham, M., Sull'elettrodinamica di Minkowski. Rend. Circ. Matem. Palermo, 1910. XXX: p. 33-46.

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B. Spin in Waves

1. Circular Polarization2. Photon Spin Paradox3. Spin Tensor for EM

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1. Circular Polarization

• 1812 Jean-Baptiste Biotdiscovers circular polarization

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1b Making Circular Polarized Light

A “quarter wave plate” will convert linearly polarized light into circular polarized.

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1c Does Circular Polarized Light have Spin? 14

• circularly polarized light passing through a quarter wave plate will come out linear polarized.

• Assuming circular polarized light has spin angular momentum, and linear polarized light does not

• Then is angular momentum transferred to the plate?

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1d Mechanical Argument15

• J.H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light", Proc. Roy. Soc. A82, 560-567 (1909);

• Barlow, G., On the Torque produced by a Beam of Light in Oblique Refraction through a Glass Plate. Proc. R. Soc. Lond., A87 (1912), p. 1.

1909 Poynting: From mechanical analogy, argues that circularly polarized light passing through a quarter wave plate (emerging linear polarized) should exert torque per area equal to the light pressure “P” times wavelength

=∆∆

πλτ2

PA

Proposes mechanism is that electrons are displaced into circular orbits around their equilibrium

Instead, you can argue that in a quarterwaveplate, the displacement vector is not parallel to the electric field so there is a torque: ED

A

rr×=

∆∆τ

2a Experimental Verification16

R. A. Beth, “Direct Detection of the Angular Momentum of Light”, Phys. Rev. 48, 471 (1935); “Mechanical Detection and Measurement of the Angular Momentum of Light”, Phys. Rev. 50, 115-125 (1936).

• 1936 Beth performs experiment

• Experimental design quadruples effect

• M is a half wave plate (turns left circular polarized to right circular polarized, hence doubles effect)

• B & T are quarter wave plates

• T is a mirror which reflects beam back through M (doubles effect).

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2b Angular Momentum in EM17

• 1914 Abraham argues that angular momentum of light would be given by,

• For infinite plane wave, this integral is ZERO, independentof polarization.

• Paradox : there should be NO torque on quarter wave plate!

M. Abraham, phys. Zs. 15 (1914), 914. A more accessible treatment is:W. Heitler, "On the Radiation Emitted by a Multipole and its Angular Momentum", Proc. Cambridge Phil Soc 32, 112-126 (1936); The Quantum Theory of Radiation (Dover, 1984) p. 401 [first published 1936]

( )∫ ××= HErdc

Jrrrr

τ2

1

0=Jr

2c The Wall Effect18

• Poynting (and others) argued that the angular momentum must be carried in the “wall” of a finite cylindrical beam of light.

• The field gradient at the edge induces longitudinal components, which induce an azimuthal momentum

• Wavefronts become helical. These“orbital angular momentum” stateshave been verified.

• Many authors view this as the end of the paradox.Others think its just “dirtied the water” by introducing a competing effect, and sidesteps the question.

For recent review, see D. Kiang and K. Young, “The angular momentum of photons in a circularly polarized beam", Am. J. Phys. 76, 1012-1014 (2008).

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3. The Spin Tensor19

• Alternative (minority) viewpoint, is that the classical electromagnetic field DOES have a spin density throughout the circularly polarized beam (plane waves DO exist).

• Change in spin would be due to dielectric (and magnetic) torques in active media (where D is not parallel to E, etc)

• Or, in more correct tensor form, the conservation law:

BHEDdt

d emrrrr

r

×+×=Σ

µσ

νσνσ

µσµνσσ FUFU −=Σ∂

3b Electromagnetic Spin Tensor

• Using Maxwell’s equations and clever integration by parts, the solution given by Rohrlich (attributed to earlier wo rk by Abraham):

• This does seem to give zero spin for linear polariz ation and nonzero spin for circular polarization.

• Problem: this appears to be gauge dependent .

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( )µανναµπ

αµν

µ

µν

AFAF

EA

AE

c −−=Σ=Σ=Σ

×=Σ=Σ

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444

4

'rr

rrr

F. Rohrlich, “Classical Charged Particles”, Addison-Wesley (1965), p94

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3c My Version

• My own derivation says that inside media, the correct form would be more like:

• The conservation law,

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HADA

HAADrrrr

rrrr

×−=Σ=Σ

−×=Σ=Σ444

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' µ

µν

F. Rohrlich, “Classical Charged Particles”, Addison-Wesley (1965), p94

JAAHEBDdt

d

JABHDEdt

d

rrrrrrr

rrrrrrr

4' −=×−×=Σ

×+×+×=Σ

ρ

C. Gravitational Analogies

1. Gravitational Energy2. Gravitational Waves3. Spin in gravity field?

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• Gauss Law shows Cavendish constant “G”plays role of permittivity

• By analogy the energy must be:

• In this room the energy density is quite big! [did I do the calculation right?]

1 Energy in Gravity Field 23

m

q

Gg

E

ρπερ

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−=•∇

=•∇

r

r

3

10

2

21

1073.5

4

mJu

G

gu

×=

=π

2. Linear Polarized Gravity Waves

+ Polarization x Polarization

Note particles oscillation linearly

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3. Circularly Polarized

• Here dots move in circles

• Will it transfer angular momentum to target?

• What is “spin tensor” for gravity? (Is it “torsion”?)

• Is there a similar “paradox” for gravity spin?

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References (Papers)26

• J.H.Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light", Proc. Roy. Soc. A82, 560 (1909).

• C. G. Darwin, “Notes on the Theory of Radiation “, Proc. Roy. Soc. A136, 36-52 (1932).• R. A. Beth, “Mechanical detection and measurement of the angular momentum of light", Phys. Rev. 50, 115-125 (1936).• R. Feynman, R. Leighton and M. Sands, The Feynman Lecture on Physics (Addison-Wesley, 1965), Chapter 17, pp. 9.

Argues circularly polarized light transfers spin to a plate.• Jackson, J.D., Classical Electrodyanmics (Wiley, 2nd ed. 1975), pp. 236-244. He relegates the question of angular

momentum of circularly polarized light to several homework problems. Without the answers, we don’t really know what interpretation he gives.

• R.I. Khrapko, “Question #79 Does plane wave not carry a spin?", Am. J. Phys. 69, 405 (2001).• V. B. Yurchenko, “Answer to Question #79 Does plane wave not carry a spin?", Am. J. Phys. 70, pp. 568-9 (2002).• L. Allen and M. J. Padgett, “Response to Question #79 Does plane wave not carry spin angular momentum?", Am. J. Phys.

70, 567-568 (2002).• R.I. Khrapko has proposed a point-like classical EM spin tensor (physics/0102084). This one agrees with mine for vacuum,

but it lacks the source terms and hence is a weak argument (which explains why this paper was rejected over 300 times). This author makes many points, but usually presents it as standard theory is wrong. Also he does not reference others who proposed similar tensors, e.g. Rohrlich

• F. Rohrlich, “Classical Charged Particles”, Addison-Wesley (1965), p94, defines a spin tensor for EM field, however he does not clearly state who did it first. Later p. 101 he attributes some of the ideas to Abraham to be the first to calculate spin of plane wave circular polarized light, and more general spherical wave to Sommerfeld.

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References (more!)27

• Recent Debate argues instead its edge-effect angular momentum on finite sized target, AJP 69, (2001), 405, 70, (2002), 565 & 567.

• R.N.C. Pfeifer, T.A. Nieminen, N.R. Heckenberg, and H. Rubinsztein-Dunlop, “Two Controversies in Classical Electromagnetism" in Proceedings of SPIE. Optical Trapping and Optical Micromanipulation III, San Diego, Ca KishanDholakia and Gabriel C. Spalding Editors, (August 2006).

• D. Kiang and K. Young, “The angular momentum of photons in a circularly polarized beam", Am. J. Phys. 76, 1012-1014 (2008). This is a recent short article, which summarizes the issue. He asserts the belief that the angular momentum resides in the edge of the beam, and does integration by parts. His argument addresses the circular motion of electrons as damped oscillators (does NOT consider point torques on dipoles, except briefly). Claims his result is general, however it seems to me he still assumes monochromatic light.

• M. Padgett, J. Courtail and L. Allen, “Light's Orbital Angular Momentum", Physics Today 57, No. 5 (May 2004) 35-40. This is a good intro article with pictures, however little formulas.

• J.C. Maxwell, A Treatise on Electricity & Magnetism, in 2 volumes (Dover edition 1954), 3rd edition 1891 Clarendon Press.• W. Pauli, Theory of Relativity, (Dover 1981), see section 35 for a summary of different forms of Stress-Energy tensor.

Anti-references28

The following standard references have been checked and don’t provide much useful for our problem.

• Griffths, Introduction to Electrodynamics: Certainly a good reference, but I don’t see anything that addresses spin of circularly polarized light.

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Things to do29

• Show mechanism for how momentum is transferred to medium