Angular momentum of the photon – experimental proposal

Jerzy Kosek, Poland

1. Introduction2. Linear and angular momentum of the photon.3. Measurement of the photon spin – Beth, 1936.4. Angular momentum of the circularly polarized photon.5. Angular momentum of the linearly polarized photon.6. Description of the photon state in Quantum Theory of Light.7. Setup of the experiment.8. List of accessories.9. Realization of experiment.

1. Introduction

The goal of this presentation is to demonstrate the fact, that photon has angular momentum (spin) - roughly saying - spinning in two possible directions: clockwise and counter clockwise relative to the direction of its propagation.

We hope that it allows students deeper understand spin - basic property of every elementary particle.

2.Linear and angular momentum of the photon

The quantum theory of light predicts that every photon in addition to its linear momentum possesses also intrinsic angular momentum (named spin) equal to 1.

•What does it mean? •What do we learn from it? •How can we measure it?

In accordance with classical electrodynamics and also with quantum mechanics light possesses linear momentum. It’s usually demonstrated with use of a small whirligig with mirrors montage in a vacuum bulb.

2. Linear and angular momentum of the photon

Projection of the photon spin on the direction of photon propagation is equal to ћ or -ћ for left or right circularly polarized light.

Projection of the photon spin; s=+1

s=-1 Direction of photon propagation

The magnitude of the total spin is given by the equation:

For photon we have quantum spin number s=1. It gives:

)1( ssS

2S

Direction of photon propagation

a). Right–circularly polarized photon

a). Left–circularly polarized photon

S

3.Measurement of the photon spin

Experimental proof of that theoretical prediction was done by R. Beth in 1936 in Princeton. As Beth announces in his paper (R. A. Beth, Mechanical Detection and Measurement of the Angular Momentum of Light, Physical Review, v. 50, July 15, 1936) he had several discussion about the experiment with Einstein. In this experiment Beth showed that when linearly polarized light is converted to circularly polarized one by doubly refracting slab, the slab experiences a reaction torque.

It is difficult to demonstrate in high school laboratory this experiment and no simple solutions were proposed so far.

Experimental setup of Beth’s experiment

4. Angular momentum of the circularly polarized photon

According to the Quantum Theory of Light every photon of circularly polarized light has the same angular momentum.

It’s expected that an object absorbing circularly polarized photons will rotate clockwise or counter clockwise with respect to the type of circular polarisation.

Absorbing plate

rotation

rotation

λ/4

λ/4

α =+45o

α =-45ob)

a)

Source of the light

Source of the light

5. Angular momentum of the linearly polarized photon

If light is linearly polarized then it doesn’t have angular momentum and no rotation of an object will be observed.

Linear polarizer

α =90o

Plate

λ/4

λ/4α =0o

c)

d)

Absorbing plate

no rotation

no rotation

Source of the light

Source of the light

6. Description of the photon state in the Quantum Theory of Light

When light is linearly polarized then every photon can be considered as a superposition of states of left and right circular polarization with equal probabilities. For instance linearly polarized light at an angle at 450 to the horizontal plane can be expressed as follows:

One of mysteries of the photon is fact that before measurement the photon in a superposition state has an undefined angular momentum. The same applies to every quantum particle.

)|(|2

1| RL

Linear polarizer

α =90o

Plate

λ/4α =0o

c)

d)

Absorbing plate

no rotation

λ/4

The above equation expresses fact that measurement of spin carried on linearly polarized light gives on average half photons having a left circular polarization state |L> and half a right |R> one. The total momentum of that ensemble of photons is equal to zero.

6. Description of the photon state in the Quantum Theory of Light

In right-circularly polarized light every photon is in the state:

having the same momentum –ћ. Total momentum transmitted from circularly polarized light to absorbing object is the sum of momentums of every photon.

R||

Absorbing plate

rotation

λ/4

λ/4

α =+45o

α =-45ob)

a)

Similarly we can describe left-circularly polarized photons.

7. Setup of the experiment

Light emerging from laser (1) is transformed by two lenses (2) into parallel beam of diameter about 2 cm. Next, the light is reflected by mirror (3), passes linear polariser (4), quarter plate (5) and is incident on absorbing plate (6), hanging on throat (7) in a plastic, tall and transparent tube (8) with the output to the vacuum pump.

The tube is supported on a two heavy retort stands. A long thread is used to minimalize contrary force momentum due to torsion of the thread. Vacuum in tube is created to minimalize rotation resistance of the plate due to environment.

laser

87

6

5

4

3

21 2

vacuum pump

8. List of accessories1.Laser – high power or mercury-vapor lamp with interference filter – 1 piece2. Lenses, focal length: f= 6 cm – 1 piecef= 30 cm – 1 piecef= 60 cm – 1 piece3. Mirror 100% - 1piece4. Linear polarizer – 1 piece5. Plate λ/4 (λ – wavelength of light beam) – 1 piece6. Absorbing plate (black) diameter < 3cm (less then tube diameter) – possibly light7. Thread – 2 m8. Plastic tube - 1 piecelength l = 2m diameter ≈3 cmwith output to vacuum pumptransparent9. Vacuum pump – 1 piece10. Mountings – retort stands: - height 2,5 m – 2 pieces (on picture shown only one retort stand),- height 0,5 m – 4 pieces11. Lens holders – 2 pieces12. Handle (for polarizer and plate λ/4, for mirror 100%) - 2 pieces

laser

87

6

5

4

3

21 2

vacuum pump

9. Realization of experiment

1. Prepare experimental setup without plate λ/4. Light have to be uniformly distributed on absorbing plate 6.

2. Insert plate λ/4. Polarizer 4 adjust in position α=00 later α=900. No rotation of absorbing plate will be observed.

3. Polarizer 4 adjust in position α=+450, later α=-450. A small rotation of absorbing plate is expected to be observed in two different directions. It will show two possible directions of the photon “rotation” – clockwise and counter clockwise in respect to its direction of propagation.

laser

87

6

5

4

3

21 2

vacuum pump