Extending Geometrical Optics: A Lagrangian Theory for ... · Introduction Basic Theory Results...

Introduction Basic Theory Results Conclusions

Extending Geometrical Optics:A Lagrangian Theory for Vector Waves

D. E. Ruiz

In close collaboration with: I. Y. Dodin and C . L. Ellison

APS DPP Meeting

November 4, 2016


EM waves in dielectrics behave as particles with spin.1

© 2008 Macmillan Publishers Limited. All rights reserved.

phase differenceffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 2 n22Þ

p=R per unit ray length (see

Supplementary Information). The anisotropy axes are naturallyattached to the Frenet trihedron (the p and s modes are polarizedalong n ¼ N and b, respectively), and it is convenient to writethe equation for the evolution of polarization in the Frenet framewhere Ap! 2T21. By introducing the phase difference betweenthe s and p modes, we arrive at a modified precession equationfor the Stokes vector19:

_~S ¼ ~V$~S; ~V ¼ ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1% n%2p

R%1; 0;%2T%1Þ: ð7Þ

Thus, the linear birefringence due to the total internal reflectioncompetes with the circular birefringence due to the Berry phase,resulting in a precession of the Stokes vector about the

inclined vector ~V (Fig. 2c). The helicity is not conserved there,S3 = const. (which violates conservation of the total angularmomentum J), but the polarization evolution is still smooth, sothat one can regard this regime as a modified adiabatic evolution.Unlike the isotropic-medium case, equations (4) and (5), theprecession of the Stokes vector influences the ray deflection inequations (6) by means of varying helicity S3(l). This causesoscillations of the light trajectory that are similar to thezitterbewegung of electrons with a spin–orbit interaction19.

EXPERIMENT

To verify the evolution equations (6) and (7), we performedan experiment involving helical light beams propagating at agrazing angle inside a glass (BK7) cylinder. The experimentalsetup is shown in Fig. 3. A linearly polarized HeNe laser beam atl0 ¼ 633 nm wavelength was either right- or left-hand circularlypolarized by a variable liquid-crystal retarder (MeadowlarkOptics). The circularly polarized beam was sent at a grazing angleinto a glass cylinder, using a right-angle prism fitted with anindex-matching gel. We used a cylinder with radius R0 ¼ 8 mmand length L0 ¼ 96 mm and an incident beam of 1 mm width.Once inside the cylinder, the beam underwent continuedinternal reflections that resulted in a helical trajectory along theglass/air interface. The number of coils was adjusted bycontrolling the beam’s angle of propagation u between t and the

cylinder axis. The output Stokes parameters and the beamposition were measured using a polarizer, quarter-wave plate andcharge-coupled device (CCD) camera (12-bit digital-cooled, PCOSensicam 370XL, 1,280 $ 1,024 pixels) imaging the outlet face ofthe cylinder through a second identical right-angle prism.

To calculate the output beam parameters, note that thehelical ray has a constant curvature R21 ¼ R0

21 sin2 u and torsionT 21 ¼ R0

21 sin u cos u. Hence, ~V ¼ const, and equations (6) and(7) can readily be integrated (see Supplementary Information).

For R- and L-polarized incident beams, ~SðR;LÞin ¼ ð0; 0;+1Þ, this

yields the output polarizations ~SðR;LÞout and the shifts of the

trajectory drout(R,L):

~SðR;LÞout ¼+ðv1v3½1% cosðVl0Þ';%v1 sinðVl0Þ;

ð1% v23Þ cosðVl0Þ þ v2

3Þ; ð8Þ

drðR;LÞout ¼+!ll0R0

sin2 u v23 þ 1% v2

3

" # sin Vl0ð ÞVl0

$ %b: ð9Þ

where ~v ¼ ~V=V and l0 ¼ L0/cos u is the total ray length in thecylinder. The relative output shift between the initially R- andL-polarized beams is Dout ¼ [drout

(R) 2 drout(L)]b. Unlike the isotropic

case, Dout is a nonlinear function of the ray length l0 due to theinfluence of the Stokes vector precession. The second, oscillatoryterm in square brackets in equation (9) describes thezitterbewegung of the light trajectory19.

Figure 4 shows the output Stokes parameters ~SðRÞout and the

relative shift Dout as theoretically predicted by equations (8) and(9) and experimentally measured at different angles ofpropagation. The angle of propagation is expressed by thenumber of turns of the helix m, as tan u ¼ 2pR0m/L0. Anexperimental error of the Stokes parameters of 0.07 was causedby the angular tolerance of the polarization elements. Thenumber of coils m was determined to a typical accuracy of0.2 turn. The Stokes parameters were measured using the four-measurements technique37, and the position of the output beamwas determined as a centre of mass (centroid) of the intensitydistribution at the output face of the cylinder. The effects of the

Cylinder

Laser

P1P2 Imaging lens

Camera

QWP

LCVR

Prism

Prism

R

L

Figure 3 Experimental setup. A laser light beam enters the glass cylinder at a grazing angle through the input prism, coils along the cylinder surface, and leaves itby means of the output prism. The liquid-crystal variable retarder (LCVR) is used for generating and switching between the circularly polarized modes, whereas thequarter-wave plate (QWP) and polarizer P2 are intended for measurement of the Stokes parameters. The inset shows a real picture of the spiral light beam inside

the cylinder.

ARTICLES

nature photonics | VOL 2 | DECEMBER 2008 | www.nature.com/naturephotonics 751

© 2008 Macmillan Publishers Limited. All rights reserved.

Hall effect of light

For EM waves innon-birefringent dielectrics

ω = c |K|/n

Geometrical opticsray equations

dX

dt=∂ω

∂KdK

dt=−

∂ω

∂X

Vector waves are analogous to quantum particles with spin. Wave “spin-orbital”interactions can be expected. GO looses these effects completely.

1K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, Nature Photonics 2, 748 (2008)


Is geometrical optics accurate enough? No.• Consider a N-dim wave Ψ(t,x) such that

i∂tΨ = H(t,x,−i∇)Ψ.

• Choose Ψ = ξe iθ.

• Assume the medium parameters vary slowly.

ε.

= max

(1

ωτ,

1

|k|`

) 1, ω(t,x)

.= −∂tθ, k(t,x)

.= ∇θ.

• In GO, |∂ξ|/|ξ| |∂θ| so that2

ωξ = H(t,x,k)ξ ⇒ det [ωIN − H(t,x,k)] = 0 ⇒ ω = H0(t,x,k).

• The ray equations aredX

dt=∂H0

∂P,

dP

dt= −∂H0

∂X.

2E. R. Tracy, A. J. Brizard, A. S. Richardson, and A. N. Kaufman, Ray Tracing and Beyond: Phase Space Methods inPlasma Wave Theory (Cambridge University Press, New York, 2014).


Polarization effects can be obtained from variational principles.

• Waves follow the least action principle δΛ = δ∫dt L = 0

L =

∫d3x

[(i/2)(Ψ†Ψ− c. c.)︸︷︷︸

symplectic part

−Ψ†H(t,x,−i∇)Ψ︸︷︷︸Hamiltonian part

]• Reduced wave theories are obtained by parameterizing Ψ and approximating

L to a certain order in ε.

• Choose Ψ = ξe iθ,

L = −∫

d3x ξ†[∂tθ + H(t,x,∇θ)

]ξ︸︷︷︸

= L0

+O(ε) ⇒ωξ = H(t,x,k)ξ∂t(ξ

†ξ) + ∇ · [ξ†(∂kH)ξ] = 0

Standard GO: ⇒ L ' L0.Extended GO: ⇒ L ' L0 + εL1.

• L1 is calculated by block diagonalizing the Hamiltonian.34

|Ψ〉 = T |ψ〉 ⇒ εL1 ∝ ||∂T ||.

Pierre de Fermat

(1601-1665)

Gerald Whitham

(1927-1980)3L. Friedland and A. N. Kaufman, Phys. Fluids 30, 3050 (1987).4R. G. Littlejohn and W. G. Flynn, Phys. Rev. A 44, 5239 (1991).


Point-particle dynamics is the dynamics of narrow wave packets.

Lsym =i

2

∫d3x (ψ†ψ − c. c.)

• Parameterize such that ψ(t,x) =√I(t,x) z(t, z) e iθ,

∫d3x I(t,x) = 1, z†z = 1.

Lsym = −∫

d3x I ∂tθ︸︷︷︸Lsym,0

+i

2

∫d3x I (z†z − z†z)︸︷︷︸

Lsym,1

• Assume the point-particle limit I(t,x) = δ3(x−X(t)).

Lsym,0 = −∫

d3x δ3(x−X(t)) ∂tθ

=

∫d3x δ3(x−X(t))∇θ(x, t) · X(t)

= P(t) · X(t), P(t).

= ∇θ(t,X(t)).

The wave packet is compressedto a point in space.


Main theoretical result: Extended geometrical optics (XGO)

The wave ray dynamics in XGO is governed by5

LXGO = P · X− H0(t,X,P)︸︷︷︸Lowest−order GO

+ (i/2)(Z †Z − c. c.) + Z †U(t,X,P)Z︸︷︷︸Polarization corrections

.

where Z ∈ CN , H0(t,x,p) is the GO Hamiltonian, U is the spin-coupling Hamiltonian

U(t,X,P).

= (i/2)[T †(DtT )− h. c.] + (i/2)[∂XT † · (H − H0) · ∂pT − h. c.],

and Dt.

= ∂t + X · ∂X + P · ∂P.

X = ∂PH0︸︷︷︸GO velocity

− Z †(∂PU)Z︸︷︷︸Spin deflection

Z † = −iZ †U

P =−∂XH0︸︷︷︸GO force

+ Z †(∂XU)Z︸︷︷︸Stern−Gerlach force

Z = iUZ︸︷︷︸Spin precession

5D. E. Ruiz and I. Y. Dodin, Phys. Lett. A 379, 2337 (2015).


Covariant extended geometrical optics

For more general waves D(xµ, i∂ν)Ψ = 0, the covariant wave ray dynamics is governed by6

LXGO = Pµ · Xµ + λ0(Xµ,Pν)︸︷︷︸Leading−order GO

− (i/2)(Z †Z − c. c.) + Z †U(Xµ,Pν)Z︸︷︷︸Polarization corrections

.

where Z ∈ CN , λ0(Xµ,Pν) = 0 is the GO dispersion surface,

U(Xµ,Pν).

= −(i/2)[T †(DτT )− h. c.]− (i/2)[(∂PµT†)(D − λ0)(∂XµT )− h. c.],

is the spin-coupling Hamiltonian, and Dτ.

= ∂τ + Xµ · ∂Xµ + Pν · ∂Pν .

dXµ

dτ= −

∂λ0

∂Pµ︸︷︷︸GO velocity

− Z†(∂U∂Pµ

)Z︸︷︷︸

Spin deflection

dZ†

dτ= iZ†U

dPν

dτ=

∂λ0

∂Xν︸︷︷︸GO force

+ Z†(∂U∂Xν

)Z︸︷︷︸

Stern−Gerlach force

dZ

dτ= −iUZ︸︷︷︸

Spin precession

6D. E. Ruiz and I. Y. Dodin, arXiv (2016), 1612.06184.


We used XGO to obtain the first point-particle Lagrangian modelfor the relativistin spin-1/2 particle.789

γµ(i∂µ − qAµ)ψ = mψ︸︷︷︸Dirac equation

XGO==⇒ LD = P · X− H0(t,X,P) +

i~2

(Z †Z − Z †Z

)− S(Z †,Z) ·ΩBMT(t,X,P)

H0.

=√

m2c4 + (Pc − qA)2 + qV S(t).

= ~Z †σZ/2

ΩBMT(t,X,P).

=q

mc

[(g

2− 1 +

1

γ

)B−

(g

2−

γ

γ + 1

)β ×E−

(g2− 1) γ

γ + 1(B · β)β

]Z↑ =

(10

)Z↓ =

(01

)

X = ∂PH0︸︷︷︸GO velocity

− ∂P(S ·ΩBMT)︸︷︷︸Spin deflection

i Z = (ΩBMT · σ)Z

P =−∂XH0︸︷︷︸GO force

+ ∂X(S ·ΩBMT)︸︷︷︸Stern−Gerlach force

S = S×ΩBMT︸︷︷︸Spin precession

Ruiz and Dodin (2015)

The same theory yields a point-particle Lagrangian of an electron with spin. It is the first Lagrangian that is formally deduced from a quantum Lagrangian and captures orbital and spin dynamics simultaneously.

cf. Gaioli & Alvarez ('98), Barut et al ('84-'93), Derbenev & Kondratenko ('73), Plahte ('67), Rubinow & Keller ('63), Pauli ('32), Frenkel ('26)...

Understanding plasma waves yields new insights to quantum problems

Slow fields: the BMT theory is made conservative

Oscillatory fields: ponderomotive Lagrangian of an electron in a vacuum relativistic EM wave, arbitrary Ode Broglie/OEM

7D. E. Ruiz and I. Y. Dodin, Phys. Lett. A 379, 2337 (2015).8V. Bargmann, L. Michel, and V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).9M. Weng, H. Bauke, and C. H. Keitel, Scientific Reports 6, 31624 (2016).


With XGO, we obtained polarization effects of EM waves in dielectrics.10

∂tE = cε∇×H

∂tH = − cµ∇×E

⇒ i∂tψ = H(t,x,−i∇)ψ, ψ.

=

(EH

).

XGO==⇒ LEM = K · X− c |K|

n+

i

2

(Z †Z − Z †Z

)+ Z †σzZ

(K⊥ ×Kc

|K⊥|2n2·∇n

),

n(X).

=√εµ, K⊥

.=(Kx ,Ky , 0

).

ZR =

(10

)right polarized

ZL =

(01

)left polarized

10D. E. Ruiz and I. Y Dodin, Phys. Rev. A 92, 043805 (2015).


Polarization effects may be important for RF waves in tokamaks.

∂tv = qmE + q

mc v ×B∂tE = −4πqn0v + c∇×B

∂tB = −c∇×E

⇒ i∂tψ = H(t,x,−i∇)ψ

XGO====⇒Ωω0

LRF = K · X− ω0(X,K) +i

2

(Z†Z − Z†Z

)+ Z†σzZ

(K⊥ ×K

|K||K⊥|2·∇ω0 +

ω2p

2ω20 |K|

Ω ·K)

ω0.=√ω2

p + c2K2

The basic effect is known from optics, but may be more important for plasma waves due to larger O/L.

Polarization, or "spin" effects alter the ray equations

Weakly magnetized cold electron plasma:

General plasma: U is found numerically

Dielectric with refraction index n(x):

cf. Libermanand Zeldovich (1992), Onoda et al (2004), Bliokh et al (2008)

Ruiz and Dodin (2015)

0.4 0.6 0.8 1 1.2 1.4

X coordinate

-1

-0.5

0

0.5

1

Zcoordinate

Magnetic field profile

0.4 0.6 0.8 1 1.2 1.4

X coordinate

-1

-0.5

0

0.5

1

Zcoordinate

Plasma density profile


With XGO, we obtained a nonperturbative ponderomotive modelfor the relativistic spin-1/2 particle in a vacuum EM field.12

Dirac particle interacting with an EM vacuum field ⇒ Volkov states11

XGO==⇒ LD = P · X− Heff(t,X,P) +

i~2

(Z †Z − Z †Z

)− S(Z †,Z ) ·Ωeff(t,X,P),

Heff.

=√

m2effc

4 + P2c2, m2effc

2 .= m2c2 + q2〈A2〉,

Ωeff(t,X,P).

=q2

4mγ(ω − k · v)(m + meff)

[∇〈A2〉 × k +

k ·∇〈A2〉ω(m + meff)

(k×P)−ω

m + meff∇〈A2〉 ×P

].

11D. M. Volkov, Z. Phys. 94, 250 (1935).12D. E. Ruiz, C. L. Ellison, and I. Y. Dodin, Phys. Rev. A 92, 062124 (2015)


Summary

1. Vector waves rays have three independent variables: X, P, and polarization Z .

2. Polarization effects include polarization precession (including mode conversion) andcorrections to wave ray dynamics.

3. We present a general Lagrangian wave theory that describes the leading-order GOdynamics and the corrective “wave-spin” dynamics.

4. The theory was applied to several physical systems of interest:• Dirac particle• EM waves in dielectrics• Waves in magnetized plasmas• Dirac particle in a strong EM vacuum field.


Questions remaining in the field

1. Polarization effects for RF ray tracing in tokamaks.

2. Polarization effects of waves in crystals.

3. Generalization to curvilinear metric (differential geometry).

4. Extending this theory to the case of waves with dissipation.13

13I. Y. Dodin, A. I. Zhmoginov, and D. E. Ruiz, arXiv (2016), 1610.05668.


Questions?

Date post:	17-Jul-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Extending Geometrical Optics: A Lagrangian Theory for ... · Introduction Basic Theory Results...

Documents