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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
Introduction Basic Theory Results Conclusions
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)
Introduction Basic Theory Results Conclusions
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).
Introduction Basic Theory Results Conclusions
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).
Introduction Basic Theory Results Conclusions
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.
Introduction Basic Theory Results Conclusions
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).
Introduction Basic Theory Results Conclusions
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.
Introduction Basic Theory Results Conclusions
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).
Introduction Basic Theory Results Conclusions
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).
Introduction Basic Theory Results Conclusions
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
Introduction Basic Theory Results Conclusions
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)
Introduction Basic Theory Results Conclusions
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.
Introduction Basic Theory Results Conclusions
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.
Introduction Basic Theory Results Conclusions
Questions?