The second term in on the RHS of equation 61 is the wedge product V∧W. The Trace of the first term is the dot product.

A×B·C = A∧B∧C

grade-flipping does not change the cardinality of the basis.

That is to say, it will have the largest possible grade, namely grade=d.

A ⌞ B:=⟨A B⟩r−s    (right contraction)     (51a)         (forward contraction) A ⌟ B:=⟨A B⟩s−r

A •H B:=⟨A B⟩|s−r|    provided r>0 and s>0     (51e)         (Hestenes inner product)

1s 1v         D=1     1s 2v 1b        D=2    1s 3v 3b 1t       D=3   1s 4v 6b 4t 1q      D=4

e earlier definition of dot product (equation 7) and the definition of components (equation 45).

There is no advantage in imagining some super-vector that has the γi vectors as its components.

γi γj = − γj γi      for all i ≠ j

In Minkowski spacetime, any such basis will have the following properties:

:= γ1γ2             B := γ2γ3            then    AB := γ1γ3

sC = s·C                 = s∧C

It is easy to calculate the geometric product: AB = 1 + γ1γ4 + γ2γ3 +  γ1 γ2 γ3 γ4

We make a point of keeping things non-chiral, to the extent possible, because it tells us something about the symmetry of the fundamental laws of physics, as discussed in reference 3.

The paintbrush picture is a little dodgy in the case where C is a scalar

Suppose P, Q, and R are grade=1 vectors. Then P ∧ (Q·R) is simple. It’s just a vector parallel to P, magnified by the scalar quantity (Q · R).

|p−q| to p+q inclusive (counting by twos)

2.15  More About the Geometric Product

w define the wedge product between any blade and any blade.

P∧Q∧R := 1  6 (PQR + QRP + RPQ − RQP − QPR − PRQ)

P∧(Q∧R) := P∧Q∧R

blade is defined to be any scalar, any vector, or the wedge product of any number of vectors.

vector,   bivector,   trivector, ⋯,  multivector

vector,   bivector,   trivector, ⋯,  blade

quantity γ0 γ1 + γ2 γ3

It is necessarily either a blade or the sum of blades, all of the same grade.

PQ = −QP = P∧Q     iff P ⊥ Q

grade≤1

A bivector can be visualized as a patch of flat surface, which has area and orientation

We add bivectors edge-to-edge,

The eigenvalues of Z are x ± iy an

we are motivatedtherefore to introduce the operation

the fol-lowing symmetries:

The statevariables are divided into two groups.

and it would haveled to ~xxfs ¼ 0

free energy being a perfect differential, and be-cause the constitutive relations (47) do not involveconvolution integrals

prospects of obser-ving a magnetic monopole are rather remote [38, 39],for now it is appropriate to regard the Tellegen med-ium as chimerical

Hence, non-zero va-lues of BðwÞ of actual materials are not know

h magnetoelectric tensors arecommonplace. Typically, such properties are exhibitedat low frequencies and low temperatures.

The constitutive functions must be characterized aspiecewise uniform

Physically, this constraint arises from the follow-ing two considerations:

generalized duality transformation

reciprocity constraint. But it is not, because itdoes not impose any transpose-symmetry requirementson ~ee, ~aa, ~bb and ~nn

D x; tð Þ ¼ e0 ~EE x; tð Þ þ ~PP x; tð Þ ; ð15Þ~HH x; tð Þ ¼ m
10 ~BB x; tð Þ
~MM x; tð Þ

Both ~PP x; tð Þ and ~MM x; tð Þ are nonunique to the extentthat they can be replaced by ~PP x; tð Þ
r  ~AA x; tð Þ and~MM x; tð Þ þ ð@=@tÞ ~AA x; tð Þ, respectively, in (12) and (13)without affecting the left sides of either equation.

r  ~MM x; tð Þ

Actually, only spatialaveraging is necessary [6], because it implies temporalaveraging due to the finite magnitude of the universalmaximum speed e0m0ð Þ

he constitutive parameters satisfy the On-sager relations

deviation
S of theentropy from its equilibrium value as the quadratic expres-sion [7]

ndicates averaging over time t

variables are supposed to be odd with respect to are-versal of velocities of the microscopic particles constitutingthe linear medium; in other words

. In crystals and molecular systems,magnetoelectricity – a phenomenon with local coexistence of electric and magnetic moments – takesplace when space inversion is locally broken [8].

The cross-polarization matrices in constitutive relations of bianisotropic media are, in fact,pseudotensors

e cross-polarization coupling vanishes in the long-wavelength limit

the trailing-shield head invented by Michael Mallary. This head offered higher field gradients

control the level of exchange-coupling between grains

oxide-segregant exchange-break between grains

uniaxial anisotropy constant Ku, which in turn is higher for a material with a higher magnetic coercivity

Perpendicular recording uses higher coercivity materials because the head's write field penetrates the medium more efficiently in the perpendicular geometry.

superparamagnetic limit

Hard disk technology with longitudinal recording has an estimated limit of 100 to 200 gigabit per square inch (16 to 31 Gb/cm2) due to the superparamagnetic effect, though this estimate is constantly changing.

19

We measure thesaturation magnetization of our Py layers using vibrating sample magnetometry

Surface magneto-optic Kerr effect

Comparison of themeasured out-of-plane Oersted field with a finite-elementcalculation of th

As expected, we find that theout-of-plane Oersted field is antisymmetric about the centerof the wire, and the equivalent spin-orbit field is approxi-mately constant across the wire width

expected change in the Kerr rotation signal isDwðmÞ ¼ aPolarDhM þ bQuadratic cos 2/polD/M: (5)A derivation of this result using a Jones-matrix calculation isgiven in the supplementary material

quarter wave plate (labeled QWP-1) to compensate a slightbirefringence of the beam splitter

mode-locked Ti:Sapphire laser working at 780 nm centerwavelength.

(4)

Ms is much larger than any of the other field terms

Permalloy, the in-plane anisotropy is negligible

This is replaced by a quarterwave plate QWP-2 for detecting the current-induced in-plane magnetizationrotation.

esulting from two orthogonal effec-tive-magnetic-field components h y0 and hz

We initially align themagnetization along the x0 direction using an external field

in-plane AC current, Iac cos xt, at 1013 Hz with Iac ¼ 10 mA

MOKE is analogous to the anomalous Hall effect (/ m z),while the quadratic MOKE is the analog of the planar Halleffect (/ m x m y).

intimate connection between the permittivity tensor andthe conductivity tensor in electromagnetism.

quadratic MOKE changes the polarization fromcircular to slightly elliptical (see supplementary material).

rotation of the polarization angle due to the mag-netization can be written as 26wðmÞ ¼ aPolarmz þ bQuadraticmxmy þ    ;

We verify the accu-racy of this method for measuring spin-orbit-induced torquesby studying a series of Pt/Permalloy (Ni81Fe19 ¼ Py) bilayers

sensitiv-ity comparable to the techniques based on magnetotranspor

Recently, Montazeri et al. demonstrated the existenceof a quadratic MOKE response in the setup with normal lightincidence.

nance (ST-FMR)6 can be used for metallic magnets with eitherperpendicular or in-plane anisotrop

econd-harmonic Halleffect measurements work well for measuring torques actingon a metallic magnetic layer with perpendicular magneticanisotropy, but for magnets with in-plane anisotropy, the needto separate out thermally induced signals makes this techniquemore difficult to apply.

nvolve current-induced torques arising from spin-orbitinteractions, either in heavy-metal (HM)/ferromagnet (FM)bilayers5,6,11,12 or topological insulator (TI)/FM bilayers

inding values in excellent agreement with spin-torque ferromagnetic resonance measurements

voids interference when sub-sequently overlapping forward- and backward-generatedpair contributions.

overlapped without interference, which effectivelytransfers the two-photon phase shift onto a single-photon state

wavelength-selective wave plate(WSWP),

gh a periodically poled potassium titanyl phosphatenonlinear crystal (PPKTP

quantum imaging with undetected photons” [

f “induced coherence without inducedemission” was introduced as an elegant way to create pho-tonic superposition states [35,36]

nonlinear interferom-etry in SU(1,1)

infers optical phase shifts through stan-dard intensity measurements while still maintaining the quantum advantage in the measurement precision

Laser interferometry enables interaction-free sensing with a precision ultimately limited by shot noise

6. J. F. Dillon, Jr.: in Physics of Magnetic Garnets, Proc. of International Schoolof Physics, Enrico Fermi, p. 379 (North-Holland Pub. Company 197

Onsager relations mean that the diagonal components of the dielectric tensorare even function of M, while the nondiagonal ones are odd function of M

e(w) can be shown to have the following form [6],

crystal has a cubic symmetry, and e(w) = € · 1, where € is the dielectricpermeability of the material and 1 is a unit tensor.

where the magnetization is zero

magnetic permeability tensorJL(w)onoptical phenomena is small, so we assume thatJL(w)= Jlol

e the magnetic domain structures inferromagnets(4

This is known asthenonreciprocal property of lightpropagation in ferromagnets.

andtherelation between MO effects andthedielectric tensor,

Kerr rotation (KR) of "' 0.5°

FR in nonmagnets istoo small to be applied for devices

w = 18300cm-1

nonmagnets as well asferromagnets.

of anoma-lously large Faraday rotation due to diamagnetic bismuth ions in ferrimagnetic gar-nets,

"Modern Magneto-Optics and Magneto-Optical Materials" by A.K. Zvezdinand V.A. Kotov

8 discusses "photo-induced magnetism", which is completely differentfrom "photo-thermal magnetism" used in an optical magnetic memory de-vice.

In Chap. 5,after introducing a phenomenological theory of the Faraday and Kerr effects,we present a microscopic theory based on the ligand-field theory and discussthe future developments.

Due to spin-orbit coupling, undoped GaAs single crystal exhibits much larger Faraday rotation than glass (SiO2).

Standard optical Kerr gate measurements

gyration vector,

anisotropic permittivity,

orthogonal velocity components,

known as the Cotton-Mouton effect and used for a Circulator.

Then, if we let g lie in the z direction for simplicity, the ε tensor simplifies to the form:

can obtain a nonlinear optical effect of magneto-optical parametric generation (somewhat analogous to a Pockels effect whose strength is controlled by the applied magnetic field).

resulting principal axes become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations

losses can be neglected, ε is a Hermitian matrix.

depending on the frequency ω of incident light.

with reversed rotation directions of the two principal polarizations, corresponding to complex-conjugate ε tensors for lossless media, are called optical isomers.

s well as Lorentz reciprocity, which is a necessary condition to construct devices such as optical isolators (through which light passes in one direction but not the other).

quasistatic magnetic field.

y inverting the Laplacian

evaluated at the vector difference x − x′

kk/k2 = ˆkˆk

since we areinterested in the optical wavelength region.16

wherethe multiple reflections could be ignored

The MOKE,fundamentally related to the spin-polarized electronic bandstructure, is manifested itself by the change of polarizationand/or intensity of incident polarized light when it is re-flected from the surface of a magnetized medium

d-wave ratherthan conventional s-wave symmetry

During that time he deployed it in crucial experiments involving polarization, birefringence, and optical rotation,[3][4][5] all of which contributed to the eventual acceptance of his transverse-wave theory of light.

k-linear

Optical injection provides a local dc source ofpolarized electrons

However, no two-dimensional crystal with intrinsic magnetismhas yet been discovered 10–14

systematic way to distinguish between them in

are closely related

distinctionbetween intrinsic and extrinsic contributions to the spin current is not useful

optically created spin polarization.

transform differently

double groups

lacks a pedagogical derivation of the mechanisms leading tospin scattering in monolayer

with often contrastingresults

heavy tran-sition metal atoms in the lattice

valley

N TOR

flow in the x-direction of spins polarized along y is transformed to a flow in the y-direction of spins polarized along x.

e magnetization field or M-field can be defined according to the following equation: M = d m d V {\displaystyle \mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}}}

t the magnetization tries to reduce the poles as much as possible

spectral decomposition

line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles, but o

rence is represented as a Gaussian function expressed as[13]

he former being an equivalent to the coherence length of the light source

ut scattering is too small to be detected. No special preparation of a biological specimen is required, and images can be obtained ‘non-contact’ or through a transparent window or membrane. It is also important to note that the laser output from the instruments used is low – eye-safe near-infrared or visible-light[29]

capture a broad band of light and focus it into each pinhole significantly increasing the amount of light directed into each pinhole and reducing the amount of light blocked by the spinning-disk.

the energies of the bands near the surface are often pinned to the Fermi level, due to the influence of surface states.[

Ea and Eb each incident at one of the inputs

A classical prediction of the intensities of the output beams for the same beam splitter and identical coherent input

This is required by the reversibility (or unitarity of the quantum evolution) of the beam splitter.

is misleading in that only a tiny fraction of spins ever have transverse phase coherence.

gives rise to polarization effects

n a particle, causing them to move at the same frequency. The particle, therefore, becomes a small radiating dipole whose radiation we see as scattered light. The particles may be individual atoms or molecules; it can occur when light travels through transparent solids and liquids, but is most prominently seen in

"circular" current through the disc

electric stirring effect

the emitted light

swapping

combined action of the direct and inverse spin Hall effect

coupling parameter ʏ

the relative motion between the magnetic moment (associated to the spin) and the electric field creates a coupling that distorts the motion of the electrons.

Two possible mechanisms

In 1983 Averkiev and Dyakonov[3] proposed a way to measure the inverse spin Hall effect under optical spin orientation in semiconductors.

they both lie in the plane "transverse"

H is determined from E by 90-degree rotation and a fixed multiplier