φm (x) ≥ cm .

em

It follows that on thehypersurface M the Rn -valued n − 1-form ∗dx is equal to the product of n and ascalar n − 1-form ν: ∗dx  nν.

µM (e1, . . . , eN−1)

)  (−1)N+1FN .

form satisfies µM,x (e1 , . . . , eN−1)  1.

f (v1 , v2 , . . . , vn ; 1) is a negatively oriented frame

volume form µM depends on the embedding of M into RN . I

If this frame is positively,resp. negatively, oriented

This defines an n-form ω on M and we must show that ω  µM

he support of α is defined as the set of all points x in M with theproperty that for every open ball B around x there is a y ∈ B ∩ M such that αy , 0

We define an orientationon M by requiring ψ to be orientation-preserving.

orientation induced

µM depends on the embedding of M into RN . It changes if we dilateor shrink or otherwise deform M

volume forms are seldom exact!

det(Dφ(u) du1 du2 · · · dun

µi  ψ∗i (µ) √det(Dψi (t)T Dψi (t)) dt1 dt2 · · · dtn

Gauss map of M

N+1

[n, e1, e2, . . . , eN−1; 1)  [e1, e2, . . . , eN ; 1].

Proof

Choose ε  ±1 such that (n(x), b1, b2, . . . , bn ; ε) is a positively orientedframe of RN .

Mach–Zehnder intensity modulator is used to generate psoptical pulses of SLED

We use a 1550 nm laser diode(SFL1550S, Thorlabs) and a super luminescent light emitting diode(SLED; DL-CS5169A, Denslight) for the generation of ps opticalpulses.

in the pulsed Sagnac interfer-ometer, lasers with long coherence lengths perform well

oblique-incidence designenables a study for longitudinal and transverse Kerr effects

all-fiber design simplifies optical alignment;17

s low noise and drift-free measurement

, limited by thephotodetector noise

superconducting states can be broken even with smalloptical powers.

ejects all the reciprocal effects, such as linear birefringence andoptical activity unlike conventional MOKE microscopy

100 nrad/√Hz sensitivityusing only a 10 μW optical power without the magnetic field modu-lation.

ime-reversal symmetry breaking (TRSB) in condensed mat-ter systems generally results in magnetism

1 μrad/√Hz sensitivityat a 3 μW

s the favorable properties of a Sagnac interferometer, such as rejection of all the reciprocal effects, a

onceptual symmetries of resistor, capacitor, inductor, and memristor

Continuous” means that for every x ∈ M there exists a localparametrization ψ : U → RN of M at x with the property that Dψ(t) : Rn → Tψ(t) Mpreserves the orientation for all t ∈ U.

bj  ∑ni1 a′i, j b′

consisting of a frame(b1, b2, . . . , bn ) together with a sign ε  ±1.

uniqueness is proved by verifying that the formula holds,

are orthogonalvectors.

gJ (x)  ∑I φ∗ ( fI )(x) det(DφI,J (x)).

For ak-form α ∈ Ωk (V) define the pullback φ∗ (α) ∈ Ωk (U) by

all k-multilinear functions of the formdxI  dxi1 dxi2 · · · dxik

β(bI )

increasing multi-indices of degree k

multi-indices I and J l

There is a nice way to construct a basis of the vector space Ak (V) starting froma basis {b1, . . . , bn } of V

alternating k-multilinear functions is denoted by Ak (V)

useful trick for producing alternating k-multilinear functions startingfrom k covectors µ1, µ2, . . . , µk ∈ V∗

µ1µ2 · · · µk (v1 , v2 , . . . , vk )  det(µi (vj ))1≤i, j≤

which is simply the Jacobi matrix D g of g! (This is the reason that many authorsuse the notation dg for the Jacobi matrix.

Using this formalism we can write for any smooth function g on U

codimension

I is not a regularvalue!

{β1, β2 , . . . , βn } of V∗ is said to be dual to the basis {b1, b2, . . . , bn } ofV

φ∗ d  dφ∗

J ∑I(ψ−1i ◦ ψj )∗ ( fI det(D(ψ−1i ◦ ψj )I,J )).

ψ−1i

ψi (Ui )

A×B=A⊕BA×B=A⊕BA\times B=A\oplus B, but in the case of the product/sum of infinitely many vector spaces they are distinct: ∏iAi≠⨁iAi∏iAi≠⨁iAi\prod_i A_i\neq \bigoplus_i A_i. This wouldn't be something covered in introductory classes. The deep distinction between the two is that one is a category theory product and one is a category theory coproduct

such as having an odd number of electrons per unit cell.

Edelstein effect

chirality represented by a pair of oppositely polarized spins

applications in the chiral spintronics2 field.

that when electrochem-ical water splitting occurs with an anode that accepts prefer-entially one spin owing to CISS, the process is enhanced andthe formation of hydrogen peroxide is diminished. [4

Two recent examples area spin filter[39] and the emergence of a Hall voltage owing tospin accumulation. [24,34]

Figure 2.

chiralmaterials. [8,19,28–31]

CISS is repeatedly found tocorrelate with optical activity [28,33,3

Third, as in CISS in transmission, alsohere the sign of the preferred spin depends on the direction ofthe molecular dipole,

with medium length

reak-junction.

ferromagnetic substrate

Energy distribution

b) Photoelectronpolarization

This conjec-ture, while reasonable, took another 12 years to verify

organized

it is easy to show that “current through a coil” arguments yieldan effect that is lower by many orders of magnitude.

Mott polarimeter

Based on the well-known connectionbetween the direction (cw or ccw) of circularly polarized lightand the spin polarization (up or down) of the excited electrons,

ink was finally found in the form of magneto-chiraldichroism, that is, a difference in the magnetic optical activity ofthe two enantiomers of a chiral medium.

t-handed configuration. However, when these crystals were separated manually, he found that they exhibited right and left asymmetry

Louis Pasteur was the first to recognize that optical activity arises from the dissymmetric arrangement of atoms in the crystalline structures or in individual molecules of certain compounds.

One common metric, a minimal-energy metric for a fixed focal-point intensity [18,104], isequivalent to a focal-point maximization metric under con-straints of fixed energy, as can be shown by comparing theLagrangian functions of each

y extent [45]. To define kloc , one could use agradient-based expression such as −i∇, yet the resul

−i∇, yet the resultingoperator would not be Hermitian, li

C. The average field valuearound the contour is given by 1/|C| ∫C |ψ(x)|2 dx =ξ † [(1/|C|) ∫C †] ξ ,

the local wave number(as measured by the spatial variations in the field),

ld to go to zero anywhereand that enforce “concentration” through other charac-teristics of an optical beam. The two properties that weconsider are the full width at half maximum (FWHM)

G = 0.21 exhibits to our knowledgethe smallest spot size of any theoretical proposal to date

We take the electric field polarized outof the plane, such that ψ can be simplified to a scalarfield solution of the Helmholtz equation

f 1.9λ and widths (diameters)ranging from 10λ to 23λ, equiv

sub-diffraction-limited solutionswhose input powers scale only polynomially with the spotsize.

diffraction limit (G = 1)

s is a weak-scattering assumption

In each case the far-zone bound islarger than that of the near zone or the mid zone, suggest-ing that the far-zone bounds of Eqs. (13) and (14) may beglobal bounds at any distance

Eq. (14) is physically achievable,

G  1

ttern: instead, divide themaximal intensity, Eq. (9), by the intensity of an uncon-strained focused beam (without the zero-field condition),which is simply ψ†1 ψ1 (which conforms to the usual Airydefinition for a circular aperture)

to e−ikz/z fo

ocusing-aperture distancemuch larger than the aperture radius

the six electric and magnetic polarizations decouple

asis, such as Fourier modes on a circularcontour or spherical harmoni

f light independent of the exit surface, simplyenforcing the condition that the light field comprises prop-agating waves.

it will be the lowest-order mode, polarizedalong the μ direction, that is most important in constrain-ing the field.

aximal intensity of an unconstrained beam wouldsimply focus as much of the effective-current radiation tothe origin, as dictated by the term 0†0 ,

ding the total intensity

and numerical evaluation of Eq. (6)within seconds on a laptop computer.

collocation [100]

generalized eigenproblem

ayleigh-quotient

subject to ν†Pν ≤ 1

6 × 6N matrix, where N is the number of degrees offreedom of the effective currents, †00 is a matrix withrank at most 6,

projecting it along an arbitrary polarization

1: ξ †ξ = 1.

is positive semidefinite (which is nonconvexunder maximization [96]

field along some spot-size contour C

power P not exceeding an input value of P0

= ξ ††ξ .

the currents comprise the degrees of freedom deter-mining the beam shape.

and currents at a single temporalfrequency ω (e−iωt time evolution)

tangential values

Refs. [54,55] (and recently in Ref. [56],

e possibility of sub-diffraction-limited spot sizes without near-field effects was recog-nized in 1952 by Toraldo di Francia [41]; stimulated byresults for highly directive antennas

t the ideal field profiles forall these metrics are nearly identical in the far zone,

yet be orthogonal to the fields emanat-ing from a current loop at the spot-size radius.

decrease proportional to G4 ,

izes G,

a ring at the

analytical upper bounds in the farzone

“Strehl ratio”

quadratic program

For floaters, the team used various spherical 1.2-tesla magnets

It has the unit volt

As the electron passes through the electric field of the nucleus, a magnetic field is produced in the reference frame of the electron.

c

G

polarization P.

/H-TaS 2 CMIS (c) and a S-MBA/H-TaS 2 CMIS (d) m

c,d

Magnetic-field-dependent

, Magnetic-field-dependent tunnelling current m

ceeding 60%,

a robust tunnellingmagnetoresistance >300%

unique vdW gaps,

Therecently emerged 2DAC

mCP-AFM

low spin selectivity or limited stability, andhave difficulties in forming robust spintronic devices5–8.

d or exchange interaction with magnetic atoms,thus preserving the time-reversal symmetry

relative to the square root of frequency and its value is usually on the order of nV/√Hz

When working with lock-in amplifiers, the input bandwidth is usually small, so the shot noise does not affect the measurements as much.

Superachromatic Wave Plates

each consists of three quartz and three magnesium fluoride (MgF2) plates that are optically cemented to maximize transmission and carefully aligned to minimize the wavelength dependence of the retardance.

We do not recommend disturbing this retaining ring, as it is likely to affect the optical alignment of the fast axis of the wave plate.

Zero-Order Achromatic Wave Plates

Mounted Achromatic

LCP zero-order wave plates produce a smaller decrease in retardance at larger AOIs.

J = L + S is what is conserved, so the spin-orbit coupling should take the form L · S

c is the specific heat of a quantum oscillator calculated earlier with naturalfrequency vsq,

vs

ΘD

this can be justified by considering that the lattice has acharacteristic length scale

all three acoustic modes

singular value.

fibre of φ

does notrefer to local parametrizations.

it turns outthat practically all abstract manifolds can be embedded into a vector space

nverse of ψ, which is a map ψ−1 : ψ(U) → U, is continuous

(ii) Dψ(t) is one-to-one for all t ∈ U;

Changing the order of integration (see Remark 5.3

α be a k − 1-

tj ,

formal linear combination ∑pq1 aq {xq }, which represents a distribution of pointcharges, and the linear combination of vectors ∑pq1 aq xq,

guage of linear algebra, the k-chains form a vector space with a basisconsisting of the k-cubes

formal linear combination

f (t1 , t2) dt2 dt1  − f (t1, t2) dt1 dt2. How can this besquared with formula (5.1)?

p∗ (h) det(Dp) ds1 ds2 · · · dsk

almost completelyunaffected

Although the image c(R) may look very different from the blockR,

, k-forms can beintegrated over k-dimensional parametrized regions

Thenwe know that α is exact.

orms this means that α  dg,

α  F · dx,

o by the substitution formula, Theorem B.9,we have ∫c◦p α  ± ∫c α, where the + occurs if p′ > 0 and the − if p′ < 0

g

Differentiationand integration are related

lmost completely

formula is seldom used to calculate pullbacks in practice

dφi1

curl(F) · dx  ∗dα.

div(F)  ∗d∗α.

ector-valued 1-form

exterior differentiation

By convention, forms of negative degree are0

exterior differential calculus

c′(0)  0,which does not span a line.

parametrizations to give a formal definition of the notion of amanifold in Chapter 6.

closed square is not a manifold, because the corners are not smooth.1

s a graded algebra.