Formulas for synchrotron radiation from bending magnets and storage rings
Light bending in radiation background
description
Transcript of Light bending in radiation background
Light bending in radiation background
Based onKim and T. Lee, JCAP 01 (2014) 002 (arXiv:1310.6800);Kim, JCAP 10 (2012) 056 (arXiv:1208.1319);Kim and T. Lee, JCAP 11 (2011) 017 (arXiv:1101.3433);Kim and T. Lee, MPLA 26, 1481 (2011) (arXiv:1012.1134).
Jin Young Kim (Kunsan National Univer-sity)
Outline
• Nonlinear property of QED vacuum
• Trajectory equation
• Bending by electric field
• Bending by magnetic field
• Bending in radiation background
• Summary
Motivation • Light bending by massive object is a useful tool in
astrophysics : Gravitational lensing
• Can Light be bent by electromagnetic field?
• At classical level, bending is prohibited by the lin-earity of electrodynamics.
• Light bending by EM field must involve a nonlinear interaction from quantum correction.
• The box diagram of QED gives such a nonlinear in-teraction : Euler-Heisenberg interaction (1936)
Non-trivial QED vacua • In classical electrodynamics vacuum is defined as
the absence of charged matter.• In QED vacuum is defined as the absence of exter-
nal currents. • VEV of electromagnetic current can be nonzero in
the presence of non-charge-like sources.
electric or magnetic field, temperature, …
• nontrivial vacua = QED vacua in presence of non-
charge-like sources• If the propagating light is coupled to this current,
the light cone condition is altered. • The velocity shift can be described as the index of
refraction in geometric optics.
Nonlinear Properties of QED Vacuum • Euler-Heisenberg Lagrangian: low-energy effective
action of multiple photon interactions
• In the presence of a background EM field, the non-linear interaction modifies the dispersion relation and results in a change of speed of light.
• Strong electric or magnetic field can cause a mate-rial-like behavior by quantum correction.
1
cnc
Velocity shift and index of refraction
• In the presence of electric field, the correction to the speed of light is given by
B E c
E))(u, planeonpolarizati(photon modelar perpendicu :14 a
• For magnetic field, • Index of refraction
• If the index of refraction is non-uniform, light ray can be bent by the gradient of index of refrac-tion.
E))(u, planeonpolarizati(photon mode parallel :8 a
Light bending by sugar solution
• Place sugar at the bottom of container and pour wa-ter.
• As the sugar dissolve a continually varying index of refraction develops.
• A laser beam in the sugar solution bends toward the bottom.
Snell’s law
1n
2n
3n
321 nnn
sinsin
2
1
1
2
2
1
vv
nn
1n
2n
21 nn
1
2
Differential bending by non-uniform refractive in-dex
• In the presence of a continually varying refrac-tive index, the light ray bends.
• Calculate the bending by differential calculus in geometric optics
1cot1sin
)sin( 1
22
2
nn
||1tantan rdnnn
rdnnn
law sSnell' : sinsin
2
1
1
2
2
1
nn
nnn 1221 ,
1
2
1n
2nn
Trajectory equation
• When the index of refraction is small, approxi-mate the trajectory equation to the leading order
order leading : dxds
) to from(photon xx
0un
uu
Bending by spherical symmetric electric charge bx parameter impact with from incomingphoton
• Total bending angle can be obtained by integra-tion with boundary condition
Bending by charged black hole
• Consider a charged non-rotating black hole
b1
4
1b
• Constraint on black hole
• Restore the physical constants
• Parameterize the charge as
Order-of-magnitude estimation • Black hole with ten solar mass• Since the calculation is based on flat space
time, impact parameter should be large enough
mode) : 14,1( a
• Ratio of bending angles at
Light bending by electrically charged BHs seems not negligible compared to the gravitational bend-ing.
mode) : 14,1.0( a
(for heavier BH, the relative bend-ing becomes weaker )
Bending by magnetic dipole • Contrary to Coulomb case, the bending by a mag-
netic dipole depends on the orientation of dipole relative to the direction of the incoming photon.
• Locate the dipole at origin.• Take the direction of incoming photon as +x axis.• Define the direction cosines of dipole relative to
the incoming photon.
M̂
y
x
z
M
Bending by magnetic dipole
x
z
y
vh
br
B
Bending angles
0)()( ; 0)( ,)( :conditionsboundary zyzby
x
z
y
vh
br
B
)( ),( :angles bending zy vh
6
2
bM
Special cases
b
y
x
Br
z
1 ,0
i) z direction, passing the equa-tor
M̂
y
x
z
M
Special cases
0 ,1
ii) -x direction (parallel or anti-parallel)
b
y
x
r B
Special cases
1 ,0
y
x
B
r
z
b
iii) axis along +y direction, light passing the north pole
• The gradient of index of refraction is maximal along this direction, giving the maximal bending
Order-of-magnitude estimation
• Maximal possible bending angles for strongly magnetized NS with solar mass
• Parameterize the impact param-eter
rad 104.1 rad;59.0 ,14 T,10 4mg
9S
aB
) 1(
• Up to , the bending by magnetic field can not dominate the gravitational bending.
T109S B
rad 104.1 rad; 109.5 ,14 T,10 2m
2g
13S
aB
•
10
Validity of Euler-Heisenberg action
• Critical values for vacuum polarization
• Screening by electron-positron pair creation above the critical field strength
V/m103.1E T;104.4B 1832
C9
22
C ecm
ecm
• Since the Euler-Heisenberg effective action is rep-resented as an asymptotic series, its application is confined to weak field limits.
• When the magnetic field is above the critical
limit, the calculation is not valid.
Light bending under ultra-strong EM field
• Analytic series representation for one-loop effective action from Schwinger’s integral form [Cho et al, 2006]
• Index of refraction
Upper limit on the magnetic field • No significant change of index of refraction by ultra-
strong electric field.
• Physical limit to the B-field of neutron star:
T1010 1412
• B-field on the surface of magnetar:
T1011
• Up to the order of , the index of refraction is close to one
)200/( T1012 CBB
• To be consistent with one-loop
430// CBB
Light bending under ultra-strong magnetic field • Photon passing the equator of the dipole • Index of refraction
• Trajectory equation
b
y
x
Br
z
• Bending angle
Order-of-magnitude estimation
• Maximal possible bending angles for strongly magnetized NS of solar mass
• Power dependence
T10for rad 108.1 11S
2m B
) 1(
rad59.0 g
T10for rad 18.0 12Sm B
•
) 2(
rad3.0 g T10for rad 103.2 12S
2m B
Speed of light in general non-trivial vacua • Light cone condition for photons traveling in
general non-trivial QED vacua
effective action charge[Dittrich and Gies
(1998)]
• For small correction, , and average over the propagation direction
• For EM field, two-loop corrected velocity shift agrees with the result from Euler-Heisenberg la-grangian
Light velocity in radiation background• Light cone condition for non-trivial vacuum in-
duced by the energy density of electromagnetic radiation
null propagation vec-tor
system coordinatepolar sphericalin )0,0,1,1(U
• Velocity shift averaged over polarization
Bending by a spherical black body
• As a source of lens, consider a spherical BB emitting energy in steady state.
• In general the temperature of an astronomical ob-ject may different for different surface points.
• For example, the temperature of a magnetized neu-tron star on the pole is higher than the equator.
• For simplicity, consider the mean effective surface temperature as a function of radius assuming that the neutron star is emitting energy isotropically as a black body in steady state.
Index of refraction as a function of radius• Energy density of free photons emitted by a BB at
temperature T (Stefan’s law)
• Dilution of energy den-sity:
• Index of refraction, to the leading order,
• can be replaced by (critical temperature of QED)
Trajectory equation
• Take the direction of incoming ray as +x axis on the xy-plane. • Index of refraction: • Trajectory equation: • Boundary condition:
Bending angle • Leading order solution with
• Bending angle from
b
y
x
Bending by a cylindrical BB• Take the axis of cylinder as z-axis.
• Energy density:• Index of refraction:
• Trajectory equation: • Solution:
• Bending angle:
Order-of-magnitude estimation
• Surface temperature:
• Surface magnetic field: • Mass:
• The magnetic bending is bigger than the thermal bending for , while the thermal bending is bigger than the magnetic bending for .
• However, both the magnetic and thermal bending angles are still small compared with the gravita-tional bending.
Dependence on the impact parameter
• Dependence on impact parameter is imprinted by the dilution of energy density
How to observe?
• The bending of perpendicular polarization is 1.75(14/8) times larger than the bending of par-allel polarization.
• Even in the region where the bending by mag-netic field is weak, by eliminating the overall gravitational bending, the polarization depen-dence can be tested if the allowed precision is sufficient enough.
Birefringence
Power dependence • Measure the total bending angles for different
values of the impact parameter (may be possi-ble by extraterrestrial observational facilities)
• Check the power dependence by fitting to
How to observe?
• Use the neutron star binary system with nonde-generate star (<100).
• Assume the two have the same mass.• Bending angles at time t=0 and t=T/2 are the
same if we consider only the gravitational bending.• The bending angle will be different by magnetic
field
Neutron star binary system
0
2/T