Post on 30-Jan-2021
Quasi-geometrical Optics Approximation in
Gravitational Lensing
Ryuichi Takahashi Department of Physics, Kyoto University
Kyoto, Japan
Ray path in geometrical optics
Ray path formed by diffraction effect
Diffracted ray passing through the lens center is newly formed
source
observer lens
§1.Introduction
M
lens
Er
1/2E MDr Einstein radius :
●Gravitational lensing
D
observer
Gravitational lensing of light is usually treated in geometrical optics
・double slit
1
2Er
D
double slit
path length difference : M 21
21E MDr/
M Wave effect (diffraction)
monochromatic waves
(Nakamura 1998)
●Wave effects in gravitational lensing (Schneider et al. 1992; Nakamura & Deguchi 1999 )
・long wavelength
gravitational waves λ 1AU
for space interferometer
screen
The wave does not feel the existence of the lens
lens
・λ >M Diffraction effect * source
・λ ≪M Geometrical optics approximation
・λ <M Quasi-geometrical optics approximation
Geometrical 1st order perturbation optics in powers of λ /M
(λ →0) (correction term arising from diffraction )
Difference between geometrical optics and wave optics
1
8101
M
M
AUM
§2.Wave Optics in Gravitational Lensing
(Misner et al. 1973; Schneider et al. 1992)
Background metric
dxdxgdUdtUdsB)(222 2121 r
:gravitational potential of the lens )(rU 1U
Wave propagation
02 )(;;
hRhB :Background
Riemann tensor
)(BR
Lensed waveform )(~
Lh
)(~
)()(~
hFhL
Unlensed waveform
),(exp2),( 2 ηξξη d
LSL
S tidiDD
DF
)(2
),(
2
ξηξ
ηξ
SLLS
SLd
DDD
DDt 8, 2ξ
Amplification factor
),(exp2),( 2 yxxy wTid
i
wwF
Amplification factor
L
S
O ηLDLSD
SD
ξ
SL DD 00 , ηyξx
20LLS
S
DD
Dw
(ξ :normalization constant of the length)
Nondimensional quantities
:nondimensional frequency
)(2
1),(
2xyxyx T : nondimensional time delay
d
S
LLS tD
DDT 20
Er0
Expanding F (w,y) in powers of 1/w
1,1 Tyx
Mw Mλ1/w
Setting (Einstein radius)
●Geometrical optics approximation w
Stationary points of the T (x,y) contribute to
the above integral :
Image positions are determined
j
jjgeo iwTF exp2/1
j
jdj
L tthth ),(),( ,2/1
rr
Amplification factor
0),( xyxT
:lens equation
jx
Lensed wave in time domain
),(exp2),( 2 yxxy wTid
i
wwF
(Fermat’s principle)
(Schneider et al. 1992; Nakamura & Deguchi 1999 )
)(xxy
Expanding T(x,y) around jx
)~(~~),(),( 321 xxxTTT bajbaj yxyx
jjj TT xxxyx ~),,(
●Quasi-geometrical optics approximation 1w
j
tt
jj
L
geo
L thtdththd
),(~
),(),(2/1
rrr
Amplification factor
),(exp2),( 2 yxxy wTid
i
wwF
Expanding T(x,y) to higher order
)~(~~~~),(
~~~),(
~~),(),(
5
241
61
21
xxxxxT
xxxT
xxTTT
dcbajdcba
cbajcba
bajbaj
yx
yx
yxyx
Effects on magnifications of the images
)(
exp
2/3
241
6
12122
21
wxxxxT
xxxTxxTideF
dcbajdcbaw
cbajcbawbajba
iwT
ji
j x
)(1 22/1
we
w
ijiwT
jjj
2
)3(
2
2)3(
3
)4(
2
11
12
5
2
1
16
1
jjj
jj
j
jj
j
j
j
j
j
xx
jjjjj x 1,1 2121jiwTjj e
w
idF
2/1
Lensed wave in time domain
Amplification factor
),(exp2),( 2 yxxy wTid
i
wwF
Central cusp of the lens
)0(2 rrInner density profile
x ,1
)(),(2
21 xyxyx T
x0)( x
)(
1 22/32
0
2
2/2
wyw
edF
iwy
)/1(exp2
2
2
2
101
22/2
wxxxyidwi
eF
iwy
x
observer
lens
source
(e.g. Singular Isothermal Sphere)
・Point mass lens
w≫1: geometrical optics limit
Twi
geo eiF
2/12/1
TwFgeo sin22/12
)()( 2 ξξ M
§3.Results
・Singular isothermal sphere
TwFgeo sin22/12
2/)( 2v ξw≫1: geometrical optics limit
・Isothermal sphere with a core
x
Deflection potential 22
cxx
Contribution from the lens center
)/1()(exp2
22
2
2
101
22/2
wwxxxxyidwi
edF c
iwy
x
For wx <1, the behavior is similar to that in SIS
§4.Conclusion
We discussed the effects of diffraction in
wave optics of gravitational lensing. We
studied first order perturbation in powers
of λ /M .
1.The magnification μ is changed to
μ + dμ , where dμ is of the order of
λ /M .
2. Diffracted ray passing through the lens center is newly formed .